Derive an expression for equivalent form of Ohm's law.

Question

Derive the vector form of ohm's law \(\vec j = \sigma \vec E\)

Or Derive an expression for equivalent form of Ohm's law.

Answer 1

Let I be the current in the conductor due to potential difference V across the conductor , then according to ohm's law

V = IR .............(i)

The electric field E produced in the conductor is given by 

\(E = \frac{V}{L}\)

\(\therefore\) V = EL ............(ii)

From (i) and (ii) we get

EL = IR

EL = \(I \times \rho\frac{L}{A}\)                 \((R = \rho\frac{L}{A})\)

EL = \(\rho JL\)                        \((\frac{I}{A} = J)\)

E = \(\rho J\)

J = \(\frac{E}{\rho}\)                              \((\frac{1}{\rho} = \sigma)\)

\(\vec J = \sigma \vec E\) \(\rightarrow\) This is the vector form of ohm's law