Question

Find an expression for the electric field e⃗ at the center of the semicircle. hint: a small piece of arc length δs spans a small angle δθ=δs /r, where r is the radius.

Answer 1

Let l = Q/L = linear charge density. The semi-circle has a length L which is half the circumference of the circle. So w can relate the radius of the circle to L by 

C = 2L = 2*pi*R ---> R = L/pi 

Now define the center of the semi-circle as the origin of coordinates and define a as the angle between R and the x-axis. 

we can define a small charge dq as 

dq = l*ds = l*R*da 

So the electric field can be written as: 

dE =kdq*(cos(a)/R^2 I_hat + sin(a)/R^2 j_hat) 

dE = k*I*R*da*(cos(a)/R^2 I_hat + sin(a)/R^2 j_hat) 

E = k*I*(sin(a)/R I_hat - cos(a)/R^2 j_hat) 

E = pi*k*Q/L(sin(a)/L I_hat - cos(a)/L j_hat)