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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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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.

Answer

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
\(M=M_{12}=M_{21}=\frac{\mu _0\pi r_1^2}{2r_2}\)  
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