An infinite number of electric charges each equal to 5 nano-coulombs (magnitude) are placed along X-axis at x = 1 cm, x = 2 cm, x = 4 cm, x = 8 cm ....... and so on. In this setup if the consecutive charges have opposite sign, then the electric field at x = 0
$(\frac{1}{4 {πε}_{0}} = 9 \times 10^{9} N - m^{2} / c^{2})$

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$12 \times 10^{4} N / C$

$48 \times 10^{4} N / C$

$24 \times 10^{4} N / C$

$36 \times 10^{4} N / C$

answer is C.

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Detailed Solution

$\begin{matrix} E = \frac{1}{4 π ε_{0}} [\frac{5 \times 10^{- 9}}{{(1 \times 10^{- 2})}^{2}} - \frac{5 \times 10^{- 9}}{{(2 \times 10^{- 2})}^{2}} + \frac{5 \times 10^{- 9}}{(4 \times 10^{- 2})} + \dots . .] \\ = \frac{5 \times 10^{- 9}}{4 π ε_{0} \times 10^{- 4}} [1 - \frac{1}{4} + \frac{1}{16} - \frac{1}{64} + \dots \dots] \\ = 9 \times 10^{9} \times 5 \times 10^{- 9} \times 10^{4} (\frac{1}{1 + \frac{1}{4}}) \\ = 36 \times 10^{4} N / C \end{matrix}$