Question

An infinite number of charges each of magnitude $q$ are placed on x - axis at distances of 1,2, 4, 8, ... 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)

$\frac{q}{3\pi {\epsilon }_{{}_0}}$

We know $E=\frac{kq}{r^2}$
Now electric field at $0=\frac{kq}{1^2}+\frac{kq}{2^2}+\frac{kq}{4^2}+\frac{kq}{8^2}+----$
$=kq(1+\frac{1}{2^2}+\frac{1}{2^4}+\frac{1}{2^6}+----)$
We know in G. P sum to infinite $=\frac{a}{1-r}$
$=kq\cdot \left(\frac{1}{1-\frac{1}{4}}\right)$
$=\frac{q}{4\pi {\epsilon }_0}\cdot \frac{4}{3}=\frac{q}{3\pi {\epsilon }_0.}$