Question

Two thin rods of linear charge density $\lambda$ Cm${}^{-1}$ are separated by a distance d metre. The force on unit length of each rod is :

• A. $\lambda /2\pi {\epsilon }_{0}d$
• B. ${\lambda }^2/2\pi {\epsilon }_{0}d$
• C. $\lambda /4\pi {\epsilon }_{0}d$
• D. ${\lambda }^2/4\pi {\epsilon }_{0}d$

Answer 1

The correct option is B ${\lambda }^2/2\pi {\epsilon }_{0}d$

We have linear charge density, $\lambda =\frac{Q}{l}$ where $Q$ is the charge and $l$ is the length.

$⟹Q=\lambda l$

Also, electric field, $E=\frac{\lambda }{2\pi {\epsilon }_{o}d}$, where $d$ is the separation between the rods.

The force $F=EQ$

$⟹F=\frac{\lambda Q}{2\pi {\epsilon }_{o}d}$

$⟹F=\frac{\lambda \left(\lambda l\right)}{2\pi {\epsilon }_{o}d}$

Since we are considering unit length $l=1$.

Hence $⟹F=\frac{{\lambda }^2}{2\pi {\epsilon }_{o}d}$