Figure shows two semicircular loops of radii $R_1$ and $R_2$ carrying current $I$. The magnetic field at the…

Figure shows two semicircular loops of radii $R_1$ and $R_2$ carrying current $I$. The magnetic field at the common centre ' $\mathrm{O}$ ' is
  1. $\frac{\mu_0 I}{4}\left(\frac{1}{R_1}+\frac{1}{R_2}\right)$
  2. $\frac{\mu_0 I}{4}\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$
  3. $\frac{\mu_0 I}{2 \pi}\left(\frac{1}{R_1}+\frac{1}{R_2}\right)$
  4. $\frac{\mu_0 I}{2 \pi}\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

Solution

For semicircular arc the magnetic field is given as $B=\frac{\mu_0 i}{4 R}$ The equivalent magnetic field at the centre is $\begin{aligned} & B_{\text {eq }}=\frac{\mu_0 I}{4 R_1}+\frac{\mu_0 I}{4 R_2} \\ & B_{\text {eq }}=\frac{\mu_0 I}{4}\left(\frac{1}{R_1}+\frac{1}{R_2}\right) \end{aligned}$

Asked in: MHT CET 2023 (14 May Shift 2)

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