Mutual inductance

Consider a circuit consisting of a capacitor of capacitance $C$ and a coil with $N$ turns per unit length, cross sectional area $S$ and length $d$, where $d^2\gg S$. There is another coil of length $d/2$, cross sectional area $S/2$ and $2N$ turns per unit length completely inside the larger coil, as shown in the figure. The ends of this smaller coil are connected with each other by an insulated conducting wire. The self-inductance of the larger coil is $L$. Neglecting edge effects and all the Ohmic resistances, the resonant frequency of the circuit is:

A circular current loop of radius $R$ is placed inside square loop of side length $L$ ($L\gg R$) such that they are co-planar and their centers coincide. The permeability of free space is $\mu_0$. The mutual inductance between circular loop and square loop is ________.

The dimension of mutual inductance is :

Two coils of self inductance $L_1$ and $L_2$ are connected in series combination having mutual inductance of the coils as M. The equivalent self inductance of the combination will be :

A small square loop of wire of side $l$ is placed inside a large square loop of wire L ($L\gg l$). Both loops are coplanar and their centres coincide at point O as shown in figure. The mutual inductance of the system is :

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 ______ mJ.

A conducting loop in the shape of a right angled isosceles triangle of height 10 cm is kept such that the 90° vertex is very close to an infinitely long conducting wire (see the figure). The wire is electrically insulated from the loop. The hypotenuse of the triangle is parallel to the wire. The current in the triangular loop is in counterclockwise direction and increased at a constant rate of $10\ \mathrm{A\,s^{-1}}$. Which of the following statement(s) is(are) true?

A circular wire loop of radius $R$ is placed in the $x$-$y$ plane centered at the origin $O$. A square loop of side $a$ $(a\ll R)$ having two turns is placed with its center at $z=\sqrt3R$ along the axis of the circular wire loop, as shown in figure. The plane of the square loop makes an angle of 45° with respect to the $z$-axis. If the mutual inductance between the loops is given by $\dfrac{\mu_0a^2}{2^{p/2}R}$, then the value of $p$ is