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An infinite number of charges each numerically equal to q and of the same sign are placed along the $x-$ axis at $x = 1,2,4,8.... \,metres$. Then the electric potential at $x = 0$ due to this set of charges is
$\frac{q}{{4\pi {\varepsilon _0}}}$
$\frac{q}{{3\pi {\varepsilon _0}}}$
$\frac{q}{{2\pi {\varepsilon _0}}}$
$\frac{q}{{\pi {\varepsilon _0}}}$
Solution
$\mathrm{V}= \frac{1}{4 \pi \varepsilon_{0}}\left[\frac{\mathrm{q}}{1}+\frac{\mathrm{q}}{2}+\frac{\mathrm{q}}{4}+\ldots . .\right]$
$=\frac{\mathrm{q}}{4 \pi \varepsilon_{0}}\left[\frac{1}{1}+\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\ldots \ldots\right] $
$ =\frac{\mathrm{q}}{4 \pi \varepsilon_{0}}\left[\frac{1}{1-\frac{1}{2}}\right] \quad \therefore \mathrm{V}=\frac{2 \mathrm{q}}{4 \pi \varepsilon_{0}}=\frac{\mathrm{q}}{2 \pi \varepsilon_{0}} $
