Find the magnetic field at the origin due to the combination of two semi infinite wires and a semicircular wire as shown.

Explanation

The magnetic field at the centre O due to the two straight sent infinite parts of conductor is
`vec(B_(1)) = (mu_(0))/(4 pi) . (i)/(a) hat(k) + (mu_(0))/(4 pi). (i)/(a) hat(k) = (mu_(0))/(4 pi). (2i)/(a) hat(k)` …(1)
The magnetic field due to the semicircular part is
`vec(B_(2)) = (mu_(0))/(4 pi) . (pi i)/(a) (- hat(j))` ....(2)
`:.` The net magnetic field at the centre O is
`vec(B) = vev(B_(1)) + vec(B_(2)) = (mu_(0))/(4 pi). (i)/(a) (2 hat(k) - pi hat(j))` ......(3)