Question

An infinite number of charges each of magnitude $q$ are placed on $x$-axis at distances of $1,2,4,8,\dots$ meter from the origin. The intensity of the electric field at origin is

• A. $\frac{q}{3\pi {\epsilon }_0}$
• B. $\frac{q}{6\pi {\epsilon }_0}$
• C. $\frac{q}{2\pi {\epsilon }_0}$
• D. $\frac{q}{4\pi {\epsilon }_0}$

Answer 1

Answer: (A)

$E=\frac{1}{4\pi {\epsilon }_0}\left[\frac{q}{1}+\frac{q}{2^2}+\frac{q}{4^2}+\frac{q}{8^2}-\right]$
$=\frac{1}{4\pi {\epsilon }_0}q\left[1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+--\right]$
$=\frac{1}{4\pi {\epsilon }_0}q\frac{1}{\left(1-\frac{1}{4}\right)}$
$=\frac{q}{3\pi {\epsilon }_0}$