Question

An infinite number of equal charges (q each) are placed at distance x $=$ $1,2,4,8,$ ..... $\infty$. Then the electric potential at the origin (x $=$ $0$) will be :

• A. zero
• B. infinite
• C. $\frac{q}{4\pi {\epsilon }_0}$
• D. $\frac{2q}{4\pi {\epsilon }_0}$

Answer 1

The correct option is D $\frac{2q}{4\pi {\epsilon }_0}$

The electric potential at origin is $V=\frac{q}{4\pi {ϵ}_0}[1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...]$

This is a GP series and Sum , $S_{\infty }=\frac{a}{1-r}$, where $a=$ first term and $r=$ common ratio.

here $a=1,r=1/2$

Thus, $V=\frac{q}{4\pi {ϵ}_0}\frac{1}{(1-1/2)}=\frac{2q}{4\pi {ϵ}_0}$