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Q.

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

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a

14

b

24

c

12

d

5

answer is C.

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Detailed Solution

Let q_1=1C and q_2=1μC
Force on 1C charge due to infinite charges can be written as
$F = k q_{1} q_{2} [1 + \frac{1}{(2)^{2}} + \frac{1}{(4)^{2}} + \frac{1}{(8)^{2}} + \dots .]$

Sum of infinite $G P = \frac{a}{1 - r}$

here a= first term and r= common ratio

$\begin{array}{l} \Rightarrow F = k q_{1} q_{2} [\frac{1}{1 - \frac{1}{4}}] = (k q_{1} q_{2}) \frac{4}{3} \\ \Rightarrow F = (9 \times 10^{9}) (1) (1 \times 10^{- 6}) \frac{4}{3} \\ \Rightarrow F = 12 \times 10^{3} N \end{array}$

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