A torus with mean radius $r_0$ and uniform cross sectional area $a\times a$ carries a coil with $N$  turns wound uniformly around it. The medium in the torus is air. A straight, long wire carrying current $I$ passes through the centre of the torus. The wire is perpendicular to the plane of the torus. Assuming $r_0\gg a$, the mutual inductance between the coil and the wire is

Obtain an expression for the mutual inductance of two long co-axial solenoids.

When the current flowing through one coil changes from $0 \mathrm{Amp}$ to $15 \mathrm{Amp}$ in $0.2 \mathrm{s}$, an emf of $750 \mathrm{V}$ is induced in an adjacent coil. Calculate the coefficient of mutual inductance of the two coils.

The inductors of two $LR$ circuits are placed next to each other, as shown in the figure. The values of the self-inductance of the inductors, resistances, mutual-inductance and applied voltages are specified in the given circuit. After both the switches are closed simultaneously, the total work done by the batteries against the induced $EMF$ in the inductors by the time the currents reach their steady-state values is_______$\mathrm{mJ}$.

Consider two coils of radii $r_1$ and $r_2$, placed at a distance $d$ away as shown in the figure. Assuming the length of the coils $l_1,l_2\gg d$ with $n_1,n_2$ the number of turns per unit length of the two coils respectively, the mutual inductance of the two coils is

Obtain an expression for the mutual inductance of two long solenoids.