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=(El)²/(pl/a)

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.