A charge Q is uniformly distributed over a long rod AB of length L as shown in the figure. The electric potential at the point O lying at distance L from the end A is

Options

A. `(Q)/(2 pi in_(0) L)`
B. `(3Q)/(4 pi in_(0) L)`
C. `(3Q)/(4 pi in_(0) L in 2)`
D. `(Q in 2)/(4 pi in_(0) L)`

Answer

D

Explanation

Consider a small element of length `dx` of the rod at a distance `x` from `O`, Fig. Charge on the element
`dq = (Q)/(L) dx`

Potential at `O` due to this element `= (k dq)/(x)`
Potential at `O` due to the rod,
`V = int_(L)^(2L)(k dq)/(x) = (KQ)/(L) int_(L)^(2L) (dx)/(x) = k (Q)/(L) [log x]_(L)^(2L)`
`= ((Q)/(L) In 2)/(4pi in_(0)) = (Q in 2)/(4pi in_(0) L)`