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_0I)/(2pi^2R)`
B. `(mu_0I)/(2piR)`
C. `(mu_0I)/(4pi^2R)`
D. `(mu_0I)/(pi^2R)`

Answer

D

Explanation

Consider two elements on semicircular cross-section of long wire at C and D, each of length `dl` symmetrically situated with respect to point O cross-section as shown in figure. Current through each element,
`dI=(I)/(pir)dl` and `dl=r d theta`,
`:. dI=(I)/(pir)rd theta=(I)/(pid) theta`

Magnitude of magnetic field induction at O due to current `dI` in element at C or D is
`dB=(mu_0)/(4pi)(2dI)/(r)=(mu_0dI)/(2pir)`
Net magnetic field induction at O due to both the elements carrying current `dI` is
`=2(dB)cos theta=2xx(mu_0)/(2pi)(dI)/(r)cos theta`
`=(mu_0)/(pir)xx(I/pid theta)cos theta=(mu_0I)/(pi^2r)cos theta d theta`
Total magnetic field induction at O due to current through the entire wire is
`B=(mu_0I)/(pi^2r)int_(0)^(pi//2)cos theta d theta=(mu_0I)/(pi^2r)[sin pi/2-sin 0]=(mu_0I)/(pi^2r)`