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Two concentric circular coils, one of small radius $$r_1$$ and the other of large radius $$r_2,$$  such that $$r_1\ll r_2$$ , are placed co-axially with centers coinciding. Obtain the mutual inductance of the arrangement.
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## QuestionPhysicsClass 12

Two concentric circular coils, one of small radius $$r_1$$ and the other of large radius $$r_2,$$  such that $$r_1\ll r_2$$ , are placed co-axially with centers coinciding. Obtain the mutual inductance of the arrangement.

Let current $$I_2$$  is flowing through larger coil. The magnetic field produced is uniform throughout its area, and the value is given by:
$$B_2=\frac{\mu _0I_2}{2r_2}$$
Since the inner coil is co-axially placed with the outer coil, so the corresponding flux linkage with smaller coil is given by:
$${\emptyset}_1=A_1\times B_2$$
$$=\pi r_1^2\times \frac{\mu _0I_2}{2r_2}$$
$$=\frac{\mu _0\pi r_1^2}{2r_2}I_2$$
$$=M_{12}I_2$$
Where,
$$M_{12}=\frac{\mu _0\pi r_1^2}{2r_2}$$; The mutual inductance of the smaller coil with respect to outer coil
Now, we know that,
$$M_{12}=M_{21}=\frac{\mu _0\pi r_1^2}{2r_2}$$
Therefore, the mutual inductance of the arrangement is given by
$$M=M_{12}=M_{21}=\frac{\mu _0\pi r_1^2}{2r_2}$$