Let's discuss a problem on Infinite Number of Charges from Physics Olympiad. Try the problem first, and then read the solution here.
The Problem: Infinite Number of Charges
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 $$ V=q/2\pi\epsilon_0$$
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