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An infinitely long straight conductor is bent into the shape as shown below. It carries a current of I ampere and the radius of the circular loop is R metre. Then, the magnitude of magnetic induction at the centre of the circular loop is
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The correct answer is:
$\frac{\mu_0 I}{2 \pi R}(\pi-1)$
Magnetic field due to long wire at $O$ point
$B_1=\frac{\mu_0}{2 \pi}\left(\frac{I}{R}\right)$
(upward direction)
Magnetic field due to loop at $O$ point
$\begin{aligned}
& B_2=\frac{\mu_0}{4 \pi} \cdot \frac{I \cdot 2 \pi R}{R^2} \\
& B_2=\frac{\mu_0}{2} \cdot \frac{I}{R} \quad \text { (in upward direction) }
\end{aligned}$
Hence, resultant magnetic field at centre $O$
$\begin{aligned}
B & =B_1+B_2 \\
B & =\frac{\mu_0 I}{2 \pi \cdot R}(\pi+1) \mathrm{T}
\end{aligned}$
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