Question

Two concentric circular coils of radii r and R are placed coaxially with centres coiciding. If R >> r then calculate the mutual inductance between the coils.

Answer 1

Let current $I_2$ flow in the outer coil.
Hence magnetic field at its center=$\frac{{\mu }_0{I}_2}{2R}$

Since $r<<R$, this magnetic field can be considered to be constant over the entire face of smaller coil.

Hence flux through the coil=$\varphi =\frac{{\mu }_0{I}_2}{2R}\pi r^2$

By definition of mutual inductance,

$\varphi =M_{12}{I}_2=\frac{{\mu }_0{I}_2}{2R}\pi r^2$

$⟹M_{12}=\frac{{\mu }_0\pi r^2}{2R}$