Q82 / 120 · VITEEE 2011 · Physics
An infinite number of charge, each of charge $1 \mu \mathrm{C}$ are placed on the $\mathrm{x}$-ax is with coord inates $\mathrm{x}=1,2$, $4,8, \ldots \ldots \infty$. If a charge of $1 \mathrm{C}$ is kept at the origin, then what is the net force acting on $1 \mathrm{C}$ charge?
- A$9000 \mathrm{~N}$
- Correct answerB$12000 \mathrm{~N}$
- C$24000 \mathrm{~N}$
- D$36000 \mathrm{~N}$
Solution
In figure there is the schematic diagram of distribution of charges on $x$-axis
Using principle of superposition total force acting on 1 C charge
$$
\begin{aligned}
F &=\frac{1}{4 \pi \varepsilon_{0}}\left[\frac{1 \times 1 \times 10^{-6}}{(1)^{2}}+\frac{1 \times 1 \times 10^{-6}}{(2)^{2}}\right.\\
&\left.+\frac{1 \times 1 \times 10^{-6}}{(4)^{2}}+\frac{1 \times 1 \times 10^{-6}}{(8)^{2}}+\ldots \infty\right] \\
=& \frac{10^{-6}}{4 \pi \varepsilon_{0}}\left[\frac{1}{1}+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\ldots \infty\right] \\
=& 9 \times 10^{-6} \times 10^{9}\left(\frac{1}{1-\frac{1}{4}}\right) \\
=& 9 \times 10^{9} \times 10^{-6} \times \frac{4}{3}=12000 \mathrm{~N}
\end{aligned}
$$
Hence, the correct answer is option: 'B'.