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.

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)