Ohm's law
Ohm's law (named after its discoverer Georg Ohm [1]) states that the ratio of the potential difference (or voltage drop) between the ends of a conductor (and resistor) to the current flowing through it is a constant, provided the temperature doesn't change:
- <math>
{V \over I} = R
<math>
where V is the voltage and I is the current; the equation yields the proportionality constant R, which is the electrical resistance of the device.
The law is strictly true only for resistors whose resistance does not depend on the applied voltage, which are called ohmic or ideal resistors or ohmic devices. Fortunately, the conditions where Ohm's law holds are very common (Ohm's law is never completely accurate [if R is assumed to be constant] for "real world" devices because no real device is an ohmic device).
The relation V / I = R even holds for non-ohmic devices, but then the resistance R depends on V and is no longer a constant. To check whether a given device is ohmic or not, one plots V versus I and compares the graph to a straight line through the origin.
The Ohm's law equation is often stated as
- <math>V = I \cdot R<math>
in part because that is the variation very commonly used with resistors.
Physicists often use the so-called microscopic form of Ohm's Law:
- <math>
\mathbf{j} = \sigma \cdot \mathbf{E}
<math>
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. This is the form Ohm originally stated. The common form V = I·R used in circuit design is the macroscopic, averaged-out version.
It is important to note that Ohm's law is not an actual mathematically derived law, but one that is supported very well by empirical evidence. There are times when Ohm's law does break down, however, because it is really a simplification. The primary causes of resistance to electrical flow in a metal include imperfections, impurities, and the fact that electrons bounce off the atoms themselves. When the temperature of the metal increases, that third factor increases, so that when a substance heats up because of the electricity flowing through it, as in the filament in a light bulb, the resistance actually increases. The resistance of a device is given by:
- <math>
R = \frac{L}{A} \cdot \rho = \frac{L}{A} \cdot \rho_0 (\alpha (T - T_0) + 1)
<math>
where ρ is the resistivity, L is the length of the conductor, A is its cross-sectional area, T is its temperature, <math>T_0<math> is a reference temperature (usually room temperature), and <math>\rho_0<math> and <math>\alpha<math> are constants specific to the material of the conductor.
The equation for the propagation of electricity formed on Ohm's principles is identical with that of Jean-Baptiste-Joseph Fourier for the propagation of heat; and if, in Fourier's solution of any problem of heat-conduction, we change the word temperature to electric potential and write electric current instead of flux of heat, we have the solution of a corresponding problem of electric conduction. The basis of Fourier's work was his clear conception and definition of conductivity. But this involves an assumption, undoubtedly true for small temperature-gradients, but still an assumption, viz, that, all else being the same, the flux of heat is strictly proportional to the gradient of temperature. An exactly similar assumption is made in the statement of Ohm's law, i.e. that, other things being alike, the strength of the current is at each point proportional to the gradient of electric potential. It happens, however, that with our modern methods it is much more easy to test the accuracy of the assumption in the case of electricity than in that of heat.
See also: Poiseuille's law
References
[1] Die galvanische Kette, mathematisch bearbeitet (Mathematical work on the electrical circuit, 1827)
