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(C) The electric potential $V$ at a point due to a system of point charges is the algebraic sum of the potentials due to individual charges.Given char

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$\frac{1}{4 \pi \varepsilon_0} \left( \frac{2q}{3} \right)$

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(C) The electric potential $V$ at a point due to a system of point charges is the algebraic sum of the potentials due to individual charges.Given charges $q_1 = q$ at $x_1 = 1 \, m$,$q_2 = -q$ at $x_2 = 2 \, m$,$q_3 = q$ at $x_3 = 4 \, m$,$q_4 = -q$ at $x_4 = 8 \, m$,and so on.The potential at $x = 0$ is given by $V = \sum \frac{k q_i}{r_i}$,where $k = \frac{1}{4 \pi \varepsilon_0}$.$V = k \left( \frac{q}{1} - \frac{q}{2} + \frac{q}{4} - \frac{q}{8} + \dots \right)$$V = kq \left( 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \dots \right)$The term in the bracket is a geometric series with first term $a = 1$ and common ratio $r = -1/2$.The sum of an infinite geometric series is $S = \frac{a}{1 - r}$.$S = \frac{1}{1 - (-1/2)} = \frac{1}{1 + 1/2} = \frac{1}{3/2} = \frac{2}{3}$.Therefore,$V = kq \left( \frac{2}{3} \right) = \frac{1}{4 \pi \varepsilon_0} \left( \frac{2q}{3} \right)$.

Electric charges having the same magnitude $q$ are placed at $x = 1 \, m, 2 \, m, 4 \, m, 8 \, m, \dots$ and so on. If any two consecutive charges have opposite signs but the first charge is necessarily positive,what will be the potential at $x = 0$?

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