Self 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 30 cm long solenoid has 10 turns per cm and area of 5 cm². The current through the solenoid coil varies from 2 A to 4 A in 3.14 s. The e.m.f. induced in the coil is $\alpha\times10^{-5}$ V. The value $\alpha$ is ________.

A $10\ \Omega$, 20 mH coil carrying constant current is connected to a battery of 20 V through a switch. Now after switch is opened current becomes zero in 100 $\mu$s. The average e.m.f. induced in the coil is ______ V.

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 :

When the dipole $m$ is placed at a distance $r$ from the center of the loop (as shown in the figure), the current induced in the loop will be proportional to

When current in a coil changes from 5 A to 2 A in 0.1 s, an average voltage of 50 V is produced. The self - inductance of the coil is :