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Ohm's law
We all know the simplest version of Ohm's law:

(609) 
where is the voltage drop across a resistor of resistance when a current
flows through it. Let us generalize this law so that it is expressed in terms
of and , rather than and . Consider a length
of a conductor of uniform crosssectional area with a current
flowing down it. In general, we expect the electrical
resistance of the conductor to be proportional to its length, and inversely
proportional to its area (i.e., it is harder to push an electrical
current down a long
rather than a short wire, and it is easier to push a current down a wide rather
than a narrow conducting channel.) Thus, we can write

(610) 
The constant is called the resistivity, and is measured in
units of ohmmeters. Ohm's law becomes

(611) 
However, (supposing that the conductor is aligned along the axis)
and , so the above equation reduces to

(612) 
There is nothing special about the axis (in an isotropic conducting medium), so the
previous formula immediately generalizes to

(613) 
This is the vector form of Ohm's law.
A charge which moves through a voltage drop acquires an energy from the
electric field. In a resistor, this energy is dissipated as heat. This type of heating
is called ohmic heating. Suppose that charges per unit time pass through a
resistor. The current flowing is obviously . The total energy gained by the
charges, which appears as heat inside the resistor, is

(614) 
per unit time. Thus, the heating power is

(615) 
Equations (614) and (615) generalize to

(616) 
where is now the power dissipated per unit volume in a resistive medium.
Next: Conductors
Up: Electrostatics
Previous: Electrostatic energy
Richard Fitzpatrick
20060202