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An infinite number of charges, each equal to $$q$$, are placed along the x-axis at $$x=1$$,$$x=2$$,$$x=4$$,$$x=8$$ etc. Find the potential and electric field at the point x=0 due to the set of the charges.

Discussion:

An infinite number of charges, each equal to $$q$$, are placed along the x-axis at $$x=1$$,$$x=2$$,$$x=4$$,$$x=8$$ etc.

Electric potential $$V=\frac{1}{4\pi\epsilon_0}(\frac{q}{1}+\frac{q}{2}+\frac{q}{4}+\frac{q}{8}+…)$$ $$=\frac{1}{4\pi\epsilon_0}(1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+….)$$
The terms in brackets form a geometric progression of infinite terms whose sum is $$S= \frac{a}{1-r}=\frac{1}{1-1/2}=2$$

Hence the potential $$V=q/2\pi\epsilon_0$$