An infinite number of charges, each numerical | vedclass

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(C) The electric potential $V$ at a point due to a system of charges is given by $V = \sum \frac{1}{4\pi\varepsilon_0} \frac{q_i}{r_i}$.Here, the char

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$\frac{q}{2\pi \varepsilon_0}$

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(C) The electric potential $V$ at a point due to a system of charges is given by $V = \sum \frac{1}{4\pi\varepsilon_0} \frac{q_i}{r_i}$.Here, the charges are $q$ at distances $r = 1, 2, 4, 8, \dots \, 	ext{m}$.$V = \frac{1}{4\pi\varepsilon_0} \left[ \frac{q}{1} + \frac{q}{2} + \frac{q}{4} + \frac{q}{8} + \dots ight]$$V = \frac{q}{4\pi\varepsilon_0} \left[ 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots ight]$The term in the bracket is a geometric progression with first term $a = 1$ and common ratio $r = 1/2$.The sum of an infinite geometric series is $S = \frac{a}{1-r} = \frac{1}{1 - 1/2} = \frac{1}{1/2} = 2$.Therefore, $V = \frac{q}{4\pi\varepsilon_0} 	imes 2 = \frac{2q}{4\pi\varepsilon_0} = \frac{q}{2\pi\varepsilon_0}$.

An infinite number of charges, each numerically equal to $q$ and of the same sign, are placed along the $x-$axis at $x = 1, 2, 4, 8, \dots \, \text{meters}$. The electric potential at $x = 0$ due to this set of charges is:

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