)
We need only add the components due to the different part circles.
For a full circle of radius R' we have B = (mu I)/(2R') as per the attachment.
For BC, which is only a quarter circle of radius 2R:
B_(BC) = 1/4 * (mu I)/(2(2R)) = 1/16 (mu I)/(R)
For DA, we do likewise
B_(DA) = 3/4 * (mu I)/(2R) = 3/8 (mu I)/(R)
And adding implies 7/16 (mu I)/(R)
NB
We can do it this simply because of the consat radius and because d vec L times hat r is the same all along a circle: it is the tangent d vec L vector crossed with unit radial hat r vector. So we are merely adding the components of the d vec B 's for each circle
The direction of the vec B field will be into the page as the right hand rule would indicate when applied to this cross product.
