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 infinite number of charges placed on the x-axis as shown in Fig. 20.40, is

$V=\frac{1}{4\pi {ϵ}_0}[\frac{q}{1}+\frac{q}{2}+\frac{q}{4}+\frac{q}{8}+....to\infty ]V=\frac{q}{4\pi {ϵ}_0}(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+....to\infty )=\frac{q}{4\pi {ϵ}_0}\left[\frac{1-(\frac{1}{2}{)}^{\infty }}{1-\frac{1}{2}}\right]$
$=\frac{q}{4\pi {ϵ}_0}\times \frac{(1-0)}{\left(\frac{1}{2}\right)}=\frac{q}{2\pi {ϵ}_0}$, which is choice (b).