question_answer

An infinite number of charge, each of charge 1 \[\mu \]C are placed on the x-axis with coordinates \[x=1,\text{ }2,\text{ }4,\text{ }8,\,\,...\infty .\]If a charge of 1 C is kept at the origin, then what is the net force acting on 1C charge?

A)  9000 N        

B)  12000 N

C)  24000 N

D)  36000 N

Correct Answer: B

Solution :

The schematic diagram of distribution of charges on. x-axis is shown in figure Total force acting on 1 C charge is given by \[F=\frac{1}{4\pi {{\varepsilon }_{0}}}\left[ \frac{1\times 1\times {{10}^{-6}}}{{{(1)}^{2}}} \right.+\frac{1\times 1\times {{10}^{-6}}}{{{(2)}^{2}}}\] \[\left. +\frac{1\times 1\times {{10}^{-6}}}{{{(4)}^{2}}}+\frac{1\times 1\times {{10}^{-6}}}{{{(8)}^{2}}}+.....\infty  \right]\] \[=\frac{{{10}^{-6}}}{4\pi {{\varepsilon }_{0}}}\left[ \frac{1}{1}+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+....\infty  \right]\] \[=9\times {{10}^{-6}}\times {{10}^{9}}\left( \frac{1}{1-\frac{1}{4}} \right)\] \[=9\times {{10}^{9}}\times {{10}^{-6}}\times \frac{4}{3}\] \[=9\times \frac{4}{3}\times {{10}^{3}}=12000N\]