An assmebly of charges `+q, -q , +q, -q…..` are at positions `x = 1 m, 2 m, 4 m, 8 m, `…. And so on from origin. What is the potential at `x = 0`, due to these charges ?

Options

A. `(q)/(4pi in_(0))`
B. `(-q)/(4pi in_(0))`
C. `(q)/(6pi in_(0))`
D. `(-q)/(6pi in_(0))`

Answer

C

Explanation

Let potential at `x = 0`, due to all positive charge be `V_(1)`
`:. V_(1) = (1)/(4pi in_(0)) (q)/(1) + (1)/(4pi in_(0)) (q)/(4) + (1)/(4pi in_(0)) (q)/(16) +…..`
`= (q)/(4pi in_(0)) [1 + (1)/(4) + (1)/(16) + …..]`
`(q)/(4pi in_(0)) [(1)/(1 - 1//4)] = (q)/(3 pi in_(0))`
Now potential at `x = 0`, due to all negative charges
`V_(2) = (-q)/(4 pi in_(0) (2)) - (q)/(4pi in_(0) (8)) - (q)/(4pi in_(0) (32)) + ...`
`= - (q)/(8 pi in_(0)) [1 + (1)/(4) + (1)/(16) + .....]`
`= (q)/(8pi in_(0)) [(1)/(1 - 1//4)] = - (q)/(6pi in_(0))`
Therefore, potential at `x = 0`, due to all negative charges
`V = V_(1) + V_(2) = (q)/(3pi in_(0)) - (q)/(6 pi in_(0)) = (q)/(6 pi in_(0))`