question_answer

An infinite number of charges, each of charge 1 mC, are placed on the x-axis with co-ordinates x = 1, 2, 4, 8, ....¥. If a charge of 1 C is kept at the origin, then what is the net force acting on 1 C charge                                              [DCE 2004]

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 below: Total force acting on 1 C charge is given by \[F=\frac{1}{4\pi {{\varepsilon }_{0}}}\left[ \frac{1\times 1\times {{10}^{-6}}}{{{(1)}^{2}}}+\frac{1\times 1\times {{10}^{-6}}}{{{(2)}^{2}}} \right.\]\[\left. +\frac{1\times 1\times {{10}^{-6}}}{{{(4)}^{2}}}+\frac{1\times 1\times {{10}^{-6}}}{{{(8)}^{2}}}+....\infty  \right]\] \[=\frac{{{10}^{-6}}}{4\pi {{\varepsilon }_{0}}}\left( \frac{1}{1}+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+...\infty  \right)=\,9\times {{10}^{9}}\times {{10}^{-6}}\left( \frac{1}{1-\frac{1}{4}} \right)\] \[=9\times {{10}^{9}}\times {{10}^{-6}}\times \frac{4}{3}=9\times {{10}^{3}}\times \frac{4}{3}\]= 12000 N