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$1\, \mu C$ વિદ્યુતભારોને $x-$ અક્ષ પર $x = 1, 2,4, 8, .... \infty$ મૂકવામાં આવે છે. તો ઉગમ બિંદુ પર રહેલ $1\, C$ વિદ્યુતભાર પર કેટલા .....$N$ બળ લાગે?
$9000$
$12000$
$24000$
$36000$
Solution
(b) The schematic diagram of distribution of charges on $x-$axis is shown in figure below :
Total force acting on $1 \,C$ charge is given by
$F = \frac{1}{{4\pi {\varepsilon _0}}}\left[ {\frac{{1 \times 1 \times {{10}^{ – 6}}}}{{{{(1)}^2}}} + \frac{{1 \times 1 \times {{10}^{ – 6}}}}{{{{(2)}^2}}}} \right.$
$\left. { + \frac{{1 \times 1 \times {{10}^{ – 6}}}}{{{{(4)}^2}}} + \frac{{1 \times 1 \times {{10}^{ – 6}}}}{{{{(8)}^2}}} + ….\infty } \right]$
$ = \frac{{{{10}^{ – 6}}}}{{4\pi {\varepsilon _0}}}\left( {\frac{1}{1} + \frac{1}{4} + \frac{1}{{16}} + \frac{1}{{64}} + …\infty } \right) = \,9 \times {10^9} \times {10^{ – 6}}\left( {\frac{1}{{1 – \frac{1}{4}}}} \right)$
$ = 9 \times {10^9} \times {10^{ – 6}} \times \frac{4}{3} = 9 \times {10^3} \times \frac{4}{3}$$= 12000\,N$
