Q

# Help me solve this - Magnetic Effects of Current and Magnetism - JEE Main

A current $\dpi{100} I$ flows in an infinitely long wire with cross-section in the form of a semicircular ring of radius $\dpi{100} R$ The magnitude of the magnetic induction along its axis is

• Option 1)

$\frac{\mu _{0}I}{\pi ^{2}R}$

• Option 2)

$\frac{\mu _{0}I}{2\pi ^{2}R}$

• Option 3)

$\frac{\mu _{0}I}{2\pi R}$

• Option 4)

$\frac{\mu _{0}I}{4\pi R}$

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As we learnt in

For Finite Length -

$\phi_{1}=\phi_{2}=\phi$

$B=\frac{\mu_{o}}{4\pi}\:\frac{i}{r}\:(2\sin\phi)$

- wherein

Current in a small element dI

$dI = \frac{d\Theta }{\pi } I$

Magnetic field due to the element

$dB = \frac{\mu_0}{4\pi}.2\frac{dI}{R}$

$dB_{net}= dB Sin\Theta$

$B_{net}= \int dB Sin\Theta =\frac{\mu_I}{2\pi ^2R}\int_{0}^{\pi}sin\Theta d\Theta$

$=\frac{\mu_I}{\pi ^2R}$

Option 1)

$\frac{\mu _{0}I}{\pi ^{2}R}$

Correct

Option 2)

$\frac{\mu _{0}I}{2\pi ^{2}R}$

Incorrect

Option 3)

$\frac{\mu _{0}I}{2\pi R}$

Incorrect

Option 4)

$\frac{\mu _{0}I}{4\pi R}$

Incorrect

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