Question

Derive $$E=J\rho$$ or $$\overrightarrow{J}=\sigma \overrightarrow{E}$$ or derive the expression for the current density in terms of electric field and conductivity of the material using Ohm’s law.

Answer 1

Hint: Recall the expression for the current in terms of resistance and potential difference using Ohm’s law. Substitute the expression for the potential in terms of electric field and resistance of the material in terms of resistivity of the material in the Ohm’s law. The ratio of current flow and area of cross-section is the current density.

Formula used:
$$I={\displaystyle \frac{V}{R}}$$
Here, I is the current, V is the potential difference and R is the resistance of the material.
$$R=\rho {\displaystyle \frac{l}{A}}$$
Here, $$\rho$$ is the resistivity, l is the length of the material and A is the area of cross-section of the material.

Complete step by step answer:
We have to derive the expression for the current density in terms of electric field and conductivity using Ohm’s law. We have, from Ohm’s law,
$$I={\displaystyle \frac{V}{R}}$$ …… (1)
Here, I is the current, V is the potential difference and R is the resistance of the material.
We know that the potential difference between the two points is the product of electric field and the distance between the two point charges. Therefore, we can write,
$$V=El$$ …… (2)
Here, E is the electric field and l is the distance between the two points.

We also have the expression for the resistance of the material in terms of the resistivity of the material.
$$R=\rho {\displaystyle \frac{l}{A}}$$ …… (3)
Here, $$\rho$$ is the resistivity, l is the length of the material and A is the area of cross-section of the material.
Substituting equation (2) and equation (3) in equation (1), we get,
$$I={\displaystyle \frac{El}{\rho {\displaystyle \frac{l}{A}}}}$$
$$\Rightarrow I={\displaystyle \frac{EA}{\rho }}$$
$$\Rightarrow {\displaystyle \frac{I}{A}}={\displaystyle \frac{E}{\rho }}$$

We know the current density is the ratio of current to the area of cross-section. Therefore, the above equation becomes,
$$J={\displaystyle \frac{E}{\rho }}$$
$$\Rightarrow E=J\rho$$
We also know that the reciprocal of the resistivity is the conductivity of the material. Therefore, we can express the current density in terms of conductivity as follows,
$$\overrightarrow{J}=\sigma \overrightarrow{E}$$
Here, $$\sigma$$ is conductivity.
The above expression is the required equation for the current density.

Note:The potential at a point at a distance d from the point charge is inversely proportional to the distance and the electric field at that point is inversely proportional to the square of the distance. Therefore, the potential is equal to the product of electric field and distance d. Students can also derive the expression for the current density using the formula for current flowing through the material in terms of drift velocity of the electrons.