Question

Two long thin charged rods with charge density $\lambda$ each are placed parallel to each other at a distance d apart. The force per unit length exerted on one rod by the other will beWhere $(K=\frac{1}{4\pi {\epsilon }_0})$:

• A. $\frac{K2\lambda }{d}$
• B. $\frac{K2{\lambda }^2}{d}$
• C. $\frac{K2\lambda }{d^2}$
• D. $\frac{K2{\lambda }^2}{d^2}$

Answer 1

The correct option is B $\frac{K2{\lambda }^2}{d}$

Given,

Charge density of each rod, $\lambda$ (charge per unit length)

$K=\frac{1}{4\pi {\epsilon }_o}$

Separation between rod, $d$

Electric field (at distance d) by a uniform charged rod, $E=\frac{1}{4\pi {\epsilon }_o}\frac{2\lambda }{d}=\frac{2K\lambda }{d}$

Total charge on second rod $\text{=}\text{charge}\text{density}\times \text{length}\text{=}\lambda L$

Force on charged rod placed in electric field, $F=\left(\lambda L\right)E=\frac{2KL{\lambda }^2}{d}$

Force per unit length due to rod is $\frac{F}{L}=\frac{2K{\lambda }^2}{d}$