An infinitely long conductor as shown in figure, carrying a current I with a semicircular loop on X-Y plane and two straight parts, one parallel to X-axis and another coinciding with Z-axis. What is the magnetic field induction at the centre C of the semi-circular loop.

Explanation

The magnetic field induction at C due to current through straight part of the conductor parallel to X-axis is
`B_1=(mu_0)/(4pi)(I)/((r//2))[sin.pi/2+sin0^@]=(mu_0)/(4pi)(2I)/(r)`
acting along +Z direction.
i.e., `vecB_1=(mu_0)/(4pi)(2I)/(r)hatk`
The magnetic field induction at C due to current through the semi-circular loop in X-Y plane is
`B_2=(mu_0)/(4pi)(I)/((r//2))(pi)=(mu_0)/(4pi)(2Ipi)/(r)`
acting along +Z-direction
i.e., `vecB_2=(mu_0)/(4pi)(2piI)/(r)hatk`
The magnetic field induction at C due to current through the straight part of the conductor coinciding with Z-axis is
`B_3=(mu_0)/(4pi)(I)/(r//2)[sin .pi/2+ sin 0]=(mu_0)/(4pi)(2I)/(r)` acting along negative X-axis
i.e. `vecB_3=(mu_0)/(4pi)(2I)/(r)(-hati)`
Total magnetic field induction at C is given by
`vecB=vecB_1+vecB_2+vecB_3=(mu_0)/(4pi)(2I)/(r)hatk+(mu_0)/(4pi)(2piI)/(r)hatk-(mu_0)/(4pi)(2I)/(r)hati=(mu_0)/(4pi)(2I)/(r)[(1+pi)hatk-hati]`