Self-inductance of the solenoid is the flux associated with the solenoid when the unit amount of a current passes through it.
Consider a solenoid of self-inductance L, length e, area of cross-section A.
Let alternating voltage
ε = ε₀ sin ωt applied across it, and back emf induced.

∴ e = -L \(\frac { dI }{ dt } \)
⇒ dL = \(\frac { e }{ L } \) dt
Small  amount of work done i.e
 dW = eI dt
Total work done 
 W = \(\int { el\ dt } \)
= \(\int _{ 0 }^{ 1 }{ LI\ dI\ =\frac { 1 }{ 2 } } \)
This work done will be stored in the form of magnetic energy in inductor
 U = \(\frac { 1 }{ 2 } \) LI²

Capacitance, C = 30μF = 30 × 10⁻⁶F
Inductance, L = 27 mH = 27 × 10⁻³ H
Angular frequency is given as:

\({ \omega }_{ r }=\frac { 1 }{ \sqrt { LC } } =\frac { 1 }{ \sqrt { 27\times { 10 }^{ -3 }\times 30\times { 10 }^{ -6 } } } =\frac { { 10 }^{ 4 } }{ 9 } =1.1\times { 10 }^{ 3 }rad/s\)
Hence, the angular frequency of free oscillations of the circuit is 1.11 × 10³ rad/s.

\(Here, \ L=2.0H, \ C=32\mu F=32\times { 10 }^{ -6 }F \ and \ R=10ohm, \ { \omega }_{ r }=?, \ Q=?\)
\({ \omega }_{ r }=\frac { 1 }{ \sqrt { LC } } =\frac { 1 }{ \sqrt { 2.0\times 32\times { 10 }^{ -6 } } } =\frac { { 10 }^{ 3 } }{ 8 } =125rad/s\)
\(\\ Q=\frac { 1 }{ R } \sqrt { \frac { L }{ C } } =\frac { 1 }{ 10 } \sqrt { \frac { 2.0 }{ 32\times { 10 }^{ -6 } } } =\frac { { 10 }^{ 3 } }{ 40 } =25\)

\(Here, \ R=100\Omega , \ { E }_{ v }=220V, \ v=50Hz\)
\((a) \ { I }_{ v }=?, \ { I }_{ v }=\frac { { E }_{ v } }{ R } =\frac { 220 }{ 100 } =2.2A\)
\((b) \ Net \ power \ consumed \ over \ a \ full \ cycle,\)
\(P={ E }_{ v }{ I }_{ v }=220\times 2.2=484W\)

(a) According to Lenz's law south pole develops at q, current induced must be clockwise at q.
\(\therefore \) In the coil, induced current is from p to q.
(b) Coil pq in this case would develop S-pole at q and coil xy would also develop S pole at x. Therefore, induced current in the coil pq will be from y to x.
(c) Induced current in the right loop will be along xyz.
(d) Induced current in the left loop will be along xyz.
(e) Induced current in the right coil is from x to y.
(f) No induced current since fixed lines lie in the plane of the loop.

An ideal inductor is a coil consisting of large number of turns wound in such a way that magnetic flux linked with it due to current passing through it is confined to a small region only. The ohmic resistance of such a coil is zero.

Self-inductance is the phenomenon of production of opposing emf in a coil. When the current through the coil changes. The self-inductance depends upon
(i) size of coil (area of cross-section and number of turns/length).
(ii) Relative magnetic permeability of the core of the coil on which winding is done.

They are wound over themselves to reduce the back emf. The current in the two limbs of the coil flows in opposite directions and so the net flux linked with the coil continues to be nearly zero, and no back emf is produced.

(a) I_o = 3.23A
(b) 1.55ms