Two points are at distance `a` and `b (a lt b)` from a long string of charge per unit length `lambda`. The potential difference between th epoints is proportional to

Options

A. `(b)/(a)`
B. `(b^(2))/(a^(2))`
C. `sqrt((b)/(a))`
D. `"log"_(e)(b)/(a)`

Answer

D

Explanation

`E = - (dV)/(dr) = (lambda)/(2pi epsilon_(0)r)`
or `dV = - (lambda)/(2pi epsilon_(0)r) dr` or `underset(V_(a))overset(V_(b))int dV = - (lambda)/(2pi epsilon_(0)) underset(a)overset(b)int (1)/(r ) dr`
or `V_(b) - V_(a) = -(lambda)/(2pi epsilon_(0)) [log_(e) b - log_(e)a]`
or `V_(a) - V_(b) = (lambda)/(2pi epsilon_(0)) log_(e) (b)/(a) rArr V_(a) - V_(b) prop "log"_(e) (b)/(a)`