Question

An infinite number of charges, each equal to q, are placed along the x-axis at x =1, x=2, x=8, ... and so on. The potential at the point x =0 due to this set of charges is

• A. $\frac{q}{\pi {ϵ}_0}$
• B. $\frac{q}{2\pi {ϵ}_0}$
• C. $\frac{q}{3\pi {ϵ}_0}$
• D. $\frac{q}{4\pi {ϵ}_0}$

Answer 1

The correct option is B $\frac{q}{2\pi {ϵ}_0}$

The potential at x = 0 due to a set of the infinite number of charges placed on the x-axis as shown in Fig. 20.40, is

Electric potential is given as $\frac{1}{4\pi {\epsilon }_0}\frac{q}{r}$

In this case the electric potential will be given as

$V=\frac{1}{4\pi {\epsilon }_0}q[1+\frac{1}{2}+\frac{1}{4}.....]$

This can be written as

$V=\frac{1}{4\pi {\epsilon }_0}q\left[\frac{1}{1-\frac{1}{2}}\right]$ ( sum of infinite GP)

After solving this equation we get

$V=\frac{q}{2\pi {\epsilon }_0}$

Hence, the potential at the origin is $\frac{q}{2\pi {\epsilon }_0}$