Question

A uniform charge wire, linear charge density d, is laid in the form of a semi circle of radius r. Find the electric field generated by the semi circle of the center

Answer 1

The electric field is

$$E_x=\frac{kd}{r}\int^\pi_0\cos\theta\text d\theta=\frac{kd}r\Big[\sin\theta\Big]^\pi_0=0,\\\space\\ E_y=\frac{kd}{r}\int^\pi_0\sin\theta\text d\theta=\frac{kd}r\Big[\cos\theta\Big]^\pi_0=-\frac{2kd}r.$$

E_x is the field along the diameter, E_y is the field along the radius of the semi-circle (vertical radius perpendicular to the base of the semi-circle if we put it branches down).