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 × 10⁴ N/C

24 × 10⁴ N/C

36 × 10⁴ N/C

48 × 10⁴ N/C

answer is C.

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Detailed Solution

$\begin{aligned} E = \frac{1}{4 π ϵ_{0}} [\frac{5 \times 10^{- 9}}{{(1 \times 10^{- 2})}^{2}} - \frac{5 \times 10^{- 9}}{{(2 \times 10^{- 2})}^{2}} + \frac{5 \times 10^{- 9}}{{(4 \times 10^{- 2})}^{2}} - \frac{(5 \times 10^{- 9})}{{(8 \times 10^{- 2})}^{2}} + \dots \dots] \\ \Rightarrow E = \frac{9 \times 10^{9} \times 5 \times 10^{- 9}}{10^{- 4}} [1 - \frac{1}{(2)^{2}} + \frac{1}{(4)^{2}} - \frac{1}{(8)^{2}} + \dots \dots] \\ \Rightarrow E = 45 \times 10^{4} [1 + \frac{1}{(4)^{2}} + \frac{1}{(16)^{2}} + \dots \dots] - 45 \times 10^{4} [1 + \frac{1}{(2)^{2}} + \frac{1}{(8)^{2}} + \frac{1}{(32)^{2}} \dots \dots] \\ \Rightarrow E = 48 \times 10^{4} - 12 \times 10^{4} = 36 \times 10^{4} N / C \end{aligned}$