Ohm's law

Ohm's law (named after Georg Ohm, who discovered it [1]) states that the voltage drop <math>V</math> across a resistor is proportional to the current <math>I</math> running through it:

<math>V = IR</math>

where the proportionality constant <math>R</math> is the electrical resistance of the device. The law is strictly true only for resistors whose resistance does not depend on the applied voltage; these resistors are called ohmic or ideal resistors. Fortunately, the conditions where Ohm's Law holds are very common.

The relation <math>V = IR</math> even holds for non-ohmic devices, but then the resistance <math>R</math> depends on <math>V</math> and is not a constant anymore. To check whether a given device is ohmic or not, one plots <math>V</math> versus <math>I</math> and compares the graph to a straight line through the origin.

The funny thing about Ohm's law is that it 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 an oversimplification. 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 is heating up because of the electricity flowing through it, like the filament in a light bulb, the resistance actually increases. The resistance of a device is given by:

<math>R = \frac{L}{A} \rho = \frac{L}{A} \rho_0 (\alpha (T - T_0) + 1)</math>

where <math>\rho</math> is the resistivity, <math>L</math> is the length of the conductor, <math>A</math> is its cross-sectional area, <math>T</math> is its temperature, <math>T_0</math> is a reference temperature (usually room temperature, 298 K), and <math>\rho_0</math> and <math>\alpha</math> are constants specific to the material of the conductor.

Originally, Ohm formulated his law in the form

<math>\mathbf{j} = \sigma \mathbf{E}</math>

where <math>\mathbf{j}</math> is current density, <math>\sigma</math> is conductivity and E is electric field strength, but it is the simplified form "<math>V = IR</math>" that is most commonly referred to as Ohm's law by electrical engineers.

Relation to heat conduction

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.

Relation to hydrodynamic laminar stationary flow

See: Poiseuille's law

References [1] Die galvanische Kette, mathematisch bearbeitet (Mathematical work on the electrical circuit, 1827)