b
(b) 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
\(F = \frac{1}{{4\pi {\varepsilon _0}}}\left[ {\frac{{1 \times 1 \times {{10}^{ - 6}}}}{{{{(1)}^2}}} + \frac{{1 \times 1 \times {{10}^{ - 6}}}}{{{{(2)}^2}}}} \right.\)
\(\left. { + \frac{{1 \times 1 \times {{10}^{ - 6}}}}{{{{(4)}^2}}} + \frac{{1 \times 1 \times {{10}^{ - 6}}}}{{{{(8)}^2}}} + ....\infty } \right]\)
\( = \frac{{{{10}^{ - 6}}}}{{4\pi {\varepsilon _0}}}\left( {\frac{1}{1} + \frac{1}{4} + \frac{1}{{16}} + \frac{1}{{64}} + ...\infty } \right) = \,9 \times {10^9} \times {10^{ - 6}}\left( {\frac{1}{{1 - \frac{1}{4}}}} \right)\)
\( = 9 \times {10^9} \times {10^{ - 6}} \times \frac{4}{3} = 9 \times {10^3} \times \frac{4}{3}\)\(= 12000\,N\)