How to use the gradient to find Electric field

[howell]

1. A rod carrying a uniform charge distribution is bent into a semi circle with the center on the orgin and a radius R. Calcualte the Electric field at the center of the semi circle using the electric potential expression found in part a



2. E = -(gradient)V



3. The electric potential at the center is V = kQ/r or k (pi) (lambda) where lambda is the charge density. The correct expression for the electric field at the center of the circle is 2k(lambda)/r. However, I'm finding that simply taking the gradient of the electric potential function gives kQ/r^2 or k(lambda)(pi)/r, which is not correct. I'm afraid I'm making a fundamental mistake with the definition of the gradient and how to account for the changing electric field direction at the center of the semi circle.

 

[rl.bhat]

Here you have taken the gradient along the radius.
First you have to finds out delta Ex( due to a small element dl) = -dV/dx and dEy = - dv/dy where dx = dr*cos (theta) and dy = dr*sin(theta). Take summation over the whole semicircular rod to find the field.