A rod of length L lies along the x-axis with its left end at the origin. It has a non-uniform charge density `lamda=alphax`, where a is a positive constant.
(a) What are the units of `alpha`?
(b) Calculate the electric potential at point A where `x = - d` .

Explanation

`dq = beta x dx`
`dV_(p) = (1)/(4 pi in_(0)) .(dq)/((1 + x)) = (beta)/(4 pi in_(0)) (xdx)/((d + x))`
`V_(P) = (beta)/(4 pi in_(0)) int_(0)^(L) (x dx)/(d + x)`
`int_(0)^(L) (xdx)/(d + x) = int_(0)^(L) ((d + x) - d)/((d + x)) dx = int_(0)^(L) dx - d int_(0)^(L) (dx)/(d + x)`
`= |x|_(0)^(L) - d|log_(e ) (d + L) - log_(e )(d)]`
`= L - d log_(e ) (1 + (L)/(d))`
`V_(P) = (beta)/(4 pi in_(0)) [ L - d log_(e) (1 + (L)/(d))]`