A current `I` flows an infinitely long wire with cross section in the form of a semi - circular ring of radius `R`. The magnitude of the magnetic induction along its axis is :

Options

A. `(mu_(0)I)/(2pi^(2)R)`
B. `(mu_(0)I)/(2piR)`
C. `(mu_(0)I)/(4piR)`
D. `(mu_(0)I)/(pi^(2)R)`

Answer

D

Explanation

Consider the wire to be made up of large number of thin wires of infinite length. Consider such wire of thickness dl subtending an angle `d theta` at centre.
Current through this wire,

`dI=(d theta)/(pi) 1 :.` Magnetic field at centre due to this portion,
`dB=(mu_(0))/(4pi) . (2dI)/R=(mu_(0)I)/(2pi^(2)R) d theta`
Net magnetic field at the centre,
`B= int_(-pi//2)^(pi//2)d B cos theta =(mu_(0)I)/(2pi^(2)R), int_(-pi//2)^(2pi//2) cos theta d theta =(mu_(0)I)/(pi^(2)R)`