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An assembly of charge $+q, -q, +q, -q ……..$ are placed at distance $x = 1m, x = 2m, x = 4m, x = 8m$ and so on, from the origon, in a plane. The potential at $x = 0,$ due to the charges would be ______.

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Positive potential $V_+ =\large\frac{q}{4 \pi \in_0 (1)} +\frac{q}{4 \pi \in_0 (4)}+\frac{q}{4 \pi \in_0 (16)}....1\frac{q}{4 \pi \in_0} \bigg(1+\large\frac{1}{4} +\frac{1}{4} +\frac{1}{16} +....... \infty \bigg)$
$\qquad= \large\frac{q}{4 \pi \in_0 } \bigg( \large\frac{1}{1- (1/4) }\bigg) =\frac{q}{3 \pi \in_0}$
Negative potential $V_- =\large\frac{-q}{4 \pi \in_0 (2)} -\frac{q}{4 \pi \in_0 (8)}+\frac{q}{4 \pi \in_0 (32)}....$
$\qquad= \large\frac{q}{8 \pi \in_0} \bigg( 1+ \large\frac{1}{4} +\frac{1}{16} +,,,, \bigg)$
$\qquad= \large\frac{-q}{6 \pi \in_0}$
Net potential $V_{net}= \large\frac{q}{3 \pi \in_0} -\frac{q}{6 \pi \in_0}$
$\qquad= \large\frac{q}{6 \pi \in_0}$
answered Jun 7, 2014 by meena.p

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