Question

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

• A. $\frac{q}{3 \pi \varepsilon_{0}}$
• B. $\frac{q}{6 \pi \varepsilon_{0}}$
• C. $\frac{q}{2 \pi \varepsilon_{0}}$
• D. $\frac{q}{4 \pi \varepsilon_{0}}$

Answer 1

Answer: (A)

$E=\frac{1}{4 \pi \varepsilon_{0}}\left[\frac{q}{1}+\frac{q}{2^{2}}+\frac{q}{4^{2}}+\frac{q}{8^{2}}-\right]$
$=\frac{1}{4 \pi \varepsilon_{0}} q\left[1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+--\right]$
$=\frac{1}{4 \pi \varepsilon_{0}} q \frac{1}{\left(1-\frac{1}{4}\right)}$
$=\frac{q}{3 \pi \varepsilon_{0}}$