A current `I` flows in a long straight wire with cross-section haviing the form of a thin half-ring of radius `R` (Fig). Find the induction of the magnitude field at the point `O`.

Explanation

Consider a wire of angle `d theta` at angle `theta`,
Current in this wire, `i'=i/(piR) R d theta=i/pi d theta`
Let current is in upward direction. Small magnetic field at `O` due to this wire.
`dB=(mu_(0)i')/(2piR)=(mu_(0)i)/(2pi^(2)R) d theta`
`B_(x)=int_(-pi//2)^(pi//2) dB sin theta=(mu_(0)i)/(2pi^(2)R) int_(-pi//2)^(pi//2) sin theta d theta=0`
`B_(y) =int_(-pi//2)^(pi//2) dB cos theta=(mu_(0)i)/(2pi^(2) R) int_(-pi//2)^(pi//2) cos theta d theta`
`=(mu_(0)i)/(2pi^(2)R)|sin theta|_(-pi//2)^(pi//2)`
`=(mu_(0)i)/(2pi^(2)R){sin(pi//2)-sin(-pi//2)}`
`=(mu_(0)i)/(pi^(2) R)`