An infinite number of point charges, each carrying $1 μ C$ charge, are placed along the y –axis at $y = 1 m, 2 m, 4 m, 8 m, .......$ . The total force on a 1 C point charge, placed at the origin, is $x \times 10^{3} N$ . The value of x, to the nearest integer, is ______
[ Take $\frac{1}{4 π ε_{0}} = 9 \times 10^{9} N m^{2} / C^{2}]$

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answer is 12.

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Detailed Solution

Let $q_{1} = 1 C$ and $q_{2} = 1 μ C$
Force on 1C charge due to infinite charges can be written as
$F = k q_{1} q_{2} [1 + \frac{1}{{(2)}^{2}} + \frac{1}{{(4)}^{2}} + \frac{1}{{(8)}^{2}} + ....]$

$S u m o f i n f i n i t e G P = \frac{a}{1 - r} h e r e a = f i r s t t e r m a n d r = c o m m o n r a t i o$

$\Rightarrow F = k q_{1} q_{2} [\frac{1}{1 - \frac{1}{4}}] = (k q_{1} q_{2}) \frac{4}{3}$

$\Rightarrow F = (9 \times 10^{9}) (1) (1 \times 10^{- 6}) \frac{4}{3} \Rightarrow F = 12 \times 10^{3} N$

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