Solution:

Self-inductance is that property of a coil by virtue of which it opposes any change in the magnitude of current passing through it by inducing an emf in itself Consider an inductor of inductance L, carrying alternating current through it. Suppose at any instant of time an emf induced in the inductor is

` epsilon = -L (dI)/(dt)`

To maintain the growth of current through the inductor, power has to be supplied from external source.

` (dW)/(dt) = - epsilon I = L (dI)/(dt) I => dW = L I d I`

Total amount of work done to build up current from zero to `I` is

` W = L int_0^I I d I = 1/2 LI^2`

This work done gets stored in the inductor in the from of magnetic energy.

Thus, ` U = W = 1/2 LI^2`

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Solution:

(a) Mutual inductance of two coils is equal to the magnetic flux linked with one coil when a unit current is passing through the other coil.

i.e. `phi = MI`

Its `SI` unit is henry `(H)` or `WbA^(-1)`.

(b) Mutual inductance between two long co-axial solenoids is defined as the magnetic flux linked to the second coil when unit current is flowing in the first coil.

It is the phenomenon by virtue of which a coil resists any change in the strength of current in its neighbouring coil. Suppose current `I_ 2` is flowing through outer solenoid. Magnetic field at a point on the axis of the solenoid is

`B_2 = mu_0 n_2 I_2`

where `n_2` is number of turns per unit length of the solenoid.

Flux linked with the inner solenoid is

` phi_(12) = B_2 pi r_1^2 (n_1 l)`

where `r_1` is the radius of inner solenoid and `n_1` is the number of turns per unit length of the 1st solenoid.

`phi_(12) = ( mu_0 n_1 n_2 pi r_1^2 l) I`

Here `M_(12) = mu_0 n_1 n_2 pi r_1^2 l`

`M_( 12)` is mutual inductance of the solenoids.

(c) Suppose a magnetic flux (`phi`) is linked with coil `c_1` due to current (`I_2`) in coil `c_2`.

` phi_1 alpha I_2`

` phi_1 = M I_2`

` => (d phi_1)/(dt) = M (dI_2)/(dt)`

` :. epsilon = - M (dI_2)/(dt)`

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Solution:

Lenz's Law: The current induced in a circuit always flows in such a direction that it opposes the change or the that produces it.

Mathematically, ` epsilon = - (d phi)/(dt)`

No, it does not violate the principle of conservation of energy.

Justification : Lenz's law complies with the principle of conservation of energy. For example, when the N-pole of a bar magnet is pushed into a coil as shown, the direction of induced current in the coil G will be such that the end `2` of the coil will act as N-pole. Thus, work has to be done against the magnetic repulsive force to push the magnet into the coil. The electrical energy produced in the coil is at the expense of this work done.

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Solution:

Eddy currents: The induced circulating current produced in bulk piece of conductor, when subjected to a changing magnetic flux is called eddy current. When a conductor in bulk is kept in changing magnetic field, eddy currents are induced.

Applications:

(i) Magnetic braking in trains: In some electrically powered trains, strong electromagnets are situated above the rails. When these magnets are activated, eddy currents are induced in the rails. These currents oppose the motion of the train. Due to the absence of mechanical linkage, the breaking effect is strong.

(ii) Electromagnetic damping: Certain galvanometers have fixed core made of non-magnet metallic materials. When the coil of galvanometer oscillates, eddy currents generated on the core oppose its motion. As a result, the coil comes to rest quickly.

(iii) Electric power meters: The shiny metal disc in the electric power meter rotates due to the eddy currents. Electric currents are induced in the disc by magnetic fields produced by sinusoidally varying current (i.e. ac) in the coil.

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