Given \(l=30cm=0.3m\)
\(B=0.05T\)
\(\upsilon =20{ ms }^{ -1 }\)
\(\therefore \) Induced e.m.f.
\(e=Bl\upsilon =0.05\times 0.3\times 20=3 \ V\)

\(Given \ L=44mH=44\times { 10 }^{ -3 }H\)
\( { \ E }_{ v }=220V,v=50Hz\)
\({ \ I }_{ v }=?\)
\(Since \ { I }_{ v }=\frac { { E }_{ v } }{ { X }_{ L } } =\frac { 220 }{ \omega L }\)
\( \therefore \ { I }_{ v }=\frac { 220 }{ 2\pi vL } \)
\(or \ { I }_{ v }=\frac { 2220 }{ 2\times 3.142\times 50\times 44\times { 10 }^{ -3 } } \)
\(=15.9A\)

The back emf in the coil opposes the change in the current as per Lenz's law.

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

Self-Inductance When the current in a coil is changed, a back emf is induced in the same coil. This phenomenon is called self-inductance. If Lis self-inductance of coil, then
\(N\phi \propto I\Rightarrow N\phi =LI\Rightarrow L=\frac { N\phi }{ I } \)
The SI unit of self-inductance is Henry (H).
(ii) Mutual inductance of solenoid coil system
\(M=\frac { { \mu }_{ 0 }{ N }_{ 1 }{ N }_{ 2 }{ A }_{ 2 } }{ l } \)
Here, N₁ = 15, N₂ = 1, l = 1cm = 10^-2m,
A = 2.0cm² = 20 x 10^-4m²
∴ \(M=\frac { 4\pi \times { 10 }^{ -7 }\times 15\times 1\times 2.0\times { 10 }^{ -4 } }{ { 10 }^{ -2 } } \)
\(=120\pi \times { 10 }^{ -9 }H\)
Induced emf in the loop
\({ \varepsilon }_{ 2 }=M\frac { { \Delta I }_{ 1 } }{ { \Delta t } } (numerically)=20\pi \times { 10 }^{ -9 }\frac { \left( 4-2 \right) }{ 0.1 } \)
\(=120\times 3.14\times { 10 }^{ -9 }\times \frac { 2 }{ 0.1 } =7.5\times { 10 }^{ -6 }V=7.5\mu V\)

Applied voltage, V = 140 \(sin\ 100\pi t\ V\)
\(C \ = \ \frac { 50 }{ \pi } \mu F=\frac { 50 }{ \pi } \times { 10 }^{ -6 }F.\\ L \ = \ \frac { 5 }{ \pi } H, \ R=400\Omega \)
Comparing with V = V₀ \(sin\ \omega t,\) we get
V₀ = 140 V and \(\omega =100\pi \)
Inducting reactance, 
\({ X }_{ L }=\omega L=100\pi \times \frac { 5 }{ \pi } =500\Omega \)
Capacitive reactance,
\({ X }_{ C }=\frac { 1 }{ \omega C } =\frac { 1 }{ 100\pi \times \frac { 50 }{ \pi } \times { 10 }^{ -6 } } \)
\(=200\Omega \quad \)
Impedance of the circuit, 
\(Z=\sqrt { { R }^{ 2 }+({ X }_{ L }-{ X }_{ C })^{ 2 } } \)
\(=\sqrt { ({ 400) }^{ 2 }+({ 500-200) }^{ 2 } } \)
\( =\sqrt { 1600+900 } \ = \ 500\Omega \)
Maximum current in the circuit,
\({ I }_{ 0 }=\frac { { V }_{ 0 } }{ Z } =\frac { 140 }{ 500 }\)
\( { I }_{ rms }=\frac { { I }_{ 0 } }{ \sqrt { 2 } } =\frac { 140 }{ 500\times \sqrt { 2 } } =0.2\quad A\)
\({ V }_{ rms }\) across resistor, V_R = I_rms R
\(=0.2\times 400\ =\ 80V\)
Vrms across capacitor, V_C = I_rms X_L
\(=0.2\times 200\ =\ 40V\)
Now,   \(V\neq { V }_{ R }+{ V }_{ L }+{ V }_{ C }\)
Because V_C,V_L and V_R are not in same phase, instead
 \(V=\sqrt { { V }_{ R }^{ 2 }+({ V }_{ L }-{ V }_{ C })^{ 2 } } \)
\(=\sqrt { { 80 }^{ 2 }+({ 100-40) }^{ 2 } } =100V\)
Which is same as that of applied rms voltage.

Electromagnetic induction
Electromagnetic induction may be defined as the phenomenon of production of electric current (or e.m.f.) in a closed coil (or circuit), when magnetic flux linked with the coil (or circuit) is changed.
Magnetic flux \(\left( \phi \right) \). The total number of lines of induction passing normally through a given area is called magnetic flux. Consider a surface A. Take a small element of area dA. If \(\overrightarrow { B } \) makes an angle \(\theta \) with the area vector \(d\overrightarrow { A } \), then magnetic flux linked with area dA will be
\(d\phi =\overrightarrow { B } .d\overrightarrow { A } \)
(where area vector is a vector \(d\overrightarrow { A } \) having magnitude dA and direction normal to \(dA\))

Total magnitude flux linked with the whole surface S will be
\(\phi =\int { \overrightarrow { B } .d\overrightarrow { A } } \)
If the surface is plane and having area S, then
\(\phi =\overrightarrow { B } .\overrightarrow { A } \)
or  \(\phi =BA\cos { \theta } \)    .....(1)
Units of \(\phi \)
If \(B=1T,A=1{ m }^{ 2 },\theta ={ 0 }^{ ° },then\quad \phi =1Wb.\)
Hence magnetic flux is said to be 1 weber (1Wb) if a magnetic field of 1 tesla is normal to the area of 1 \({ m }^{ 2 }.\)
So  \(1Wb=1T{ m }^{ 2 }\)
Magnetic induction (B). It is defined as the magnetic flux associated per unit area.
i.e. \(B=\frac { \phi }{ A } \)
Units. In S.I. the unit of B is Wb \({ m }^{ 2 }\) or tesla (T).
Positive and negative flux
Since \(\phi =BA\cos { \theta } \)
So (I) \(If \ \theta >{ 90 }^{ ° },\phi =-ve\left( i.e. \ \phi \ is \ inward \ the \ surface \right) \)
(ii) \(If \ \theta <{ 90 }^{ ° },\phi =+ve\left( i.e. \ \phi \ is \ outward \ to \ the \ surface \right) \)
(iii) \(If \ \theta ={ 90 }^{ ° },\phi =BS\left( i.e. \ maximum \ flux \right) \)
Dimensional formula of \(\phi \)
\(\phi =BA\cos { \theta =\frac { FA }{ q\upsilon } \cos { \theta \left[ \therefore F=qB\upsilon \right] } } \)

\(=\frac { \left[ ML{ T }^{ -2 } \right] \left[ { L }^{ 2 } \right] }{ \left[ AT \right] \left[ L{ T }^{ -1 } \right] } \)

\(=\left[ M{ L }^{ 2 }{ T }^{ -2 }{ A }^{ -1 } \right] \)

(a) 1.0 J, yes
(b) \(\omega \) = 10³ rad s^-1, v = 159 Hz
(c) 1.0 J

As heat produced is the same, rms voltage of a.c. source = d.c. voltage = 110 V

(i) Output of three phase a.c. is almost constant.
(ii) Output of three phase a.c. is greater than the output of one phase and two phase systems.
(iii) Three phase a.c. system is more economical.

The slip ring arrangement in an a.c. generator is replaced by split ring arrangement or commutator arrangement in d.c. generator.

No, a transformer cannot work on d.c.

The resistance coils are double wound to avoid induction effects. Magnetic field due to current in one half of the coil is cancelled by magnetic field due to current in the other half of the coil (which is in opposite direction).

No, as every material has some resistance. Yes, we can coil a wire to have resistance with inductance.

Induced e.m.f., \(e=V=\frac { work }{ charge } =\frac { M{ L }^{ 2 }{ T }^{ -2 } }{ AT } \)
\(=[{ M }^{ 1 }{ L }^{ 2 }{ T }^{ -3 }{ A }^{ -1 }]\)
Rate of change of magnetic flux\(=\frac { d\Phi }{ dt } =\frac { BA }{ t } \)
 \(=\frac { FA }{ q\upsilon t } =\frac { { (MLT }^{ -2 })({ L }^{ 2 }) }{ (AT)({ L }T^{ -1 })(T) } \ (\because F=Bq\upsilon )\\ \ \ =[{ M }^{ 1 }{ L }^{ 2 }T^{ -3 }A^{ -1 }]\)
Both have the same units/dimensions.

No, current is induced. This is because orbiting satellite intercepts only the vertical component of earth's magnetic field, which is zero at the equator.

\(M=\frac{\mu_0 N_1 N_2 A}{l}\), Here, A is area of cross section of inner solenoid and l is length of longer solenoid.  

They are both, useful and harmful.

\(X_L=\omega L=2 \pi \nu L=2 \pi \times 50 \times 0.25\)
\(=25 \pi \mathrm{ohm}\)

\(e=-L \frac{d l}{d t}\)
Where symbols have usual meaning.

This is because induced e.m.f. opposes the current due to the actual source e.m.f.

Weber, Tesla.

No, because the pole intercepts neither H nor V.

Due to change in magnetic flux, e.m.f. is always induced, but induced current will flow only when tghe circuit is complete.

1 weber = 10⁸ Maxwell