Question

An infinite number of charges each equal to ${}^{\prime }{q}^{\prime }$ are placed along the X-axis at $x=1$, $x=2$, $x=4$, $x=8$ The potential at the point $x=0$ due to this set of charges is :

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

Answer 1

Answer: (B)

$\frac{2Q}{4\pi {ϵ}_0}$

We know V due to charge Q $=\frac{kQ}{r.}$
V is a scalar quantity
$⟹$ total potential at origin is just sum of potential due to all charges

$\therefore V=\frac{kq}{1}+\frac{kq}{2}+\frac{kq}{4}+.......$

$=kq\left(\frac{1}{1-\frac{1}{2}}\right)$ (sum of infinite $G.P=\frac{a}{1-r})$

$\therefore V=\frac{2Q}{4\pi {\epsilon }_0}$