Question

An infinite number of charges each equal to Q are placed along the X-axis at $X=4,X=8$ and so on. The potential at the point $X=0$ due to set of charges is :

• A. $\frac{Q}{8\pi {ϵ}_0}$
• B. $\frac{Q}{3\pi {ϵ}_0}$
• C. $\frac{1}{4\pi {ϵ}_0}\frac{2}{3}Q$
• D. $\frac{1}{4\pi {ϵ}_0}\frac{Q}{3}$

Answer 1

The correct option is A $\frac{Q}{8\pi {ϵ}_0}$

here, the required potential $V=\frac{1}{4\pi {ϵ}_0}[\frac{Q}{4}+\frac{Q}{8}+\frac{Q}{16}+....]$
or $V=\frac{Q}{16\pi {ϵ}_0}[1+\frac{1}{2}+\frac{1}{4}+....]$
or $V=\frac{Q}{16\pi {ϵ}_0}.\frac{1}{1-1/2}$ (this is GP sum, $S=\frac{a}{1-r},$ here first term $a=1$ and common ratio $r=1/2$)
$=\frac{Q}{16\pi {ϵ}_0}\times 2=\frac{Q}{8\pi {ϵ}_0}$