Let us discuss the heat developed in a current carrying conductor and derive a relation for the same.
According to Joule’s law, the heat developed in a conducting wire is given by I2 R,
Where I is current flowing through the wire having resistance R.
If p is the electrical resistivity of wire,
L is the length of wire and
A is the area of cross section of wire
Then, Heat developed W=I2R
Or W=v2/R 1)
[because V=IR ( from OHM’s law)]
As V=El (2)
And R=pl/a (3)
Where V is the applied potential,
E is the electric field developed across the wire of length l and resistance R.
By putting equations (2) and (3) in equation(1),we get
W=σE2lA ( because σ=1/p)
Thus ,heat developed per unit volume (lA) per second is
W= σ E2
Or W=JE (because from point form of Ohm’s law J= σ E)
Where J is the current density.
If J is in ampere per m2and E is in volts per m then the units of W will be watts per m3.