c
(c) \(E = \frac{1}{{4\pi {\varepsilon _0}}}.\left[ {\frac{{5 \times {{10}^{ - 9}}}}{{{{(1 \times {{10}^{ - 2}})}^2}}} - \frac{{5 \times {{10}^{ - 9}}}}{{{{(2 \times {{10}^{ - 2}})}^2}}} + \frac{{5 \times {{10}^{ - 9}}}}{{{{(4 \times {{10}^{ - 2}})}^2}}}} \right.\) \(\left. { - \frac{{(5 \times {{10}^{ - 9}})}}{{{{(8 \times {{10}^{ - 2}})}^2}}} + .....} \right]\)
\( \Rightarrow E = \frac{{9 \times {{10}^9} \times 5 \times {{10}^{ - 9}}}}{{{{10}^{ - 4}}}}\left[ {1 - \frac{1}{{{{(2)}^2}}} + \frac{1}{{{{(4)}^2}}} - \frac{1}{{{{(8)}^2}}} + ...} \right]\)
\( \Rightarrow E = 45 \times {10^4}\left[ {1 + \frac{1}{{{{(4)}^2}}} + \frac{1}{{{{(16)}^2}}} + ...} \right]\)
\( - 45 \times {10^4}\left[ {\frac{1}{{{{(2)}^2}}} + \frac{1}{{{{(8)}^2}}} + \frac{1}{{{{(32)}^2}}} + ...} \right]\)
\( \Rightarrow E = 45 \times {10^4}\left[ {\frac{1}{{1 - \frac{1}{{16}}}}} \right] - \frac{{45 \times 10{\,^4}}}{{(2){\,^2}}}\left[ {1 + \frac{1}{{{4^2}}} + \frac{1}{{{{(16)}^2}}} + ..} \right]\)
\(E = 48 \times 10^4 -12 \times 10^4 = 36 \times 10^4 \,N/C\)