An infinite number of point charges,each carr | vedclass

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Question

(C) The force between two point charges is given by Coulomb's Law: $F = \frac{1}{4 \pi \epsilon_{0}} \frac{q_{1} q_{2}}{r^{2}}$.Here,$q_{1} = 1 \,C$ a

Vedclass · Accepted Answer

$12$

Vedclass · Answer

(C) The force between two point charges is given by Coulomb's Law: $F = \frac{1}{4 \pi \epsilon_{0}} \frac{q_{1} q_{2}}{r^{2}}$.Here,$q_{1} = 1 \,C$ at the origin,and $q_{2} = 1 \,\mu C = 10^{-6} \,C$ at various positions $y$.The total force $F$ is the sum of forces from all charges:$F = \sum \frac{k q_{1} q_{2}}{y^{2}} = k q_{1} q_{2} \left( \frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{4^{2}} + \frac{1}{8^{2}} + \ldots ight)$$F = (9 	imes 10^{9}) 	imes (1) 	imes (10^{-6}) 	imes \left( 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \ldots ight)$This is an infinite geometric series with first term $a = 1$ and common ratio $r = \frac{1}{4}$.The sum $S = \frac{a}{1-r} = \frac{1}{1 - 1/4} = \frac{1}{3/4} = \frac{4}{3}$.$F = 9 	imes 10^{3} 	imes \frac{4}{3} = 12 	imes 10^{3} \,N$.Comparing with $x 	imes 10^{3} \,N$,we get $x = 12$.

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