Physics Definition Based Problems

### Definition Based Problems

Q 3282345237

Define the term self-inductance of a solenoid. Obtain the expression for the magnetic energy stored in an inductor of self-inductance L build up a current I through it.
Or
Define self-inductance of a coil. Show that magnetic energy required to build up the current I in a coil of self-inductance L, is given by 1/2 LI^2 .

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
Q 3282445337

(a) Define mutual inductance and write its SI units.
(b) Derive an expression for the mutual inductance of two long co-axial solenoids of same length wound one over the other.
(c) In an experiment, two coils c_1 and c_2 are placed dose to each other. Find out the expression for the emf induced in the coil c_1 due to a change in the current through the coil c_2.

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)
Q 3232445332

State Lenz's Law. Does it violate the principle of conservation of energy. Justify your answer.

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.
Q 3212445339

What are eddy currents? How are they produced ? Describe briefly three main useful applications of eddy currents.

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.