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Q.

A current I flows in an infinitely long wire with cross-section in the form of a semicircular ring of radius R. The magnitude of the magnetic induction at point on its axis is

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a

$\frac{μ_{0} I}{2 π^{2} R}$

b

$\frac{μ_{0} I}{4 π R}$

c

$\frac{μ_{0} I}{2 π R}$

d

$\frac{μ_{0} I}{π^{2} R}$

answer is A.

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Detailed Solution

Question Image

$d I = \frac{I}{π R} R d θ = \frac{I}{π} d θ$

$B = \int_{0}^{π} d B \sin θ = \int \frac{μ_{0} d I}{2 π R} \sin θ$

Here $d I = \frac{I}{π R} R d θ$ ; $B = \frac{μ_{0} I}{2 π^{2} R} \int_{0}^{π} \sin θ d θ$

$B = \frac{μ_{0} I}{π^{2} R}$

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