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Q.

A current carrying wire is placed in the, grooves of an insulating semicircular disc of radius R, as shown. The current enters at point A and leaves from point B. Determine the magnetic field at point D.

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a

$\frac{\sqrt{3} μ_{0} i}{4 πR}$

b

none of these

c

$\frac{μ_{0} i}{4 πR \sqrt{3}}$

d

$\frac{μ_{0} i}{8 πR \sqrt{3}}$

answer is B.

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Detailed Solution

$r_{1} = 2 R \sin 30^{0} = R, r_{2} = 2 R \sin 60^{0} = R$

Question Image

$\begin{array}{l} B_{D} = B_{AC} + B_{CB} \\ = \frac{μ_{0}}{4 π} [\frac{i}{r_{1}} \sin ϕ_{1}] ⊙ + \frac{μ_{0}}{4 π} [\frac{i}{r_{2}} \sin ϕ_{2}] \otimes \\ = \frac{μ_{0}}{4 π} i [\frac{\sin 60^{0}}{r_{1}} - \frac{\sin 30^{0}}{r_{2}}] ⊙ = \frac{μ_{0}}{4 πR \sqrt{3}} ⊙ \end{array}$

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