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

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Solution

Potential due to a charge at distance r is kq/r.

Potential due to the +charge at distance 1 m is kq/1
Potential due to the -charge at distance 2 m is -kq/2
Potential due to the charge at distance 4 m is kq/4
and so on.
The total is kq * (1 - 1/2 + 1/4 - 1/8 + ...)
The thing inside parentheses is a geometric series. The sum of an infinite geometric series is a/(1 - r) where a = first term = 1 and r = common ratio = -1/2. So the infinite sum is 1/[1 - (-1/2)] = 1/(3/2) = 2/3.

Thus the total potential is kq * 2/3.

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