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 a distance L from the end A is:

Options

A. `Q/(8pi epsilon_0L)`
B. `3Q/(8pi epsilon_0L)`
C. `(Q)/( 4 pi epsi_(0) L In 2)`
D. `( Q In 2)/(4 pi epsi_(0) L)`

Answer

D

Explanation

Let an element of length dx, charge dq is at distance x from point O.
`dV=k (dq)/x` where `dq=Q/L dx`
`therefore V= int_L^(2L) (kdq)/x= int_L^(2L) ((Q/L)dx)/x= Q/(4 pi e_0 L) int_L^(2L) (1/x) dx`
`=Q/(4 pi e_0 L) [log_e x] _L^(2L) =Q/(4 pi e_0L) [log_e 2L- log_e L]=Q/(4 pi e_0L) [log_e^2 (2L)/2] =Q/(4 pi e_0L) In (2)`