The current I is related to the surface integral of the current density J by
where use has been made of the relation
with
the conductivity and E the electric field.If is constant, then
Now, for a wire, E and
are constant and perpendicular, so
where A is the cross sectional area, V is the voltage, and L is the length of the wire. Plugging in gives
Defining the resistance
and solving for V gives Ohm's law,
(1)
(2)
where use has been made of the relation
(3)
with
the conductivity and E the electric field.If is constant, then (4)
Now, for a wire, E and
are constant and perpendicular, so(5)
where A is the cross sectional area, V is the voltage, and L is the length of the wire. Plugging in gives
(6)
Defining the resistance
(7)
and solving for V gives Ohm's law,
(8)
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