Ohm's law

A voltage source, V, drives an electric current I through resistor, R, the three quantities obeying Ohm's law: V = IR

Ohm's law, named after its discoverer Georg Ohm ^[1], states that the potential difference between two points along a connected path (i.e. a electrical network), or equivalently the voltage drop from one point to a second point within the connected path, usually designated by V (or sometimes E or U), on a metallic conductor and the current I flowing through it are proportional at a given (i.e. fixed) temperature:

$V=I\cdot R$

where R is a constant called the electrical resistance of the conductor.

The unit of resistance is the ohm, which is equal to one volt per ampere, or one volt-second per coulomb.

The inverse of resistance, 1/R, is conductance, and its SI unit is the siemens (also unofficially call the mho). Conductance, or its alternating current (and frequency-dependent) analog admittance, is widely used in certain types of electrical and electronic analysis.

An elementary explanation of electrical circuits, the above diagram, and how Ohm's Law is used

Electrical circuits consist of electrical devices connected by wires (or other suitable conductors) (See the article electrical circuits for some basic combinations). The above diagram is about as simple an electrical circuit as can be constructed. One electrical device in the electrical circuit is shown as a circle with + and - terminals. This symbol represents a voltage source such as a battery. The other electrical device in the electrical circuit is illustrated by a zig-zag symbol and has an R beside it. This symbol represents a resistor, and the value of its resistance is designated by the R. The + or positive terminal of the voltage source is connected to one of the terminals of the resistor using a wire of negligible resistance, and through this wire a current I is shown to be passing, and in a specified direction illustrated by the arrow. The other terminal of the resistor is connected to the - or negative terminal of the voltage source by a second wire. A complete circuit is formed by this configuration because all the current that leaves one terminal of the voltage source must return to the other terminal of the voltage source. (While not shown, because electrical engineers know that it would be redundant, there is an implied current I, and an arrow pointing to the left, associated with the second wire.)

Voltage is the term used to describe the force that moves charge (electrons) through wires and electrical devices, current is the rate of flow of charge, and resistance is the property of a resistor that limits current to the amount that must flow under the applied voltage. So, for a voltage source producing a voltage V, and a resistor of resistance R, Ohm's law provides the equation (I=V/R) for computing the current I.

The term 'conductor' is used in the statement of Ohm's law to indicate that the circuit element across which a voltage is to be measured conducts electricity. Resistors are conductors that limit the passage of electricity to some degree. A resistor with a high value of resistance, say above 10 megaohms, is a poor conductor, while a resistor with a low value of resistance, say below 0.1 ohm, is a good conductor. (Insulators are electrical devices that, for most practical purposes, do not conduct electricity.) The only circuit element in the above diagram that may be identified as the 'conductor' of Ohm's law is the resistor, and the two points across which a voltage is measured are the resistor's leads or terminal interconnection conductors. These interconnection points are not explicitly shown in circuit diagrams such as the one above because including all interconnection points in a circuit diagram, in order to illustrate more completely the interconnection features of an actual circuit, would introduce unnecessary detail to the diagram.

Overview

The law as published by Ohm applied specifically to his experiments with conduction in metallic wires. Later, when electronic circuits were created that required a wider range of resistances in a compact form, resistors were manufactured from nonmetals that obeyed Ohm's Law. Metallic and nonmetallic resistors are called ohmic devices, because they obey Ohm's Law, at least within certain limits of voltage and current.

Outside these limits, the resistance of an ohmic device varies with the voltage and current. Finally, at extremely high voltages, the device may suffer from electric breakdown or arcing, causing a short circuit; or, at high currents, the device may overheat and melt, causing an open circuit. The resistance of most devices also varies with their temperature, and more specialised devices have resistances that vary with magnetic field intensity, light intensity or many other stimuli.

The relation $V=IR$ can also be applied to non-ohmic devices, but it then ceases to represent Ohm's Law. In non-ohmic cases, R depends on V and is no longer a constant of proportionality but a variable called differential resistance. To check whether a given device is ohmic or not, one plots V versus I and checks that the curve is a straight line.

Physics

Physicists often use the continuum form of Ohm's Law:

$\mathbf {J} =\sigma \cdot \mathbf {E}$

where J is the current density (current per unit area), σ is the conductivity (which can be a tensor in anisotropic materials) and E is the electric field.

The common form $V=I\cdot R$ used in circuit design is the macroscopic, averaged-out version.

The equation above is only valid in the reference frame of the conducting material. If the material is moving at velocity v relative to a magnetic field B, a term must be added as follows

$\mathbf {J} =\sigma \cdot \left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)$

The analogy to the Lorentz force is obvious, and in fact Ohm's law can be derived from the Lorentz force and the assumption that there is a drag on the charge carriers proportional to their velocity.

A perfect metal lattice would have no resistivity, 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.

Electrical and electronic engineering

Most electrical engineers use Ohm's Law every working day. One can not be a functioning electrical engineer without understanding this law intimately, and when and how to apply it. Virtually all electronic circuits have resistive elements which are much more often than not considered ideal ohmic devices, i.e. they obey Ohm's Law. From the engineer's point of view, resistors (devices that "resist" the flow of electrical current) develop a voltage across their terminal conductors (e.g. the two wires emerging from the device) proportional to the amount of current flowing through the device.

More specifically, the voltage measured across a resistor at a given instant is strictly proportional to the current passing through the resistor at that instant. When a functioning electrical circuit drives a current I, measured in amperes, through a resistor of resistance R, the voltage that develops across the resistor is I R, the value of R serving as the proportionality factor. Thus resistors act like current to voltage converters (just as springs act like displacement to force converters).

Similarly, resistors act like voltage to current converters when a desired voltage is established across the resistor because a current I equal to 1/R times V must be flowing through the resistor. That current must have been supplied by a circuit element functioning as a current source and it must be passed on to a circuit element that serves as a current sink.

The DC resistance of a resistor is always a positive quantity, and the current flowing through a resistor generates (waste) heat in the resistor as it does in one of Ohm's wires. Voltages can be either positive or negative, and are always measured with respect to a reference point. When we say that a point in a circuit has a certain voltage, it is understood that this voltage is really a voltage difference (a two terminal measurement) and that there is an understood, or explicitly stated, reference point, often called ground or common. Currents can be either positive or negative, the sign of the current indicating the direction of current flow. Current flow in a wire consists of the slow drift of electrons due to the influence of a voltage established between two points on the wire.

Hydraulic analogy

While the terms voltage, current and resistance are fairly intuitive terms, beginning students of electrical engineering might find the analog terms for water flow helpful. Water pressure, typically measured in pounds per square inch, is the analog of voltage because establishing a water pressure difference between two points along a (horizontal) pipe causes water to flow. Water flow rate, as in gallons of water per minute, is the analog of current, as in coulombs per second. Finally, flow restrictors such as apertures placed in pipes between points where the water pressure is measured are the analog of resistors. We say that the rate of water flow through an aperture restrictor is proportional to the difference in water pressure across the restrictor. Similarly, the rate of flow of electrical charge, i.e. the electrical current, passing through an electrical resistor is proportional to the difference in voltage measured across the resistor.

Sheet resistance

Thin metal films, ususally deposited on insulating substrates, are used for various purposes, the electrical current traveling parallel to the plane of the film. When describing the electrical resistivity of such devices, the term ohms-per-square is used. See sheet resistance.

Temperature effects

When the temperature of the conductor increases, the collisions between electrons and atoms increase. Thus as a substance heats up because of electricity flowing through it (or by any heating process), the resistance will usually increase. The exception is semiconductors. The resistance of an Ohmic substance depends on temperature in the following way:

$R={\frac {L}{A}}\cdot \rho ={\frac {L}{A}}\cdot \rho _{0}(\alpha (T-T_{0})+1)$

where ρ is the resistivity, L is the length of the conductor, A is its cross-sectional area, T is its temperature, $T_{0}$ is a reference temperature (usually room temperature), and $\rho _{0}$ and $\alpha$ are constants specific to the material of interest. In the above expression, we have assumed that L and A remain unchanged within the temperature range.

It is worth mentioning that temperature dependence does not make a substance non-ohmic, because at a given temperature R does not vary with voltage or current ($V/I=\mathrm {constant}$).

Intrinsic semiconductors exhibit the opposite temperature behavior, becoming better conductors as the temperature increases. This occurs because the electrons are bumped to the conduction energy band by the thermal energy, where they can flow freely and in doing so they leave behind holes in the valence band which can also flow freely.

Extrinsic semiconductors have much more complex temperature behaviour. First the electrons (or holes) leave the donors (or acceptors) giving a decreasing resistance. Then there is a fairly flat phase in which the semiconductor is normally operated where almost all of the donors (or acceptors) have lost their electrons (or holes) but the number of electrons and the number of electrons that have jumped right over the energy gap is negligible compared to the number of electrons (or holes) from the donors (or acceptors). Finally as the temperature increases further the carriers that jump the energy gap becomes the dominant figure and the material starts behaving like an intrinsic semiconductor.

Strain (mechanical) effects

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 strain), which means to mechanically stretch 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 (the other form of strain), 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.

AC circuits

For an AC circuit Ohm's law can be written $\mathbf {V} =\mathbf {I} \cdot \mathbf {Z}$, where V and I are the oscillating phasor voltage and current respectively and Z is the complex impedance for the frequency of oscillation.

In a transmission line, the phasor form of Ohm's law above breaks down because of reflections. In a lossless transmission line, the ratio of voltage and current follows the complicated expression

$Z(d)=Z_{0}{\frac {Z_{L}+jZ_{0}\tan(\beta d)}{Z_{0}+jZ_{L}\tan(\beta d)}}$,

where d is the distance from the load impedance $Z_{L}$ measured in wavelengths, β is the wavenumber of the line, and $Z_{0}$ is the characteristic impedance of the line.

Relation to heat conduction

Ohm's principle predicts the flow of electrical charge (i.e. current) in electrical conductors when subjected to the influence of voltage differences; Jean-Baptiste-Joseph Fourier's principle predicts the flow of heat in heat conductors when subjected to the influence of temperature differences. The same equation describes both phenomena, the equation's variables taking on different meanings in the two cases. Specifically, solving a heat conduction (Fourier) problem with temperature (the driving "force") and flux of heat (the rate of flow of the driven "quantity", i.e. heat energy) variables also solves an analogous electrical conduction (Ohm) problem having electric potential (the driving "force") and electric current (the rate of flow of the driven "quantity", i.e. charge) variables. The basis of Fourier's work was his clear conception and definition of thermal conductivity. He assumed that, all else being the same, the flux of heat is strictly proportional to the gradient of temperature. Although undoubtedly true for small temperature gradients, strictly proportional behavour will be lost when real materials (e.g. ones having a thermal conductivity that is a function of temperature) are subjected to large temperature gradients. A similar assumption is made in the statement of Ohm's law: other things being alike, the strength of the current at each point is proportional to the gradient of electric potential. The accuracy of the assumption that flow is proportional to the gradient is more readily tested, using modern measurement methods, for the electrical case than for the heat case.

History ^[2]

Prior to Ohm's work, a qualitative relationship between voltage and current was worked out by Henry Cavendish. 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). Cavendish's results were unknown until Maxwell published them in 1879.

Ohm did his work on resistance in the years 1825 and 1826, and published his results in 1827. 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

$\mathbf {X} ={\frac {\mathbf {a} }{\mathbf {b} +\mathbf {l} }}$,

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 eponymous law and published his results in [1].

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" and proclaimed that Ohm was "a professor who preached such heresies was unworthy to teach science." 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.

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

Benjamin Franklin chose the sign convention used today for assigning the sign of a measured voltage, such as when measuring the voltage across a battery. He did not know what was actually "flowing."

Unstated in the above definition of Ohm's Law is the necessity of making measurements of voltage and current that are averaged over a sufficiently long period of time in order to compute accurately a resistor's actual resistance. This constraint would not have been understood by Ohm in the 1820s because instruments for making current and voltage measurements over short periods of time were not available. A hundred years later, the effect now known as Johnson noise, or thermal noise, was discovered. This thermal effect implies that measurements of current and voltage that are taken over sufficiently short periods of time will yield, usually, ratios of V/I that are different from the time averaged value of R, although the ensemble average of the ratios computed from a large number of statistically independent current and voltage measurements will yield the time-averaged value of R.

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.

In a sense, Ohm's law is not a law at all. Rather, Ohm's law may be more properly seen as the definition of resistance. (In the hierarchy of 'physical' or 'logical' terms, definitions are more fundamental than laws or theories.) Joules, coulombs and time are basic units of measure. A volt can be defined based on those units of measure. Ohm's "law" defines the ohm. An ohm, the unit of resistance, is derived from those more basic units. In essence, Ohm discovered that V/I is a constant for a given conductor at a given temperature, and over a large range of V and I.

See also

• Poiseuille's law
• Scientific laws named after people

References

[1] Ohm, G., Die galvanische Kette, mathematisch bearbeitet, 1827 (Mathematical work on the electrical circuit) [Facsimile (PDF)]

[2] Sanford P. Bordeau. Volts to Hertz...the Rise of Electricity. Burgess Publishing Company, Minneapolis, MN. pp.86-107.

External links

• Calculation of Ohm's law · The Magic Triangle
• Calculation of electric power, voltage, current and resistance