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)
