Question

What is the relation between current density and electric field?

Answer 1

Given information:

Electric field (E) and current density (J).

To find:

We have to find the relation between current density and electric field.

Calculations: We know that,

$\mathrm{V}=\mathrm{IR}$

and,

$\mathrm{R}=\mathrm{\rho }\frac{\mathrm{l}}{\mathrm{A}}$

Here,

R is the resistance.

I is the current.

V is the voltage.

$\mathrm{\rho }$ is the resistivity of the material.

L is the length of the conductor.

A is the cross-sectional area.

$$\begin{gathered} \therefore \mathrm{I}=\frac{\mathrm{V}}{\mathrm{R}} \\ =\frac{\mathrm{V}}{\mathrm{\rho }{\displaystyle \frac{\mathrm{l}}{\mathrm{A}}}} \\ =\frac{\mathrm{VA}}{\mathrm{\rho l}} \end{gathered}$$

Now,

$$\begin{gathered} \Rightarrow \frac{\mathrm{I}}{\mathrm{A}}=\frac{1}{\mathrm{\rho }}\frac{\mathrm{V}}{\mathrm{l}} \\ \Rightarrow \mathrm{J}\propto \mathrm{E} \end{gathered}$$

Therefore, the relation between current density and an electric field is $\mathrm{J}\propto \mathrm{E}$.