Resistor

A resistor is a two-terminal electronic component that produces a voltage across its terminals that is proportional to the electric current passing through it in accordance with Ohm's law:

V = IR

Resistors are elements of electrical networks and electronic circuits and are ubiquitous in most electronic equipment. Practical resistors can be made of various compounds and films, as well as resistance wire (wire made of a high-resistivity alloy, such as nickel/chrome).

The primary characteristics of a resistor are the resistance, the tolerance, maximum working voltage and the power rating. Other characteristics include temperature coefficient, noise, and inductance. Less well-known is critical resistance, the value below which power dissipation limits the maximum permitted current flow, and above which the limit is applied voltage. Critical resistance depends upon the materials constituting the resistor as well as its physical dimensions; it's determined by design.

Resistors can be integrated into hybrid and printed circuits, as well as integrated circuits. Size, and position of leads (or terminals) are relevant to equipment designers; resistors must be physically large enough not to overheat when dissipating their power
Units

The ohm (symbol: Ω) is a SI-driven unit of electrical resistance, named after Georg Simon Ohm. Commonly used multiples and submultiples in electrical and electronic usage are the milliohm (1x10−3), kilohm (1x103), and megohm (1x106).
Theory of operation
Ohm's law

The behavior of an ideal resistor is dictated by the relationship specified in Ohm's law:

V = IR

Ohm's law states that the voltage (V) across a resistor is proportional to the current (I) through it where the constant of proportionality is the resistance (R).

Series and parallel resistors
Main article: Series and parallel circuits

Resistors in a parallel configuration each have the same potential difference (voltage). To find their total equivalent resistance (Req):

A diagram of several resistors, side by side, both leads of each connected to the same wires
\frac{1}{R_\mathrm{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}

The parallel property can be represented in equations by two vertical lines "||" (as in geometry) to simplify equations. For two resistors,

R_\mathrm{eq} = R_1 \| R_2 = {R_1 R_2 \over R_1 + R_2}

The current through resistors in series stays the same, but the voltage across each resistor can be different. The sum of the potential differences (voltage) is equal to the total voltage. To find their total resistance:

A diagram of several resistors, connected end to end, with the same amount of current going through each
R_\mathrm{eq} = R_1 + R_2 + \cdots + R_n

A resistor network that is a combination of parallel and series can be broken up into smaller parts that are either one or the other. For instance,

A diagram of three resistors, two in parallel, which are in series with the other
R_\mathrm{eq} = \left( R_1 \| R_2 \right) + R_3 = {R_1 R_2 \over R_1 + R_2} + R_3

However, many resistor networks cannot be split up in this way. Consider a cube, each edge of which has been replaced by a resistor. For example, determining the resistance between two opposite vertices requires additional transforms, such as the Y-Δ transform, or else matrix methods must be used for the general case. However, if all twelve resistors are equal, the corner-to-corner resistance is 5⁄6 of any one of them.

The practical application to resistors is that a resistance of any non-standard value can be obtained by connecting standard values in series or in parallel.
Power dissipation

The power dissipated by a resistor (or the equivalent resistance of a resistor network) is calculated using the following: P = I^2 R = I V = \frac{V^2}{R}

All three equations are equivalent. The first is derived from Joule's first law. Ohm’s Law derives the other two from that.

The total amount of heat energy released is the integral of the power over time:

W = \int_{t_1}^{t_2} v(t) i(t)\, dt.

If the average power dissipated is more than the resistor can safely dissipate, the resistor may depart from its nominal resistance and may become damaged by overheating. Excessive power dissipation may raise the temperature of the resistor to a point where it burns out, which could cause a fire in adjacent components and materials. There are flameproof resistors that fail (open circuit) before they overheat dangerously.

Note that the nominal power rating of a resistor is not the same as the power that it can safely dissipate in practical use. Air circulation and proximity to a circuit board, ambient temperature, and other factors can reduce acceptable dissipation significantly. Rated power dissipation may be given for an ambient temperature of 25 °C in free air. Inside an equipment case at 60 °C, rated dissipation will be significantly less; if we are dissipating a bit less than the maximum figure given by the manufacturer we may still be outside the safe operating area, and courting premature failure.
Construction
A single in line (SIL) resistor package with 8 individual, 47 ohm resistors. One end of each resistor is connected to a separate pin and the other ends are all connected together to the remaining (common) pin - pin 1, at the end identified by the white dot.
Lead arrangements

Through-hole components typically have leads leaving the body axially. Others have leads coming off their body radially instead of parallel to the resistor axis. Other components may be SMT (surface mount technology) while high power resistors may have one of their leads designed into the heat sink.
Carbon composition

Carbon composition resistors consist of a solid cylindrical resistive element with embedded wire leads or metal end caps to which the lead wires are attached. The body of the resistor is protected with paint or plastic. Early 20th-century carbon composition resistors had uninsulated bodies; the lead wires were wrapped around the ends of the resistance element rod and soldered. The completed resistor was painted for color coding of its value.

The resistive element is made from a mixture of finely ground (powdered) carbon and an insulating material (usually ceramic). A resin holds the mixture together. The resistance is determined by the ratio of the fill material (the powdered ceramic) to the carbon. Higher concentrations of carbon, a weak conductor, result in lower resistance. Carbon composition resistors were commonly used in the 1960s and earlier, but are not so popular for general use now as other types have better specifications, such as tolerance, voltage dependence, and stress (carbon composition resistors will change value when stressed with over-voltages). Moreover, if internal moisture content (from exposure for some length of time to a humid environment) is significant, soldering heat will create a non-reversible change in resistance value. These resistors, however, if never subjected to overvoltage nor overheating were remarkably reliable.

They are still available, but comparatively quite costly. Values ranged from fractions of an ohm to 22 megohms.
Carbon film

A carbon film is deposited on an insulating substrate, and a helix cut in it to create a long, narrow resistive path. Varying shapes, coupled with the resistivity of carbon, (ranging from 90 to 400 nΩm) can provide a variety of resistances.[1] Carbon film resistors feature a power rating range of 0.125 W to 5 W at 70 °C. Resistances available range from 1 ohm to 10 megohm. The carbon film resistor has an operating temperature range of -55 °C to 155 °C. It has 200 to 600 volts maximum working voltage range.[2]
Thick and thin film

Thick film resistors became popular during the 1970s, and most SMD (surface mount device) resistors today are of this type. The principal difference between thin film and thick film resistors is not the actual thickness of the film, but rather how the film is applied to the cylinder (axial resistors) or the surface (SMD resistors).

Thin film resistors are made by sputtering (a method of vacuum deposition) the resistive material onto an insulating substrate. The film is then etched in a similar manner to the old (subtractive) process for making printed circuit boards; that is, the surface is coated with a photo-sensitive material, then covered by a pattern film, irradiated with ultraviolet light, and then the exposed photo-sensitive coating is developed, and underlying thin film is etched away.

Because the time during which the sputtering is performed can be controlled, the thickness of the thin film can be accurately controlled. The type of material is also usually different consisting of one or more ceramic (cermet) conductors such as tantalum nitride (TaN), ruthenium dioxide (RuO2), lead oxide (PbO), bismuth ruthenate (Bi2Ru2O7), nickel chromium (NiCr), and/or bismuth iridate (Bi2Ir2O7).

The resistance of both thin and thick film resistors after manufacture is not highly accurate; they are usually trimmed to an accurate value by abrasive or laser trimming. Thin film resistors are usually specified with tolerances of 0.1, 0.2, 0.5, or 1%, and with temperature coefficients of 5 to 25 ppm/K.

Thick film resistors may use the same conductive ceramics, but they are mixed with sintered (powdered) glass and some kind of liquid so that the composite can be screen-printed. This composite of glass and conductive ceramic (cermet) material is then fused (baked) in an oven at about 850 °C.

Thick film resistors, when first manufactured, had tolerances of 5%, but standard tolerances have improved to 2% or 1% in the last few decades. Temperature coefficients of thick film resistors are high, typically ±200 or ±250 ppm/K; a 40 kelvin (70 °F) temperature change can change the resistance by 1%.

Thin film resistors are usually far more expensive than thick film resistors. For example, SMD thin film resistors, with 0.5% tolerances, and with 25 ppm/K temperature coefficients, when bought in full size reel quantities, are about twice the cost of 1%, 250 ppm/K thick film resistors.
Metal film

A common type of axial resistor today is referred to as a metal-film resistor. Metal electrode leadless face (MELF) resistors often use the same technology, but are a cylindrically shaped resistor designed for surface mounting. Note that other types of resistors (e.g., carbon composition) are also available in MELF packages.

Metal film resistors are usually coated with nickel chromium (NiCr), but might be coated with any of the cermet materials listed above for thin film resistors. Unlike thin film resistors, the material may be applied using different techniques than sputtering (though that is one such technique). Also, unlike thin-film resistors, the resistance value is determined by cutting a helix through the coating rather than by etching. (This is similar to the way carbon resistors are made.) The result is a reasonable tolerance (0.5, 1, or 2%) and a temperature coefficient that is generally between 50 and 100 ppm/K.[3]
Wirewound
Types of windings in wire resistors:
1 - common
2 - bifilar
3 - common on a thin former
4 - Ayrton-Perry

Wirewound resistors are commonly made by winding a metal wire, usually nichrome, around a ceramic, plastic, or fiberglass core. The ends of the wire are soldered or welded to two caps or rings, attached to the ends of the core. The assembly is protected with a layer of paint, molded plastic, or an enamel coating baked at high temperature. Wire leads in low power wirewound resistors are usually between 0.6 and 0.8 mm in diameter and tinned for ease of soldering. For higher power wirewound resistors, either a ceramic outer case or an aluminum outer case on top of an insulating layer is used. The aluminum-cased types are designed to be attached to a heat sink to dissipate the heat; the rated power is dependent on being used with a suitable heat sink, e.g., a 50 W power rated resistor will overheat at a fraction of the power dissipation if not used with a heat sink. Large wirewound resistors may be rated for 1,000 watts or more.

Because wirewound resistors are coils they have more undesirable inductance than other types of resistor, although winding the wire in sections with alternately reversed direction can minimize inductance. Other techniques employ bifilar winding, or a flat thin former (to reduce cross-section area of the coil). For most demanding circuits resistors with Ayrton-Perry winding are used.
Foil resistor

The primary resistance element of a foil resistor is a special alloy foil several micrometres thick. Since their introduction in the 1960s, foil resistors have had the best precision and stability of any resistor available. One of the important parameters influencing stability is the temperature coefficient of resistance (TCR). The TCR of foil resistors is extremely low, and has been further improved over the years. One range of ultra-precision foil resistors offers a TCR of 0.14 ppm/°C, tolerance ±0.005%, long-term stability (1 year) 25 ppm, (3 year) 50 ppm (further improved 5-fold by hermetic sealing), stability under load (2000 hours) 0.03%, thermal EMF 0.1 μV/°C, noise -42 dB, voltage coefficient 0.1 ppm/V, inductance 0.08 μH, capacitance 0.5 pF.[4]
Ammeter shunts

An ammeter shunt is a special type of current-sensing resistor, having four terminals and a value in milliohms or even micro-ohms. Current-measuring instruments, by themselves, can usually accept only limited currents. To measure high currents, the current passes through the shunt, where the voltage drop is measured and interpreted as current. A typical shunt consists of two solid metal blocks, sometimes brass, mounted on to an insulating base. Between the blocks, and soldered or brazed to them, are one or more strips of low temperature coefficient of resistance (TCR) manganin alloy. Large bolts threaded into the blocks make the current connections, while much-smaller screws provide voltage connections. Shunts are rated by full-scale current, and often have a voltage drop of 50 mV at rated current.
Grid resistor

In heavy-duty industrial high-current applications, a grid resistor is a large convection-cooled lattice of stamped metal alloy strips connected in rows between two electrodes. Such industrial grade resistors can be as large as a refrigerator; some designs can handle over 500 amperes of current, with a range of resistances extending lower than 0.04 ohms. They are used in applications such as dynamic braking and load banking for locomotives and trams, neutral grounding for industrial AC distribution, control loads for cranes and heavy equipment, load testing of generators and harmonic filtering for electric substations.[5][6][7]

The term grid resistor is sometimes used to describe a resistor of any type connected to the control grid of a vacuum tube. This is not a resistor technology; it is an electronic circuit topology.
Negative resistors

Resistance of a conductor

DC resistance

The resistance R of a conductor of uniform cross section can be computed as

R = \rho\frac{\ell}{A} \,

where

\ell is the length of the conductor, measured in metres [m]

A is the cross-sectional area of the current flow, measured in square metres [m²]

ρ (Greek: rho) is the electrical resistivity (also called specific electrical resistance) of the material, measured in ohm-metres (Ω m). Resistivity is a measure of the material's ability to oppose electric current.

For practical reasons, any connections to a real conductor will almost certainly mean the current density is not totally uniform. However, this formula still provides a good approximation for long thin conductors such as wires.
[edit] AC resistance

If a wire conducts high-frequency alternating current then the effective cross sectional area of the wire is reduced because of the skin effect. If several conductors are together, then due to proximity effect, the effective resistance of each is higher than if that conductor were alone.
[edit] Causes of resistance
[edit] In metals

A metal consists of a lattice of atoms, each with a shell of electrons. This can also be known as a positive ionic lattice. The outer electrons are free to dissociate from their parent atoms and travel through the lattice, creating a 'sea' of electrons, making the metal a conductor. When an electrical potential difference (a voltage) is applied across the metal, the electrons drift from one end of the conductor to the other under the influence of the electric field.

Near room temperatures, the thermal motion of ions is the primary source of scattering of electrons (due to destructive interference of free electron waves on non-correlating potentials of ions), and is thus the prime cause of metal resistance. Imperfections of lattice also contribute into resistance, although their contribution in pure metals is negligible.

The larger the cross-sectional area of the conductor, the more electrons are available to carry the current, so the lower the resistance. The longer the conductor, the more scattering events occur in each electron's path through the material, so the higher the resistance. Different materials also affect the resistance.[1]
[edit] In semiconductors and insulators

In metals, the Fermi level lies in the conduction band (see Band Theory, below) giving rise to free conduction electrons. However, in semiconductors the position of the Fermi level is within the band gap, approximately half-way between the conduction band minimum and valence band maximum for intrinsic (undoped) semiconductors. This means that at 0 kelvins, there are no free conduction electrons and the resistance is infinite. However, the resistance will continue to decrease as the charge carrier density in the conduction band increases. In extrinsic (doped) semiconductors, dopant atoms increase the majority charge carrier concentration by donating electrons to the conduction band or accepting holes in the valence band. For both types of donor or acceptor atoms, increasing the dopant density leads to a reduction in the resistance. Highly doped semiconductors hence behave metallic. At very high temperatures, the contribution of thermally generated carriers will dominate over the contribution from dopant atoms and the resistance will decrease exponentially with temperature.
[edit] In ionic liquids/electrolytes

In electrolytes, electrical conduction happens not by band electrons or holes, but by full atomic species (ions) traveling, each carrying an electrical charge. The resistivity of ionic liquids varies tremendously by the concentration - while distilled water is almost an insulator, salt water is a very efficient electrical conductor. In biological membranes, currents are carried by ionic salts. Small holes in the membranes, called ion channels, are selective to specific ions and determine the membrane resistance.
[edit] Resistivity of various materials
Main article: electrical resistivities of the elements (data page)
Material Resistivity, ρ
ohm-metre
Metals 10–8
Semiconductors variable
Electrolytes variable
Insulators 1016
Superconductors 0 (exactly)
[edit] Band theory simplified
Electron energy levels in an insulator

Quantum mechanics states that the energy of an electron in an atom cannot be any arbitrary value. Rather, there are fixed energy levels which the electrons can occupy, and values in between these levels are impossible. The energy levels are grouped into two bands: the valence band and the conduction band (the latter is generally above the former). Electrons in the conduction band may move freely throughout the substance in the presence of an electrical field.

In insulators and semiconductors, the atoms in the substance influence each other so that between the valence band and the conduction band there exists a forbidden band of energy levels, which the electrons cannot occupy. In order for a current to flow, a relatively large amount of energy must be furnished to an electron for it to leap across this forbidden gap and into the conduction band. Thus, even large voltages can yield relatively small currents.
[edit] Differential resistance

When the current–voltage dependence is not linear, differential resistance, incremental resistance or slope resistance is defined as the slope of the V-I graph at a particular point, thus:

R = \frac {\mathrm{d}V} {\mathrm{d}I} \,

This quantity is sometimes called simply resistance, although the two definitions are equivalent only for an ohmic component such as an ideal resistor. For example, a diode is a circuit element for which the resistance depends on the applied voltage or current.

If the V-I graph is not monotonic (i.e. it has a peak or a trough), the differential resistance will be negative for some values of voltage and current. This property is often known as negative resistance, although it is more correctly called negative differential resistance, since the absolute resistance V/I is still positive. An example of such an element is the tunnel diode.

Differential resistance is only useful to compare a nonlinear device with a linear source/load in some small interval; for example if it is necessary to evaluate a zener diode's voltage stability under different current values.
[edit] Temperature dependence

Near room temperature, the electric resistance of a typical metal increases linearly with rising temperature, while the electrical resistance of a typical semiconductor decreases with rising temperature. The amount of that change in resistance can be calculated using the temperature coefficient of resistivity of the material using the following formula:

R = R_0 [\alpha (T - T_0) + 1]\,\!

where T is its temperature, T0 is a reference temperature (usually room temperature), R0 is the resistance at T0, and α is the percentage change in resistivity per unit temperature. The constant α depends only on the material being considered. The relationship stated is actually only an approximate one, the true physics being somewhat non-linear, or looking at it another way, α itself varies with temperature. For this reason it is usual to specify the temperature that α was measured at with a suffix, such as α15 and the relationship only holds in a range of temperatures around the reference.[4]

At lower temperatures (less than the Debye temperature), the resistance of a metal decreases as T5 due to the electrons scattering off of phonons. At even lower temperatures, the dominant scattering mechanism for electrons is other electrons, and the resistance decreases as T2. At some point, the impurities in the metal will dominate the behavior of the electrical resistance which causes it to saturate to a constant value. Matthiessen's Rule (first formulated by Augustus Matthiessen in the 1860s; the equation below gives its modern form) [5][6] says that all of these different behaviors can be summed up to get the total resistance as a function of temperature,

R = R_\text{imp} + a T^2 + b T^5 + cT \,

where Rimp is the temperature independent electrical resistivity due to impurities, and a, b, and c are coefficients which depend upon the metal's properties. This rule can be seen as the motivation to Heike Kamerlingh Onnes's experiments that lead in 1911 to discovery of superconductivity. For details see History of superconductivity.

Intrinsic semiconductors become better conductors as the temperature increases; the electrons are bumped to the conduction energy band by thermal energy, where they flow freely and in doing so leave behind holes in the valence band which also flow freely. The electric resistance of a typical intrinsic (non doped) semiconductor decreases exponentially with the temperature:

R= R_0 e^{-aT}\,

Extrinsic (doped) semiconductors have a far more complicated temperature profile. As temperature increases starting from absolute zero they first decrease steeply in resistance as the carriers leave the donors or acceptors. After most of the donors or acceptors have lost their carriers the resistance starts to increase again slightly due to the reducing mobility of carriers (much as in a metal). At higher temperatures it will behave like intrinsic semiconductors as the carriers from the donors/acceptors become insignificant compared to the thermally generated carriers.[7]

The electric resistance of electrolytes and insulators is highly nonlinear, and case by case dependent, therefore no generalized equations are given.
[edit] Strain dependence

Just as the resistance of a conductor depends upon temperature, the resistance of a conductor depends upon strain. By placing a conductor under tension (a form of stress that leads to strain in the form of stretching of the conductor), the length of the section of conductor under tension increases and its cross-sectional area decreases. Both these effects contribute to increasing the resistance of the strained section of conductor. Under compression (strain in the opposite direction), the resistance of the strained section of conductor decreases. See the discussion on strain gauges for details about devices constructed to take advantage of this effect.
[edit] Measuring resistance

An instrument for measuring resistance is called an ohmmeter. Simple ohmmeters cannot measure low resistances accurately because the resistance of their measuring leads causes a voltage drop that interferes with the measurement, so more accurate devices use four-terminal sensing.

History of ohms law

In January 1781, before Georg Ohm's work, Henry Cavendish experimented with Leyden jars and glass tubes of varying diameter and length filled with salt solution. He measured the current by noting how strong a shock he felt as he completed the circuit with his body. Cavendish wrote that the "velocity" (current) varied directly as the "degree of electrification" (voltage). He did not communicate his results to other scientists at the time,[26] and his results were unknown until Maxwell published them in 1879.[27]

Ohm did his work on resistance in the years 1825 and 1826, and published his results in 1827 as the book Die galvanische Kette, mathematisch bearbeitet (The galvanic Circuit investigated mathematically).[28] He drew considerable inspiration from Fourier's work on heat conduction in the theoretical explanation of his work. For experiments, he initially used voltaic piles, but later used a thermocouple as this provided a more stable voltage source in terms of internal resistance and constant potential difference. He used a galvanometer to measure current, and knew that the voltage between the thermocouple terminals was proportional to the junction temperature. He then added test wires of varying length, diameter, and material to complete the circuit. He found that his data could be modeled through the equation

x = \frac{a}{b + l},

where x was the reading from the galvanometer, l was the length of the test conductor, a depended only on the thermocouple junction temperature, and b was a constant of the entire setup. From this, Ohm determined his law of proportionality and published his results.

Ohm's law was probably the most important of the early quantitative descriptions of the physics of electricity. We consider it almost obvious today. When Ohm first published his work, this was not the case; critics reacted to his treatment of the subject with hostility. They called his work a "web of naked fancies"[29] and the German Minister of Education proclaimed that "a professor who preached such heresies was unworthy to teach science."[30] The prevailing scientific philosophy in Germany at the time, led by Hegel, asserted that experiments need not be performed to develop an understanding of nature because nature is so well ordered, and that scientific truths may be deduced through reasoning alone. Also, Ohm's brother Martin, a mathematician, was battling the German educational system. These factors hindered the acceptance of Ohm's work, and his work did not become widely accepted until the 1840s. Fortunately, Ohm received recognition for his contributions to science well before he died.

In the 1850s, Ohm's law was known as such, and was widely considered proved, and alternatives such as "Barlow's law" discredited, in terms of real applications to telegraph system design, as discussed by Samuel F. B. Morse in 1855.[31]

While the old term for electrical conductance, the mho (the inverse of the resistance unit ohm), is still used, a new name, the siemens, was adopted in 1971, honoring Ernst Werner von Siemens. The siemens is preferred in formal papers.

In the 1920s, it was discovered that the current through an ideal resistor actually has statistical fluctuations, which depend on temperature, even when voltage and resistance are exactly constant; this fluctuation, now known as Johnson–Nyquist noise, is due to the discrete nature of charge. This thermal effect implies that measurements of current and voltage that are taken over sufficiently short periods of time will yield ratios of V/I that fluctuate from the value of R implied by the time average or ensemble average of the measured current; Ohm's law remains correct for the average current, in the case of ordinary resistive materials.

Ohm's work long preceded Maxwell's equations and any understanding of frequency-dependent effects in AC circuits. Modern developments in electromagnetic theory and circuit theory do not contradict Ohm's law when they are evaluated within the appropriate limits.

Other versions of Ohm's law

Ohm's law, in the form above, is an extremely useful equation in the field of electrical/electronic engineering because it describes how voltage, current and resistance are interrelated on a "macroscopic" level, that is, commonly, as circuit elements in an electrical circuit. Physicists who study the electrical properties of matter at the microscopic level use a closely related and more general vector equation, sometimes also referred to as Ohm's law, having variables that are closely related to the V, I, and R scalar variables of Ohm's law, but are each functions of position within the conductor. Physicists often use this continuum form of Ohm's Law:[21]

\mathbf{E} = \rho \mathbf{J}

where "E" is the electric field vector with units of volts per meter (analogous to "V" of Ohm's law which has units of volts), "J" is the current density vector with units of amperes per unit area (analogous to "I" of Ohm's law which has units of amperes), and "ρ" (Greek "rho") is the resistivity with units of ohm·meters (analogous to "R" of Ohm's law which has units of ohms). The above equation is sometimes written[22] as J = σE where "σ" is the conductivity which is the reciprocal of ρ.
Current flowing through a uniform cylindrical conductor (such as a round wire) with a uniform field applied.

The potential difference between two points is defined as:[23]

{\Delta V} = -\int {\mathbf E \cdot d \mathbf l}

with d \mathbf l the element of path along the integration of electric field vector E. If the applied E field is uniform and oriented along the length of the conductor as shown in the figure, then defining the voltage V in the usual convention of being opposite in direction to the field (see figure), and with the understanding that the voltage V is measured differentially across the length of the conductor allowing us to drop the Δ symbol, the above vector equation reduces to the scalar equation:

V = {E}{l} \ \ \text{or} \ \ E = \frac{V}{l}

Since the E field is uniform in the direction of wire length, for a conductor having uniformly consistent resistivity ρ, the current density J will also be uniform in any cross-sectional area and oriented in the direction of wire length, so we may write:[24]

J = \frac{I}{a}

Substituting the above 2 results (for E and J respectively) into the continuum form shown at the beginning of this section:

\frac{V}{l} = \frac{I}{a}\rho \qquad \text{or} \qquad V = I \rho \frac{l}{a}

The electrical resistance of a uniform conductor is given in terms of resistivity by:[24]

{R} = \rho \frac{l}{a}

where l is the length of the conductor in SI units of meters, a is the cross-sectional area (for a round wire a = πr2 if r is radius) in units of meters squared, and ρ is the resistivity in units of ohm·meters.

After substitution of R from the above equation into the equation preceding it, the continuum form of Ohm's law for a uniform field (and uniform current density) oriented along the length of the conductor reduces to the more familiar form:

{V}={I}{R} \

A perfect crystal lattice, with low enough thermal motion and no deviations from periodic structure, would have no resistivity,[25] but a real metal has crystallographic defects, impurities, multiple isotopes, and thermal motion of the atoms. Electrons scatter from all of these, resulting in resistance to their flow.

The more complex generalized forms of Ohm's law are important to condensed matter physics, which studies the properties of matter and, in particular, its electronic structure. In broad terms, they fall under the topic of constitutive equations and the theory of transport coefficients.

Ohm's law

This article is about the law related to electricity. For other uses, see Ohm's acoustic law.
V, I, and R, the parameters of Ohm's law.

In electrical circuits, Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference or voltage across the two points, and inversely proportional to the resistance between them.[1]

The mathematical equation that describes this relationship is:[2]

I = \frac{V}{R}

where V is the potential difference measured across the resistance in units of volts; I is the current through the resistance in units of amperes and R is the resistance of the conductor in units of ohms. More specifically, Ohm's law states that the R in this relation is constant, independent of the current.[3]

The law was named after the German physicist Georg Ohm, who, in a treatise published in 1827, described measurements of applied voltage and current through simple electrical circuits containing various lengths of wire. He presented a slightly more complex equation than the one above (see History section below) to explain his experimental results. The above equation is the modern form of Ohm's law.

In physics, the term Ohm's law is also used to refer to various generalizations of the law originally formulated by Ohm. The simplest example of this is:

\boldsymbol{J} = \sigma \boldsymbol{E},

where J is the current density at a given location in a resistive material, E is the electric field at that location, and σ is a material dependent parameter called the conductivity. This reformulation of Ohm's law is due to Gustav Kirchhoff.[4]

Other generation methods

Various other technologies have been studied and developed for power generation. Solid-state generation (without moving parts) is of particular interest in portable applications. This area is largely dominated by thermoelectric (TE) devices, though thermionic (TI) and thermophotovoltaic (TPV) systems have been developed as well. Typically, TE devices are used at lower temperatures than TI and TPV systems. Piezoelectric devices are used for power generation from mechanical strain, particularly in power harvesting. Betavoltaics are another type of solid-state power generator which produces electricity from radioactive decay. Fluid-based magnetohydrodynamic (MHD) power generation has been studied as a method for extracting electrical power from nuclear reactors and also from more conventional fuel combustion systems. Osmotic power finally is another possibility at places where salt and sweet water merges (e.g. deltas, ...)

Electrochemical electricity generation is also important in portable and mobile applications. Currently, most electrochemical power comes from closed electrochemical cells ("batteries") [8], which are arguably utilized more as storage systems than generation systems, but open electrochemical systems, known as fuel cells, have been undergoing a great deal of research and development in the last few years. Fuel cells can be used to extract power either from natural fuels or from synthesized fuels (mainly electrolytic hydrogen) and so can be viewed as either generation systems or storage systems depending on their use.

Turbines

All turbines are driven by a fluid acting as an intermediate energy carrier. Many of the heat engines just mentioned are turbines. Other types of turbines can be driven by wind or falling water.

Sources include:

* Steam - Water is boiled by:
o Nuclear fission,
o The burning of fossil fuels (coal, natural gas, or petroleum). In hot gas (gas turbine), turbines are driven directly by gases produced by the combustion of natural gas or oil. Combined cycle gas turbine plants are driven by both steam and natural gas. They generate power by burning natural gas in a gas turbine and use residual heat to generate additional electricity from steam. These plants offer efficiencies of up to 60%.
o Renewables. The steam generated by:
+ Biomass
+ The sun as the heat source: solar parabolic troughs and solar power towers concentrate sunlight to heat a heat transfer fluid, which is then used to produce steam.
+ Geothermal power. Either steam under pressure emerges from the ground and drives a turbine or hot water evaporates a low boiling liquid to create vapour to drive a turbine.
* Other renewable sources:
o Water (hydroelectric) - Turbine blades are acted upon by flowing water, produced by hydroelectric dams or tidal forces.
o Wind - Most wind turbines generate electricity from naturally occurring wind. Solar updraft towers use wind that is artificially produced inside the chimney by heating it with sunlight, and are more properly seen as forms of solar thermal energy.

[edit] Reciprocating engines

Small electricity generators are often powered by reciprocating engines burning diesel, biogas or natural gas. Diesel engines are often used for back up generation, usually at low voltages. However most large power grids also use Diesel generators, originally provided as emergency back up for a specific facility such as a hospital, to feed power into the grid during certain circumstances. Biogas is often combusted where it is produced, such as a landfill or wastewater treatment plant, with a reciprocating engine or a microturbine, which is a small gas turbine.
A coal-fired power plant in Laughlin, Nevada U.S.A. Owners of this plant ceased operations after declining to invest in pollution control equipment to comply with pollution regulations.[6]
[edit] Photovoltaic panels

Unlike the solar heat concentrators mentioned above, photovoltaic panels convert sunlight directly to electricity. Although sunlight is free and abundant, solar electricity is still usually more expensive to produce than large-scale mechanically generated power due to the cost of the panels. Low-efficiency silicon solar cells have been decreasing in cost and multijunction cells with close to 30% conversion efficiency are now commercially available. Over 40% efficiency has been demonstrated in experimental systems.[7] Until recently, photovoltaics were most commonly used in remote sites where there is no access to a commercial power grid, or as a supplemental electricity source for individual homes and businesses. Recent advances in manufacturing efficiency and photovoltaic technology, combined with subsidies driven by environmental concerns, have dramatically accelerated the deployment of solar panels. Installed capacity is growing by 40% per year led by increases in Germany, Japan, California and New Jersey.

Electric motor

An electric motor uses electrical energy to produce mechanical energy, very typically through the interaction of magnetic fields and current-carrying conductors. The reverse process, producing electrical energy from mechanical energy, is accomplished by a generator ordynamo. Many types of electric motors can be run as generators, and vice versa. For example a starter/generator for a gas turbine or Traction motors used on vehicles often perform both tasks.
Electric motors are found in applications as diverse as industrial fans, blowers and pumps, machine tools, household appliances, power tools, and disk drives. They may be powered bydirect current (for example a battery powered portable device or motor vehicle), or by alternating current from a central electrical distribution grid. The smallest motors may be found in electric wristwatches. Medium-size motors of highly standardized dimensions and characteristics provide convenient mechanical power for industrial uses. The very largest electric motors are used for propulsion of large ships, and for such purposes as pipeline compressors, with ratings in the millions of watts. Electric motors may be classified by the source of electric power, by their internal construction, by their application, or by the type of motion they give.
The physical principle of production of mechanical force by the interactions of an electric current and a magnetic field was known as early as 1821. Electric motors of increasing efficiency were constructed throughout the 19th century, but commercial exploitation of electric motors on a large scale required efficient electrical generators and electrical distribution networks.
Some devices, such as magnetic solenoids and loudspeakers, although they generate some mechanical power, are not generally referred to as electric motors, and are usually termed actuators[1] and transducers,[2] respectively.

Waveform

Waveform means the shape and form of a signal such as a wave moving in a solid, liquid or gaseous medium.

In many cases the medium in which the wave is being propagated does not permit a direct visual image of the form. In these cases, the term 'waveform' refers to the shape of a graph of the varying quantity against time or distance. An instrument called an oscilloscope can be used to pictorially represent the wave as a repeating image on a CRT or LCD screen.

By extension of the above, the term 'waveform' is now also sometimes used to describe the shape of the graph of any varying quantity against time.

Direct current

Direct current (DC) is the unidirectional flow of electric charge. Direct current is produced by such sources as batteries, thermocouples, solar cells, and commutator-type electric machines of the dynamo type. Direct current may flow in a conductor such as a wire, but can also be through semiconductors, insulators, or even through a vacuum as in electron or ion beams. The electric charge flows in a constant direction, distinguishing it from alternating current (AC). A term formerly used for direct current was Galvanic current.
Types of direct current.

Direct current may be obtained from an alternating current supply by use of a current-switching arrangement called a rectifier, which contains electronic elements (usually) or electromechanical elements (historically) that allow current to flow only in one direction. Direct current may be made into alternating current with an inverter or a motor-generator set.

The first commercial electric power transmission (developed by Thomas Edison in the late nineteenth century) used direct current. Because of the advantage of alternating current over direct current in transforming and transmission, electric power distribution today is nearly all alternating current. For applications requiring direct current, such as third rail power systems, alternating current is distributed to a substation, which utilizes a rectifier to convert the power to direct current. See War of Currents.

Direct current is used to charge batteries, and in nearly all electronic systems as the power supply. Very large quantities of direct-current power are used in production of aluminum and other electrochemical processes. Direct current is used for some railway propulsion, especially in urban areas. High voltage direct current is used to transmit large amounts of power from remote generation sites or to interconnect alternating current power grids.

Alternating current

In alternating current (AC, also ac) the movement (or flow) of electric charge periodically reverses direction. An electric charge would for instance move forward, then backward, then forward, then backward, over and over again. In direct current (DC), the movement (or flow) of electric charge is only in one direction.

Used generically, AC refers to the form in which electricity is delivered to businesses and residences. The usual waveform of an AC power circuit is a sine wave, however in certain applications, different waveforms are used, such as triangular or square waves. Audio and radio signals carried on electrical wires are also examples of alternating current. In these applications, an important goal is often the recovery of information encoded (or modulated) onto the AC signal.

Methods of generating electricity

There are seven fundamental methods of directly transforming other forms of energy into electrical energy:

* Static electricity, from the physical separation and transport of charge (examples: triboelectric effect and lightning)
* Electromagnetic induction, where an electrical generator, dynamo or alternator transforms kinetic energy (energy of motion) into electricity
* Electrochemistry, the direct transformation of chemical energy into electricity, as in a battery, fuel cell or nerve impulse
* Photoelectric effect, the transformation of light into electrical energy, as in solar cells
* Thermoelectric effect, direct conversion of temperature differences to electricity, as in thermocouples and thermopiles
* Piezoelectric effect, from the mechanical strain of electrically anisotropic molecules or crystals
* Nuclear transformation, the creation and acceleration of charged particles (examples: betavoltaics or alpha particle emission)

Static electricity was the first form discovered and investigated, and the electrostatic generator is still used even in modern devices such as the Van de Graaff generator and MHD generators. Electrons are mechanically separated and transported to increase their electric potential.

Almost all commercial electrical generation is done using electromagnetic induction, in which mechanical energy forces an electrical generator to rotate. There are many different methods of developing the mechanical energy, including heat engines, hydro, wind and tidal power.

The direct conversion of nuclear energy to electricity by beta decay is used only on a small scale. In a full-size nuclear power plant, the heat of a nuclear reaction is used to run a heat engine. This drives a generator, which converts mechanical energy into electricity by magnetic induction.

Most electric generation is driven by heat engines. The combustion of fossil fuels supplies most of the heat to these engines, with a significant fraction from nuclear fission and some from renewable sources. The modern steam turbine invented by Sir Charles Parsons in 1884 - today generates about 80 percent of the electric power in the world using a variety of heat sources.
[edit] Turbines
Large dams such as Three Gorges Dam in China can provide large amounts of hydroelectric power; it will have a 22.5 GW capability.
Susquehanna Steam Electric Station, a nuclear power plant.
A combined cycle natural gas power plant near Orem, Utah.

All turbines are driven by a fluid acting as an intermediate energy carrier. Many of the heat engines just mentioned are turbines. Other types of turbines can be driven by wind or falling water.

Sources include:

* Steam - Water is boiled by:
o Nuclear fission,
o The burning of fossil fuels (coal, natural gas, or petroleum). In hot gas (gas turbine), turbines are driven directly by gases produced by the combustion of natural gas or oil. Combined cycle gas turbine plants are driven by both steam and natural gas. They generate power by burning natural gas in a gas turbine and use residual heat to generate additional electricity from steam. These plants offer efficiencies of up to 60%.
o Renewables. The steam generated by:
+ Biomass
+ The sun as the heat source: solar parabolic troughs and solar power towers concentrate sunlight to heat a heat transfer fluid, which is then used to produce steam.
+ Geothermal power. Either steam under pressure emerges from the ground and drives a turbine or hot water evaporates a low boiling liquid to create vapour to drive a turbine.
* Other renewable sources:
o Water (hydroelectric) - Turbine blades are acted upon by flowing water, produced by hydroelectric dams or tidal forces.
o Wind - Most wind turbines generate electricity from naturally occurring wind. Solar updraft towers use wind that is artificially produced inside the chimney by heating it with sunlight, and are more properly seen as forms of solar thermal energy.

[edit] Reciprocating engines

Small electricity generators are often powered by reciprocating engines burning diesel, biogas or natural gas. Diesel engines are often used for back up generation, usually at low voltages. However most large power grids also use Diesel generators, originally provided as emergency back up for a specific facility such as a hospital, to feed power into the grid during certain circumstances. Biogas is often combusted where it is produced, such as a landfill or wastewater treatment plant, with a reciprocating engine or a microturbine, which is a small gas turbine.
A coal-fired power plant in Laughlin, Nevada U.S.A. Owners of this plant ceased operations after declining to invest in pollution control equipment to comply with pollution regulations.[6]
[edit] Photovoltaic panels

Unlike the solar heat concentrators mentioned above, photovoltaic panels convert sunlight directly to electricity. Although sunlight is free and abundant, solar electricity is still usually more expensive to produce than large-scale mechanically generated power due to the cost of the panels. Low-efficiency silicon solar cells have been decreasing in cost and multijunction cells with close to 30% conversion efficiency are now commercially available. Over 40% efficiency has been demonstrated in experimental systems.[7] Until recently, photovoltaics were most commonly used in remote sites where there is no access to a commercial power grid, or as a supplemental electricity source for individual homes and businesses. Recent advances in manufacturing efficiency and photovoltaic technology, combined with subsidies driven by environmental concerns, have dramatically accelerated the deployment of solar panels. Installed capacity is growing by 40% per year led by increases in Germany, Japan, California and New Jersey.
[edit] Other generation methods
Wind-powered turbines usually provide electrical generation in conjunction with other methods of producing power.

Various other technologies have been studied and developed for power generation. Solid-state generation (without moving parts) is of particular interest in portable applications. This area is largely dominated by thermoelectric (TE) devices, though thermionic (TI) and thermophotovoltaic (TPV) systems have been developed as well. Typically, TE devices are used at lower temperatures than TI and TPV systems. Piezoelectric devices are used for power generation from mechanical strain, particularly in power harvesting. Betavoltaics are another type of solid-state power generator which produces electricity from radioactive decay. Fluid-based magnetohydrodynamic (MHD) power generation has been studied as a method for extracting electrical power from nuclear reactors and also from more conventional fuel combustion systems. Osmotic power finally is another possibility at places where salt and sweet water merges (e.g. deltas, ...)

Electrochemical electricity generation is also important in portable and mobile applications. Currently, most electrochemical power comes from closed electrochemical cells ("batteries") [8], which are arguably utilized more as storage systems than generation systems, but open electrochemical systems, known as fuel cells, have been undergoing a great deal of research and development in the last few years. Fuel cells can be used to extract power either from natural fuels or from synthesized fuels (mainly electrolytic hydrogen) and so can be viewed as either generation systems or storage systems depending on their use.

History

Centralised power generation became possible when it was recognized that alternating current power lines can transport electricity at very low costs across great distances by taking advantage of the ability to raise and lower the voltage using power transformers.

Electricity has been generated at central stations since 1881. The first power plants were run on water power[4] or coal,[5] and today we rely mainly on coal, nuclear, natural gas, hydroelectric, and petroleum with a small amount from solar energy, tidal harnesses, wind generators, and geothermal sources.

William Henry Fox Talbot

Henry Fox Talbot is usually known today under this version of his name. However, he greatly disliked this version and always made it known that he wished to be Henry F Talbot or H F Talbot. We shall at least go some way towards his wishes and refer to him as Talbot (or Henry) throughout this article.

Henry's father was William Davenport Talbot and his mother was Elisabeth Theresa Fox-Strangways. Davenport Talbot owned Lacock Abbey in Wiltshire which had been in the possession of the Talbot family since the 1500s but this only happened because two husbands took their wife's name of Talbot to keep the Talbot family name from dying out. Lady Elisabeth Fox-Strangways was the eldest child of Henry Thomas Fox-Strangways, second earl of Ilchester, and was from a family which moved in the highest political circles. Given the status of the families one might expect Henry to be born into a wealthy but, on the contrary, when he was born, Lacock Abbey was £30,000 in debt, this being a substantial sum in those times. Financial problems was not the only difficulty the family had, for Davenport Talbot died was Henry was only five months old leaving Lady Elisabeth in a somewhat tricky situation. Henry's mother was a rather remarkable woman, however, being highly intelligent and very well educated. She was very interested in politics and fluent in French, Latin, and Greek. Her skilful management of the Lacock Abbey estate would mean that Henry would eventually face no financial problems.

Of course Lady Elizabeth's family owned a number of properties and Henry and his mother lived in a number of these at different times. She married Captain Charles Feilding (he was later promoted to a Rear-Admiral) in 1804. He was a devoted step-father to Henry who never lacked a father's love from that time. Daughters from the marriage of Elizabeth and Charles were born in 1808 and 1810. Henry met the astronomer William Herschel when he was eight years old, in the same year as he entered Rottingdean boarding school [2]:-

He was a brilliant student and eager to learn, but was painfully shy and reclusive by nature.

In 1810 Henry went to Harrow school where he remained until 1815 after which his preparation for university was completed with two years private tuition at home. As a boy he had shown great curiosity about the world and remarkably wide interests, particularly in mathematics, languages, politics, botany, optics, and astronomy. He had enjoyed chemistry at school and got himself into trouble over causing explosions while experimenting with chemicals. He entered Trinity College, Cambridge, in 1817 and there he won prizes for Greek verse, and graduated with the classical medal in 1821 being twelfth wrangler in mathematics (that is he was placed twelfth in the ranked list of First Class students). He had now reached the age of 21 and, as we mentioned above, his mother's skilful management of the Lacock Abbey estate had restored it to a healthy financial position now free from the debts it faced at the time of his birth. Talbot received a healthy income from the estate. He was elected a Fellow of the newly founded Royal Astronomical Society in 1822. He worked hard at mathematical research but also enjoyed himself travelling widely on the Continent; he was particularly pleased to visit Italy.

Talbot wrote papers on elliptic integrals, building on work of Euler, Legendre, Jacobi and Abel. By the time he met John Herschel in Munich in 1824 (the son of William Herschel whom he had met at age 8) he had published six mathematics papers and the two had many scientific interests in common, for example both were fine mathematicians and fellows of the Royal Astronomical Society. They quickly became firm friends and Talbot's interests turned more towards the study of light. David Brewster, the Scottish physicist, was also experimenting with light at this time and he published a number of Talbot's articles. Talbot was elected a Fellow of the Royal Society in 1831 for his mathematical work. In addition to his work on mathematics and physics, Talbot also published on astronomy. On 20 October 1833 he married Constance Mundy from Markeaton in Derbyshire. Shortly after this he was elected to serve as Member of Parliament for Chippenham which he did until 1835. He would continue to have an interest in politics after the election of 1835 but he chose not to stand for Parliament again in that election.

An important idea, which would change his whole life, came to him while he was in Italy in October 1833. This was part of a holiday to France, Switzerland, and Italy that he was taking with his wife Constance. He was trying to make sketches on Lake Como in the north of Italy using a camera lucida which is a drawing instrument (do not be confused by the word camera - this instrument had no relation to photography). In the introduction to The Pencil of Nature which he published in 1844, Talbot explains his thoughts in Italy on that October day:-

[I]n October, 1833, I was amusing myself on the lovely shores of the Lake of Como in Italy, taking sketches with a camera lucida, or rather, I should say, attempting to make them; but with the smallest possible amount of success ... After various fruitless attempts I laid aside the instrument and came to the conclusion that its use required a previous knowledge of drawing which unfortunately I did not possess. I then thought of trying again a method which I had tried many years before. [I] reflected on the inimitable beauty of the pictures of nature's painting which the glass lens of the camera obscura throws upon the paper in its focus - fairy pictures, creations of a moment, and destined as rapidly to fade away ... It was during these thoughts that the idea occurred to me ... how charming it would be if it were possible to cause these natural images to imprint themselves durably and remain fixed upon the paper!

He returned from Italy to continue his work as a Member of Parliament but by the spring of 1834 he was living at Lacock Abbey carrying out experiments with photography. His expertise in chemistry and light were important factors in his success. John Herschel was very interested in Talbot's experiments and devised many of the terms still in use today. He gave the name "negative" to the inverted image that Talbot was producing on his paper covered with suitably chosen chemicals, and "fixing" to the process that Talbot perfected of treating the paper with further chemicals to prevent further action by light. We have an example of a negative produced by Talbot in August 1835 which is a picture of the Oriel window in the South Gallery at Lacock Abbey. A copy of the negative is on display at Lacock where I have greatly enjoyed viewing the exhibits from Talbot's early experiments. Here is Talbot's own description:-

... I constructed [a camera obscura] out of a large box, the image being thrown upon one end of it by a good object-glass fixed at the opposite end. The apparatus being armed with a sensitive paper, was taken out in a summer afternoon, and placed about one hundred yards from a building favourably illuminated by the sun. An hour or so afterwards I opened the box and I found depicted upon the paper a very distinct representation of the building, with the exception of those parts of it which lay in the shade. A little experience in this branch of the art showed me that with a smaller camera obscura the effect would be produced in a smaller time. Accordingly I had several small boxes made, in which I fixed lenses of shorter focus, and with these I obtained very perfect, but extremely small pictures ...

Talbot was able to find a way to print back his negatives onto sensitive paper giving a correct representation of the scene pictured. Moreover he could print multiple copies from a single negative. Photography was not his only interest, however, and he continued to work on mathematics and on light over the next few years without making public his photographic discoveries. He gave the Bakerian lecture to the Royal Society in 1837 with the title Further observations on the optical phenomena of crystals and he received the Royal Medal from the Royal Society in 1838 for his mathematical achievements.

By the autumn of 1838 Talbot took up again his photographic work and began to contemplate writing a paper to be submitted to the Royal Society describing his discoveries. However in January 1839 he was shocked to read an announcement by Arago and Daguerre claiming that Daguerre had developed a means of obtaining permanent images from a camera obscura. Talbot quickly moved to publicise his own work sending examples of his photographs to the Royal Institution in London less than a week after he heard of the French announcement, and he wrote to Arago claiming priority a couple of days later. He submitted a paper Some account of the art of photogenic drawing on his photographic methods to the Royal Society on 31 January 1839. Later papers by Talbot on photography, however, the Society chose not to publish. In 1840 Talbot made further brilliant advances, discovering how to use chemicals to bring out an image from his sensitive paper after it had only been exposed for a very short time. He patented the process under the name calotype.

Two factors meant that Talbot's process would not achieve the widespread use that Daguerre's process did. One was simply that Daguerre made his process freely available while Talbot charged a fee for anyone to use his, and secondly Daguerre's process produced much sharper image. This said, however, nobody can look at the images produced by Talbot without being impressed by their great beauty. In 1844 he published Pencil of nature the first photographically illustrated book; we have quoted from the text above. Things went badly for him over his photographic patents for when he tried to use them to prevent others using similar methods, the court ruled in 1854 that he was [2]:-

... the true inventor of photography but ruled that newer processes were outside his patent. The acrimonious proceedings had stained Talbot's reputation so severely that the prejudices raised continue to surface in historical literature.

He did gain some recognition from the Royal Society for the his photographic work with the award of their Rumford medal in 1842. He went through a period of bad health towards the latter years of the 1840s but his health improved again in the next decade and he was able to undertake research with much the same vigour as he had in the 1830s. He worked on producing images in ink and took out further patents regarding his printing processes. Although he never achieved commercial success with methods he was developing, he was going down precisely the right path towards modern methods of printing from a photographic plate.

Talbot's home was Lacock Abbey but he spent much time in Scotland. He visited Edinburgh in the early 1840s and published Sun Pictures of Scotland in 1845 which contains photographs of Scotland including the Scott Monument on Princes Street in Edinburgh. From 1855 he certainly lived for part of each year in Scotland during the next 12 years or so. He was awarded an honorary degree by Edinburgh University in 1863:-

... because of his pre-eminence in literature and science, and the benefits his discoveries have conferred upon society.

His interests during his later years were not exclusively associated with photography and he became increasingly interested in archaeology being one of the first to translate the cuneiform writing from Nineveh. Schaaf writes in [2]:-

The inventor's name is preserved in various scientific fields: in mathematics, there is Talbot's curve; in physics Talbot's law and the Talbot (a unit of luminous energy); in botany two species are named after him; in astronomy a crater of the moon is named after him; and there is the persistent testimony of an art that has become so pervasive in society that its products are sometimes as invisible to us as are his latent images. In his lifetime, Talbot had published seven books and nearly sixty scientific and mathematical articles.

During his final years he suffered from heart disease and this was the cause of his death which occurred in his study at Lacock Abbey.

Michael Faraday

Michael Faraday did not directly contribute to mathematics so should not really qualify to have his biography in this archive. However he was such a major figure and his science had such a large impact on the work of those developing mathematical theories that it is proper that he is included. We say more about this below.

Faraday's father, James Faraday, was a blacksmith who came from Yorkshire in the north of England while his mother Margaret Hastwell, also from the north of England, was the daughter of a farmer. Early in 1791 James and Margaret moved to Newington Butts, which was then a village outside London, where James hoped that work was more plentiful. They already had two children, a boy Robert and a girl, before they moved to Newington Butts and Michael was born only a few months after their move.

Work was not easy to find and the family moved again, remaining in or around London. By 1795, when Michael was around five years, the family were living in Jacob's Wells Mews in London. They had rooms over a coachhouse and, by this time, a second daughter had been born. Times were hard particularly since Michael's father had poor health and was not able to provide much for his family.

The family were held closely together by a strong religious faith, being members of the Sandemanians, a form of the Protestant Church which had split from the Church of Scotland. The Sandemanians believed in the literal truth of the Bible and tried to recreate the sense of love and community which had characterised the early Christian Church. The religious influence was important for Faraday since the theories he developed later in his life were strongly influenced by a belief in a unity of the world.

Michael attended a day school where he learnt to read, write and count. When Faraday was thirteen years old he had to find work to help the family finances and he was employed running errands for George Riebau who had a bookselling business. In 1805, after a year as an errand-boy, Faraday was taken on by Riebau as an apprentice bookbinder. He spent seven years serving his apprenticeship with Riebau. Not only did he bind books but he also read them. Riebau wrote a letter in 1813 in which he described how Faraday spent his days as an apprentice (see for example [4]):-

After the regular hours of business, he was chiefly employed in drawing and copying from the Artist's Repository, a work published in numbers which he took in weekly. ... Dr Watts's Improvements of the mind was then read and frequently took in his pocket, when he went an early walk in the morning, visiting some other works of art or searching for some mineral or vegetable curiosity. ... His mind ever engaged, besides attending to bookbinding which he executed in a proper manner.

His mode of living temperate, seldom drinking any other than pure water, and when done his day's work, would set himself down in the workshop ... If I had any curious book from my customers to bind, with plates, he would copy such as he thought singular or clever ...

Faraday himself wrote of this time in his life:-

Whilst an apprentice, I loved to read the scientific books which were under my hands ...

From 1810 Faraday attended lectures at John Tatum's house. He attended lectures on many different topics but he was particularly interested in those on electricity, galvanism and mechanics. At Tatum's house he made two special friends, J Huxtable who was a medical student, and Benjamin Abbott who was a clerk. In 1812 Faraday attended lectures by Humphry Davy at the Royal Institution and made careful copies of the notes he had taken. In fact these lectures would become Faraday's passport to a scientific career.

In 1812, intent on improving his literary skills, he carried out a correspondence with Abbott. He had already tried to leave bookbinding and the route he tried was certainly an ambitious one. He had written to Sir Joseph Banks, the President of the Royal Society, asking how he could become involved in scientific work. Perhaps not surprisingly he had received no reply. When his apprenticeship ended in October 1812, Faraday got a job as a bookbinder but still he attempted to get into science and again he took a somewhat ambitious route for a young man with little formal education. He wrote to Humphry Davy, who had been his hero since he attended his chemistry lectures, sending him copies of the notes he had taken at Davy's lectures. Davy, unlike Banks, replied to Faraday and arranged a meeting. He advised Faraday to keep working as a bookbinder, saying:-

Science [is] a harsh mistress, and in a pecuniary point of view but poorly rewarding those who devote themselves to her service.

Shortly after the interview Davy's assistant had to be sacked for fighting and Davy sent for Faraday and invited him to fill the empty post. In 1813 Faraday took up the position at the Royal Institution.

In October 1813 Davy set out on a scientific tour of Europe and he took Faraday with him as his assistant and secretary. Faraday met Ampère and other scientists in Paris. They travelled on towards Italy where they spent time in Genoa, Florence, Rome and Naples. Heading north again they visited Milan where Faraday met Volta. The trip was an important one for Faraday [4]:-

These eighteen months abroad had taken the place, in Faraday's life, of the years spent at university by other men. He gained a working knowledge of French and Italian; he had added considerably to his scientific attainments, and had met and talked with many of the leading foreign men of science; but, above all, the tour had been what was most valuable to him at that time, a broadening influence.

On his return to London, Faraday was re-engaged at the Royal Institution as an assistant. His work there was mainly involved with chemical experiments in the laboratory. He also began lecturing on chemistry topics at the Philosophical Society. He published his first paper in 1816 on caustic lime from Tuscany.

In 1821 Faraday married Sarah Barnard whom he had met when attending the Sandemanian church. Faraday was made Superintendent of the House and Laboratory at the Royal Institution and given additional rooms to make his marriage possible.

The year 1821 marked another important time in Faraday's researches. He had worked almost entirely on chemistry topics yet one of his interests from his days as a bookbinder had been electricity. In 1820 several scientists in Paris including Arago and Ampère made significant advances in establishing a relation between electricity and magnetism. Davy became interested and this gave Faraday the opportunity to work on the topic. He published On some new electro-magnetical motions, and on the theory of magnetism in the Quarterly Journal of Science in October 1821. Pearce Williams writes [1]:-

It records the first conversion of electrical into mechanical energy. It also contained the first notion of the line of force.

It is Faraday's work on electricity which has prompted us to add him to this archive. However we must note that Faraday was in no sense a mathematician and almost all his biographers describe him as "mathematically illiterate". He never learnt any mathematics and his contributions to electricity were purely that of an experimentalist. Why then include him in an archive of mathematicians? Well, it was Faraday's work which led to deep mathematical theories of electricity and magnetism. In particular the remarkable mathematical theories on the topic developed by Maxwell would not have been possible without Faraday's discovery of various laws. This is a point which Maxwell himself stressed on a number of occasions.

In the ten years from 1821 to 1831 Faraday again undertook research on chemistry. His two most important pieces of work on chemistry during that period was liquefying chlorine in 1823 and isolating benzene in 1825. Between these dates, in 1824, he was elected a fellow of the Royal Society. This was a difficult time for Faraday since Davy was at this time President of the Royal Society and could not see the man who he still thought of as his assistant as becoming a Fellow. Although Davy opposed his election, he was over-ruled by the other Fellows. Faraday never held the incident against Davy, always holding him in the highest regard.

Faraday introduced a series of six Christmas lectures for children at the Royal Institution in 1826. In 1831 Faraday returned to his work on electricity and made what is arguably his most important discovery, namely that of electro-magnetic induction. This discovery was the opposite of that which he had made ten years earlier. He showed that a magnet could induce an electrical current in a wire. Thus he was able to convert mechanical energy into electrical energy and discover the first dynamo. Again he made lines of force central to his thinking. He published his first paper in what was to become a series on Experimental researches on electricity in 1831. He read the paper before the Royal Society on 24 November of that year.

In 1832 Faraday began to receive honours for his major contributions to science. In that year he received an honorary degree from the University of Oxford. In February 1833 he became Fullerian Professor of Chemistry at the Royal Institution. Further honours such as the Royal Medal and the Copley Medal, both from the Royal Society, were to follow. In 1836 he was made a Member of the Senate of the University of London, which was a Crown appointment.

During this period, beginning in 1833, Faraday made important discoveries in electrochemistry. He went on to work on electrostatics and by 1838 he [1]:-

... was in a position to put all the pieces together into a coherent theory of electricity.

The extremely high workload eventually told on Faraday's health and in 1839 he suffered a nervous breakdown. He did recover his health and by 1845 he began intense research activity again. The work which he undertook at this time was the result of mathematical developments in the subject. Faraday's ideas on lines of force had received a mathematical treatment from William Thomson. He wrote to Faraday on 6 August 1845 telling him of his mathematical predictions that a magnetic field should affect the plane of polarised light. Faraday had attempted to detect this experimentally many years earlier but without success. Now, with the idea reinforced by Thomson, he tried again and on 13 September 1845 he was successful in showing that a strong magnetic field could rotate the plane of polarisation, and moreover that the angle of rotation was proportional to the strength of the magnetic field. Faraday wrote (see for example [1]):-

That which is magnetic in the forces of matter has been affected, and in turn has affected that which is truly magnetic in the force of light.

He followed his line of experiments which led him to discover diamagnetism.

By the mid 1850s Faraday's mental abilities began to decline. At around the same time Maxwell was building on the foundations Faraday had created developing a mathematical theory which would always have been out of reach for Faraday. However Faraday continued to lecture at the Royal Institution but declined the offer of the Presidency of the Royal Society in 1857.

He continued to give the children's Christmas lectures. In 1859-60 he gave the Christmas lectures on the various forces of matter. At the following Christmas he gave the children's lectures on the chemical history of the candle. These two final series of lectures by Faraday were published and have become classics. The Christmas lectures at the Royal Institution, begun by Faraday, continue today but now reach a much greater audience since they are televised. I [EFR] have watched these lectures with great interest over many years. They are a joy for anyone interested as I am in the "public understanding of science". I particularly remember lectures by Carl Sagan on "the planets" and mathematics lectures by Chris Zeeman and Ian Stewart.

The Royal Institution literature states:-

[Faraday's] magnetic laboratory, where many of his most important discoveries were made, was restored in 1972 to the form it was known to have had in 1854. A museum, adjacent to the laboratory, houses a unique collection of original apparatus arranged to illustrate the most important aspects of Faraday's immense contribution to the advancement of science in his fifty years at the Royal Institution.

Martin, in [4], gives this indication of Faraday's character:-

He was by any sense and by any standard a good man; and yet his goodness was not of the kind that make others uncomfortable in his presence. His strong personal sense of duty did not take the gaiety out of his life. ... his virtues were those of action, not of mere abstention ...

Georg Simon Ohm

Born: 16 March 1789 in Erlangen, Bavaria (now Germany)
Died: 6 July 1854 in Munich, Bavaria, Germany
Georg Simon Ohm came from a Protestant family. His father, Johann Wolfgang Ohm, was a locksmith while his mother, Maria Elizabeth Beck, was the daughter of a tailor. Although his parents had not been formally educated, Ohm's father was a rather remarkable man who had educated himself to a high level and was able to give his sons an excellent education through his own teachings. Had Ohm's brothers and sisters all survived he would have been one of a large family but, as was common in those times, several of the children died in their childhood. Of the seven children born to Johann and Maria Ohm only three survived, Georg Simon, his brother Martin who went on to become a well-known mathematician, and his sister Elizabeth Barbara.

When they were children, Georg Simon and Martin were taught by their father who brought them to a high standard in mathematics, physics, chemistry and philosophy. This was in stark contrast to their school education. Georg Simon entered Erlangen Gymnasium at the age of eleven but there he received little in the way of scientific training. In fact this formal part of his schooling was uninspired stressing learning by rote and interpreting texts. This contrasted strongly with the inspired instruction that both Georg Simon and Martin received from their father who brought them to a level in mathematics which led the professor at the University of Erlangen, Karl Christian von Langsdorf, to compare them to the Bernoulli family. It is worth stressing again the remarkable achievement of Johann Wolfgang Ohm, an entirely self-taught man, to have been able to give his sons such a fine mathematical and scientific education.

In 1805 Ohm entered the University of Erlangen but he became rather carried away with student life. Rather than concentrate on his studies he spent much time dancing, ice skating and playing billiards. Ohm's father, angry that his son was wasting the educational opportunity that he himself had never been fortunate enough to experience, demanded that Ohm leave the university after three semesters. Ohm went (or more accurately, was sent) to Switzerland where, in September 1806, he took up a post as a mathematics teacher in a school in Gottstadt bei Nydau.

Karl Christian von Langsdorf left the University of Erlangen in early 1809 to take up a post in the University of Heidelberg and Ohm would have liked to have gone with him to Heidelberg to restart his mathematical studies. Langsdorf, however, advised Ohm to continue with his studies of mathematics on his own, advising Ohm to read the works of Euler, Laplace and Lacroix. Rather reluctantly Ohm took his advice but he left his teaching post in Gottstadt bei Nydau in March 1809 to become a private tutor in Neuchâtel. For two years he carried out his duties as a tutor while he followed Langsdorf's advice and continued his private study of mathematics. Then in April 1811 he returned to the University of Erlangen.

His private studies had stood him in good stead for he received a doctorate from Erlangen on 25 October 1811 and immediately joined the staff as a mathematics lecturer. After three semesters Ohm gave up his university post. He could not see how he could attain a better status at Erlangen as prospects there were poor while he essentially lived in poverty in the lecturing post. The Bavarian government offered him a post as a teacher of mathematics and physics at a poor quality school in Bamberg and he took up the post there in January 1813.

This was not the successful career envisaged by Ohm and he decided that he would have to show that he was worth much more than a teacher in a poor school. He worked on writing an elementary book on the teaching of geometry while remaining desperately unhappy in his job. After Ohm had endured the school for three years it was closed down in February 1816. The Bavarian government then sent him to an overcrowded school in Bamberg to help out with the mathematics teaching.

On 11 September 1817 Ohm received an offer of the post of teacher of mathematics and physics at the Jesuit Gymnasium of Cologne. This was a better school than any that Ohm had taught in previously and it had a well equipped physics laboratory. As he had done for so much of his life, Ohm continued his private studies reading the texts of the leading French mathematicians Lagrange, Legendre, Laplace, Biot and Poisson. He moved on to reading the works of Fourier and Fresnel and he began his own experimental work in the school physics laboratory after he had learnt of Oersted's discovery of electromagnetism in 1820. At first his experiments were conducted for his own educational benefit as were the private studies he made of the works of the leading mathematicians.

The Jesuit Gymnasium of Cologne failed to continue to keep up the high standards that it had when Ohm began to work there so, by 1825, he decided that he would try again to attain the job he really wanted, namely a post in a university. Realising that the way into such a post would have to be through research publications, he changed his attitude towards the experimental work he was undertaking and began to systematically work towards the publication of his results [1]:-

Overburdened with students, finding little appreciation for his conscientious efforts, and realising that he would never marry, he turned to science both to prove himself to the world and to have something solid on which to base his petition for a position in a more stimulating environment.

In fact he had already convinced himself of the truth of what we call today "Ohm's law" namely the relationship that the current through most materials is directly proportional to the potential difference applied across the material. The result was not contained in Ohm's firsts paper published in 1825, however, for this paper examines the decrease in the electromagnetic force produced by a wire as the length of the wire increased. The paper deduced mathematical relationships based purely on the experimental evidence that Ohm had tabulated.

In two important papers in 1826, Ohm gave a mathematical description of conduction in circuits modelled on Fourier's study of heat conduction. These papers continue Ohm's deduction of results from experimental evidence and, particularly in the second, he was able to propose laws which went a long way to explaining results of others working on galvanic electricity. The second paper certainly is the first step in a comprehensive theory which Ohm was able to give in his famous book published in the following year.

What is now known as Ohm's law appears in this famous book Die galvanische Kette, mathematisch bearbeitet (1827) in which he gave his complete theory of electricity. The book begins with the mathematical background necessary for an understanding of the rest of the work. We should remark here that such a mathematical background was necessary for even the leading German physicists to understand the work, for the emphasis at this time was on a non-mathematical approach to physics. We should also remark that, despite Ohm's attempts in this introduction, he was not really successful in convincing the older German physicists that the mathematical approach was the right one. To some extent, as Caneva explains in [1], this was Ohm's own fault:-

... in neither the introduction nor the body of the work, which contained the more rigorous development of the theory, did Ohm bring decisively home either the underlying unity of the whole or the connections between fundamental assumptions and major deductions. For example, although his theory was conceived as a strict deductive system based on three fundamental laws, he nowhere indicated precisely which of their several mathematical and verbal expressions he wished to be taken as the canonical form.

It is interesting that Ohm's presents his theory as one of contiguous action, a theory which opposed the concept of action at a distance. Ohm believed that the communication of electricity occurred between "contiguous particles" which is the term Ohm himself uses. The paper [8] is concerned with this idea, and in particular with illustrating the differences in scientific approach between Ohm and that of Fourier and Navier. A detailed study of the conceptual framework used by Ohm in formulating Ohm's law is given in [6].

As we described above, Ohm was at the Jesuit Gymnasium of Cologne when he began his important publications in 1825. He was given a year off work in which to concentrate on his research beginning in August 1826 and although he only received the less than generous offer of half pay, he was able to spend the year in Berlin working on his publications. Ohm had believed that his publications would lead to his receiving an offer of a university post before having to return to Cologne but by the time he was due to begin teaching again in September 1827 he was still without such an offer.

Although Ohm's work strongly influenced theory, it was received with little enthusiasm. Ohm's feeling were hurt, he decided to remain in Berlin and, in March 1828, he formally resigned his position at Cologne. He took some temporary work teaching mathematics in schools in Berlin.

He accepted a position at Nüremberg in 1833 and although this gave him the title of professor, it was still not the university post for which he had strived all his life. His work was eventually recognised by the Royal Society with its award of the Copley Medal in 1841. He became a foreign member of the Royal Society in 1842. Other academies such as those in Berlin and Turin elected him a corresponding member, and in 1845 he became a full member of the Bavarian Academy.

This belated recognition was welcome but there remains the question of why someone who today is a household name for his important contribution struggled for so long to gain acknowledgement. This may have no simple explanation but rather be the result of a number of different contributary factors. One factor may have been the inwardness of Ohm's character while another was certainly his mathematical approach to topics which at that time were studied in his country a non-mathematical way. There was undoubtedly also personal disputes with the men in power which did Ohm no good at all. He certainly did not find favour with Johannes Schultz who was an influential figure in the ministry of education in Berlin, and with Georg Friedrich Pohl, a professor of physics in that city.

Electricity was not the only topic on which Ohm undertook research, and not the only topic in which he ended up in controversy. In 1843 he stated the fundamental principle of physiological acoustics, concerned with the way in which one hears combination tones. However the assumptions which he made in his mathematical derivation were not totally justified and this resulted in a bitter dispute with the physicist August Seebeck. He succeeded in discrediting Ohm's hypothesis and Ohm had to acknowledge his error. See [10] for details of the dispute between Ohm and Seebeck.

In 1849 Ohm took up a post in Munich as curator of the Bavarian Academy's physical cabinet and began to lecture at the University of Munich. Only in 1852, two years before his death, did Ohm achieve his lifelong ambition of being appointed to the chair of physics at the University of Munich.

André Marie Ampère

Born: 20 Jan 1775 in Lyon, France
Died: 10 June 1836 in Marseilles, France

André-Marie Ampère's father, Jean-Jacques Ampère, was a prosperous man who owned a home in Lyon and a country house in Poleymieux, which is only 10 km from Lyon. Up till André-Marie was seven years old the family spent most of the year in Lyon except the summer months which were spent at Poleymieux. However, in 1782, the home at Poleymieux became their main residence since André-Marie's father wished to spend more time on his son's education. Only a short time in winter was spent at Lyon where André-Marie's father saw to his business interests.

Despite not attending school, André-Marie was to be given an excellent education. He describes this education in autobiographical writings (rather strangely referring to himself in the third person):-

His father, who had never ceased to cultivate Latin and French literature, as well as several branches of science, raised him himself in the country near the city where he was born. He never required him to study anything, but he knew how to inspire in him a desire to know. Before being able to read, the young Ampère's greatest pleasure was to listen to passages from Buffon's natural history.

Ampère read articles from L'Encyclopédie many of which, Arago remarked many years later, he could recite in full in later life. Arago also claims that Ampère read the Encyclopédie starting at volume 1 and reading the articles in alphabetical order. Whether Ampère's later desire for classification in all subjects arose from this education, or whether he enjoyed Buffon and the Encyclopédie because of a natural liking for classifying, is hard to say.

It has been claimed that Ampère had mastered all known mathematics by the age of twelve years but this seems somewhat of an exaggeration since, by Ampère's own account, he did not start to read elementary mathematics books until he was 13 years old. However Ampère was always one to feel very confident in his own abilities and he certainly began to develop his own mathematical ideas very quickly and he began to write a treatise on conic sections. Ampère had no contacts with anyone with any depth of mathematical knowledge so it is not surprising that he felt that his ideas were original.

While still only 13 years old Ampère submitted his first paper to the Académie de Lyon. This work attempted to solve the problem of constructing a line of the same length as an arc of a circle. His method involves the use of infinitesimals but since Ampère had not studied the calculus the paper was not found worthy of publication. Shortly after writing the article Ampère began to read d'Alembert's article on the differential calculus in the Encyclopédie and realised that he must learn more mathematics.

After taking a few lessons in the differential and integral calculus from a monk in Lyon, Ampère began to study works by Euler and Bernoulli. He then acquired a copy of the 1788 edition of Lagrange's Mécanique analytique and began serious study of the work. Ampère writes (again writing about himself in the third person):-

... the reading of [Mécanique analytique] had animated him with a new ardour. He repeated all the calculations in it ...

However his life was soon to be shattered. The French Revolution began with the storming of the Bastille on 14 July 1789 but the effect on the Poleymieux region was not very great at first. Ampère's father kept out of trouble until late in 1791 when he accepted the position of Justice of the Peace in Lyon. This post made it virtually impossible for him to avoid trouble but the first tragedy to hit the family was in 1792 when André-Marie's sister died. The city of Lyon refused to carry out instructions from Paris and the city was besieged for two months. On the fall of the city Ampère's father was arrested for issuing an arrest warrant for the Jacobin Chevalier who had then been put to death. Ampère's father went to the guillotine with remarkable composure writing to Ampère's mother from his cell:-

I desire my death to be the seal of a general reconciliation between all our brothers; I pardon those who rejoice in it, those who provoked it, and those who ordered it....

The effect on Ampère of his father's death was devastating. He gave up his studies of Mécanique analytique and did not return to the study of mathematics for 18 months. He only returned to something like his old self when he met a girl, Julie, who he fell deeply in love with. Julie seemed less attracted to Ampère:-

He has no manners; he is awkward, shy and presents himself poorly.

Despite this coolness they were engaged to be married in 1797 and Ampère decided he better show that he could earn a living so began tutoring mathematics in Lyon. He married Julie in 1799 and their son Jean-Jacques was born in 1800. Ampère continued tutoring mathematics until 1802 when he was appointed professor of physics and chemistry at Bourg École Centrale. This was a difficult time for Ampère since Julie became ill before he made the move to Bourg leaving her at Poleymieux.

While Ampère was in Bourg he spent much time teaching physics and chemistry but his research was in mathematics. This research resulted in him composing a treatise on probability, The Mathematical Theory of Games, which he submitted to the Paris Academy in 1803. Laplace noticed an error, explaining the error to Ampère in a letter, which Ampère was able to correct and the treatise was reprinted. In fact the treatise was modified a number of times and Ampère was reluctant to call it completed for fear that further changes might be required. This work was followed by one on the calculus of variations in 1803.

After a year in Bourg, Ampère moved closer to Poleymieux being appointed to a mathematics position at the Lycée in Lyon on Delambre's recommendation. His time spent in Lyon had been made difficult due to the continuing decline in his wife's health. Mathematically he continued to produce good work, this time an interesting treatise on analytic geometry. Like a number of other mathematicians, Ampère seemed able to concentrate on his theorems despite the personal tragedy around him and, sadly, this would be required of him throughout his unhappy life. After his wife died in July 1803, Ampère was left with feelings of guilt for he had lived apart from his wife during much of their short marriage. He decided to leave Lyon for Paris. Hofman writes in [4] regarding his feelings following his wife's death:-

His subsequent depression contributed to his decision to take the earliest opportunity to leave Lyon for new surroundings in Paris. Later he would regret this decision. The Lyon friends who attempted to fill the emotional void left by Julie's death were missed painfully. Although Ampère gradually adjusted to the priority disputes and infighting of the Parisian scientific community, he always longed for a return to the intellectual life he experienced in Lyon.

By this time Ampère had a fair reputation as both a teacher of mathematics and as a research mathematician and on the strength of this reputation he was appointed répétiteur (basically a tutor) in analysis at the École Polytechnique in 1804. Without a formal education and formal qualifications his appointment is surprising but shows that his potential was recognised at this stage. His life, already containing many tragedies, did not improve and he embarked on a disastrous marriage. Lagrange and Delambre attended his wedding to Jenny on 1 August 1806 but, before the birth of their daughter on 6 July 1807, the couple were living apart and were not on speaking terms. They were legally separated in 1808 and Ampère was given custody of their daughter Albine.

Appointed professor of mathematics at the École Polytechnique in 1809 he held posts there until 1828. Ampère and Cauchy shared the teaching of analysis and mechanics and there was a great contrast between the two with Cauchy's rigorous analysis teaching leading to great mathematical progress but found extremely difficult by students who greatly preferred Ampère's more conventional approach to analysis and mechanics. Ampère was appointed to a chair at Université de France in 1826 which he held until his death.

In Paris Ampère worked on a wide variety of topics. Although a mathematics professor, his interests included, in addition to mathematics, metaphysics, physics and chemistry. In mathematics he worked on partial differential equations, producing a classification which he presented to the Institut in 1814. This seems to have been a crucial step in his election to the Institut National des Sciences in November 1814 when he defeated Cauchy, receiving 28 of the 56 votes cast.

Ampère was also making significant contributions to chemistry. In 1811 he suggested that an anhydrous acid prepared two years earlier was a compound of hydrogen with an unknown element, analogous to chlorine, for which he suggested the name fluorine. After concentrating on mathematics as he sought admission to the Institut, Ampère returned to chemistry after his election in 1814 and produced a classification of elements in 1816.

Ampère also worked on the theory of light, publishing on refraction of light in 1815. By 1816 he was a strong advocate of a wave theory of light, agreeing with Fresnel and opposed to Biot and Laplace who advocated a corpuscular theory. Fresnel became a good friend of Ampère's and lodged at Ampère's home from 1822 until his death in 1827.

In the early 1820s, Ampère attempted to give a combined theory of electricity and magnetism after hearing about experimental results by the Danish physicist Hans Christian Orsted. Ampère formulated a circuit force law and treated magnetism by postulating small closed circuits inside the magnetised substance.

It is worth commenting on how quickly Ampère produced this theory, the inspiration striking him immediately he heard of Orsted's experimental results. Orsted's work was reported the Academy in Paris on 4 September 1820 by Arago and a week later Arago repeated Orsted's experiment at an Academy meeting. Ampère demonstrated various magnetic / electrical effects to the Academy over the next weeks and he had discovered electrodynamical forces between linear wires before the end of September. He spoke on his law of addition of electrodynamical forces at the Academy on 6 November 1820 and on the symmetry principle in the following month. Ampère wrote up the work he had described to the Academy with remarkable speed and it was published in the Annales de Chimie et de Physique.

Ampère was assisted over the next few years in his work by Felix Savary whose help in getting Ampère to write up his results was invaluable [4]:-

... beginning with the memoir he completed early in 1823, Savary now made much more creative contributions. But more than his creativity, it was Savary's discipline and ability to concentrate at length on specific problems that proved especially valuable to Ampère. There is room to speculate that, without Savary's aid. Ampère might never have found time to complete the detailed calculations required to apply his force law to magnetic phenomena.

However Ampère was not the only one to react quickly to Arago's report of Orsted's experiment. Biot, with his assistant Savart, also quickly conducted experiments and reported to the Academy in October 1820. This led to the Biot-Savart Law. Another who worked on magnetism at this time was Poisson who insisted on treating magnetism without any reference to electricity. Poisson had already written two important memoirs on electricity and he published two on magnetism in 1826.

Ampère's most important publication on electricity and magnetism was also published in 1826. It is called Memoir on the Mathematical Theory of Electrodynamic Phenomena, Uniquely Deduced from Experience and contained a mathematical derivation of the electrodynamic force law and describes four experiments. Maxwell, writing about this Memoir in 1879, says:-

We can scarcely believe that Ampère really discovered the law of action by means of the experiments which he describes. We are led to suspect, what, indeed, he tells us himself, that he discovered the law by some process which he has not shown us, and that when he had afterwards built up a perfect demonstration he removed all traces of the scaffolding by which he had raised it.

Ampère's theory became fundamental for 19th century developments in electricity and magnetism. Faraday discovered electromagnetic induction in 1821 and, after initially believing that he had himself discovered the effect in 1822, Ampère agreed that full credit for the discovery should go to Faraday. Weber also developed Ampère's ideas as did Thomson and Maxwell.

In 1826 Ampère began to teach at the Collège de France. Here he was in a position to teach courses of his own design, rather than at the École Polytechnique were the topics were set down. Ampère therefore taught electrodynamics at the Collège de France and this course was taken by Liouville in 1826-27. This was the second time Ampère had taught Liouville since Liouville had taken Ampère's courses at the École Polytechnique in the previous session. Liouville made an important contribution to Ampère's electrodynamics course by editing a set of notes taken from Ampère's lectures.

Given the tragedy in Ampère's life it might have been hoped that his children would bring him some happiness. His son certainly achieved fame as a historian and philologist who studied the cultural origins of western European languages. He was appointed to a chair of history of foreign literature at the Sorbonne in 1830. However his relationship with his father was difficult. Hofmann in [4] writes:-

Both men were temperamental and subject to long periods of brooding followed by explosive outbursts of anger. Ampère's home simply was not expansive to house both of them for any extended period of time.

Ampère had an even more difficult time with his daughter. She married one of Napoleon's lieutenants in 1827 but he was an alcoholic and the marriage soon was in trouble. Ampère's daughter fled to her father's house in 1830 and, some days later, Ampère allowed her husband to live with him also. This proved a difficult situation, led to police intervention and much unhappiness for Ampère.

Article by: J J O'Connor and E F Robertson