\(Here,\ \ e=0.5V,\)
\(dI=2.0-5.0=-3.0A\)
\(dt=300 \ millisec=3\times { 10 }^{ -1 }s,\ M=?\)
\(From \ |e|=M\frac { dI }{ dt } , \ M=\frac { |e| \ dt }{ dt } =\frac { 0.5\times 3\times { 10 }^{ -1 } }{ 3 } =0.05H\)

Here, wingspan, l = 40m, Speed, v = 1080 km hr^-1
= \(\frac { 1080\times 1000 }{ 60\times 60 } { ms }^{ -1 }\)= 300ms^-1. B = 1.75\(\times\)10^-5 T, e = ?
As  e = Blv
\(\therefore \)  e = 1.75\(\times\)10^-5\(\times\)40\(\times\)300 = 0.21 volt

\(Here, \ l=1.0m, \ e=? \ u=90km \ h^{ -1 }=\frac { 90\times 1000 }{ 60\times 60 } =25{ ms }^{ -1 }\)
\(H=0.3\times { 10 }^{ -4 }T,\ \delta ={ 60 }^{ \circ }\)
\(Magnetic\ field\ intercepted\ is\)
\(V=H\ tan\ \delta =0.3\times { 10 }^{ -4 }\sqrt { 3 } T\)
\(e=V/v=0.3\times { 10 }^{ -4 }\sqrt { 3 } \times 1.0\times 25\ =1.3\times { 10 }^{ -3 } \ V\)

\(Here,\ A=20\times 30cm^{ 2 }\)
\(=600\times { 10 }^{ -4 }{ m }^{ 2 }=6\times { 10 }^{ -2 }{ m }^{ 2 }\ B=0.3T\)
\({ \theta }_{ 1 }=\left( { 90 }^{ \circ }-{ 30 }^{ \circ } \right) =60^{ \circ };\quad { \theta }_{ 2 }={ 90 }^{ \circ }-{ 0 }^{ \circ }={ 90 }^{ \circ }\)
\({ \phi }_{ 1 }=BA \ cos{ \theta }_{ 1 }=0.3\times 6\times { 10 }^{ -2 }cos \ { 60 }^{ \circ } \ =0.9\times { 10 }^{ -2 }Wb\)
\({ \phi }_{ 2 }=BA \ cos \ { \theta }_{ 2 }=0.3\times 6\times { 10 }^{ -2 }cos \ { 90 }^{ \circ }=Zero\)

No, Lenz's law does not violate the principle of conservation of energy. When we move a magnet (say N-pole) towards a coil, the end of coil nearer to the magnet acquires the same polarity (N-pole) and hence opposes the motion of magnet Fig. So work has to be done against this force of repulsion in moving the magnet close to the coil. Similarly, when the magnet (say N-pole) is moved away from the coil, the end of coil nearer to the magnetic pole acquires opposite polarity (S-pole) and pulls the magnet towards it and again work has to be done against the force of attraction in taking the magnet away from the coil.

It is this mechanical work done by outside agency, on the magnet which is sorted in the coil in the form of electrical energy. Hence Lenz's law obeys the law of conservation of energy.

5 mH

Given \(L=0.03H\)
\(\frac { dI }{ dt } =200A{ s }^{ -1 }\)
\(e=?\)
Since \(e=L\frac { dI }{ dt } \)
\(e=0.03\times 200=6V\)

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  \(\phi =5{ t }^{ 3 }+4{ t }^{ 2 }+2t-5,\quad t=2s,\)
\(R=10\Omega \)
\(\therefore \)  Magnitude of induced e.m.f.
\(e=\left| e \right| =\frac { d\phi }{ dt } \)
or \(e=\frac { d }{ dt } \left[ 5{ t }^{ 3 }+4{ t }^{ 2 }+2t-5 \right] \)
\(=15{ t }^{ 2 }+8t+2\)
\(=15\times 4+8\times 2+2\)
or  \(e=78V\)
\(\therefore \)  \(I=\frac { e }{ R } =\frac { 78 }{ 10 } =7.8A\)

\(B=1 \ Wb{ m }^{ -2 }; \ n=80; \ A=0.01{ m }^{ 2 }; \ dt=0.1s; \ e=?\)
\(e=\frac { d\phi }{ dt } =\frac { nAB }{ dt } \)
or   \(e=\frac { 80\times 0.01\times 1 }{ 0.1 } =8V.\)

(i) Magnetic brake.
(ii) Magnetic furnace.

On the basis of his experiment, Faraday gave the following two laws:
First law Whenever magnetic flux linked with a circuit changes, an emf closed (and hence a current) is induced in it which lasts, so long as change in flux continuous.
Second law The emf induced in loop or closed circuit is directly proportional to the rate of change of magnetic flux linked with the loop \( \text { i.e. } e \propto \frac{-d \phi}{d t} \text { or } e=-N \frac{d \phi}{d t}\)

The magnitude of the emf induced in the circuit due to magnetic flux depends on the time rate of change of magnetic flux through the circuit.
\(|\varepsilon|=\frac{\Delta \phi}{\Delta t}\)

The magnetic field created by straight current carrying wire is circular along the perimeter of circular loop. Hence, change in magnetic flux associated with the coil is zero due to increase of current through the wire.
As, induced emf (e) \(\infty\)  rate of change of magnetic flux (\(\phi_{B}\)) So, induced emf=0
Hence, a change in current of wire will not create any emf in the loop.

Self inductance of a coil is numerically equal to the flux linked with the coil when the current through the coil is 1A

Self-induction of the coil is the property by virtue of which it tends to maintain the magnetic flux linked with it and opposes any change in the flux by inducing a current in it. This property of a coil is analogous to mechanical inertia. This is why self-induction is called the inertia of electricity.

When the current begins to grow through the electromagnet. This sets up eddy current in the disc begins to increase. This sets up eddy current in the disc in the same direction as that of the electromagnetic current.
Thus, if the upper surface of electromagnet acquires N-polarity, the lower surface of the disc also acquires N- polarity. As same magnetic poles repel each other, the light metallic disc is thrown up.

As we know the mutual inductance M is directly proportional to the relative permeability of space, \(\mu\)  So, it will increase on placing a thin iron sheet between the two coils.

A straight wire cannot act as an inductor as the magnetic flux linked with the wire of negligible area of cross-section is zero. The wire has to be in the form of a coil to serve as an inductor.

\(Here, \ E=648 \ J,I=9,e=?,\)
\(dI=9-0=9A,dt=0.45s\)
\(From \ E=\frac { 1 }{ 2 } { LI }^{ 2 }\)
\(648=\frac { 1 }{ 2 } { L(9) }^{ 2 }; \ L=\frac { 648\times2 }{ 9\times9 } =16H\)
\(As \ e=\frac { LdI }{ dt } \therefore e=\frac { 16(9) }{ 0.45 } =320V\)

Current through the bulb
\({ I }_{ \upsilon }=\frac { { E }_{ \upsilon } }{ Z } =\frac { { E }_{ \upsilon } }{ \sqrt { { R }^{ 2 }+{ X }_{ L }^{ 2 } } } ,where\quad { X }_{ L }=\omega L\)
\(As\quad L\propto { N }^{ 2 },\) therefore, on decreasing number of turns, L decreases; Z decreases. \({ I }_{ \upsilon }\) increases. Hence brightness of bulb increases.

(i) (d) : More rapid is the movement of bar magnet, more is the deflection observed in the galvanometer
(ii) (c) : Two circular loops carrying current in the same direction will attract each other. If they are now separated, induced currents will try to keep status quo, by increasing the current in both the coils.
(iii) (b): Acceleration of the magnet will not be equal to g. It will be less than g. This is because, as the magnet falls, amount of magnetic flux linked with the ring changes.
An induced emf is developed in the ring which opposes the downward motion of the magnet.
(iv) (c) : The magnitude of induced emf set up in the coil does not depend upon the resistance of the coil whereas induced current set up in the coil depend upon the resistance of the coil.
(v) (d) : As long as a coil of metal is kept stationary in a magnetic field, even if it is non-uniform, unless it is changing with respect to time, there will be no induced emf or current in the coil.

(i) (b) : The working of speedometers is based on eddy currents.
(ii) (a)
(iii) (a): To reduce the eddy currents in the metal armature of motors, wire is wrapped around a number of thin metal sheets called lamination.
(iv) (c) : Eddy currents also oppose the change in magnetic flux, so their direction is given by Lenz's law.
(v) (c)