# Ohm's Law This circuit shows a current, I, running through a resistor, R. On the left side there is a voltage, V. Public Domain via Wikimedia Commons

Ohm's Law is a key rule for analyzing electrical circuits, describing the relationship between three key physical quantities: voltage, current, and resistance. It represents that the current is proportional to the voltage across two points, with the constant of proportionality being the resistance.

## Using Ohm's Law

The relationship defined by Ohm's law is generally expressed in three equivalent forms:

I = R
R = V / I
V = IR

with these variables defined across a conductor between two points in the following way:

• I represents the electrical current, in units of amperes.
• V represents the voltage measured across the conductor in volts, and
• R represents the resistance of the conductor in ohms.

One way to think of this conceptually is that as a current, I, flows across a resistor (or even across a non-perfect conductor, which has some resistance), R, then the current is losing energy. The energy before it crosses the conductor is therefore going to be higher than the energy after it crosses the conductor, and this difference in electrical is represented in the voltage difference, V, across the conductor.

The voltage difference and current between two points can be measured, which means that resistance itself is a derived quantity that cannot be directly measured experimentally. However, when we insert some element into a circuit that has a known resistance value, then you are able to use that resistance along with a measured voltage or current to identify the other unknown quantity.

## History of Ohm's Law

German physicist and mathematician Georg Simon Ohm (March 16, 1789 - July 6, 1854 C.E.) conducted research in electricity in 1826 and 1827, publishing the results that came to be known as Ohm's Law in 1827. He was able to measure the current with a galvanometer, and tried a couple of different set-ups to establish his voltage difference. The first was a voltaic pile, similar to the original batteries created in 1800 by Alessandro Volta.

In looking for a more stable voltage source, he later switched to thermocouples, which create a voltage difference based to a temperature difference. What he actually directly measured was that the current was proportional to the temperature difference between the two electrical junctures, but since the voltage difference was directly related to the temperature, this means that the current was proportional to the voltage difference.

In simple terms, if you doubled the temperature difference, you doubled the voltage and also doubled the current. (Assuming, of course, that your thermocouple doesn't melt or something. There are practical limits where this would break down.)

Ohm wasn't actually the first to have investigated this sort of relationship, despite publishing first. Previous work by British scientist Henry Cavendish (October 10, 1731 - February 24, 1810 C.E.) in the 1780's had resulted in him making comments in his journals that seemed to indicate the same relationship. Without this being published or otherwise communicated to other scientists of his day, Cavendish's results weren't known, leaving the opening for Ohm to make the discovery. That's why this article isn't entitled Cavendish's Law. These results were later published in 1879 by James Clerk Maxwell, but by that point the credit was already established for Ohm.

## Other Forms of Ohm's Law

Another way of representing Ohm's Law was developed by Gustav Kirchhoff (of Kirchoff's Laws fame), and takes the form of:

J = σE

where these variables stand for:

• J represents the current density (or electrical current per unit area of cross section) of the material. This is a vector quantity representing a value in a vector field, meaning it contains both a magnitude and a direction.
• sigma represents the conductivity of the material, which is dependent upon the physical properties of the individual material. The conductivity is the reciprocal of the resistivity of the material.
• E represents the electric field at that location. It is also a vector field.

The original formulation of Ohm's Law is basically an idealized model, which doesn't take into account the individual physical variations within the wires or the electric field moving through it. For most basic circuit applications, this simplification is perfectly fine, but when going into more detail, or working with more precise circuitry elements, it may be important to consider how the current relationship is different within different parts of the material, and that's where this more general version of the equation comes into play.

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