CHAPTER -6 ELECTROMAGNETIC INDUCTION

A long solenoid ‘S’ has ‘n’ turns per meter, with diameter ‘a’. At the
centre of this coil we place a smaller coil of ‘N’ turns and diameter
‘b’ (where b < a). If the current in the solenoid increases linearly,
with time, what is the induced emf appearing in the smaller coil.
Plot graph showing nature of variation in emf, if current varies as
a function of mt2 + C.

A metallic ring of mass m and radius l (ring being horizontal) is
falling under gravity in a region having a magnetic field. If z is the
vertical direction, the z-component of magnetic field is Bz = Bo
(1+λ z). If R is the resistance of the ring and if the ring falls with a
velocity v, find the energy lost in the resistance. If the ring has
reached a constant velocity, use the conservation of energy to
determine v in terms of m, B, λ and acceleration due to gravity g.

A magnetic field B is confined to a region r ≤ a and points out of
the paper (the z-axis), r = 0 being the centre of the circular region.
A charged ring (charge = Q) of radius b, b > a and mass m lies in
the x-y plane with its centre at the origin. The ring is free to rotate
and is at rest. The magnetic field is brought to zero in time Δt.
Find the angular velocity ω of the ring after the field vanishes.

There are two coils A and B seperated by some distance. If a current
of 2 A flows through A, a magnetic flux of 10-2 Wb passes through
B (no current through B). If no current passes through A and a
current of 1 A passes through B, what is the flux through A?

Consider a metallic pipe with an inner radius of 1 cm. If a cylindrical
bar magnet of radius 0.8cm is dropped through the pipe, it takes
more time to come down than it takes for a similar unmagnetised
cylindrical iron bar dropped through the metallic pipe. Explain.

A solenoid is connected to a battery so that a steady current flows
through it. If an iron core is inserted into the solenoid, will the
current increase or decrease? Explain.

A wire in the form of a tightly wound solenoid is connected to a DC
source, and carries a current. If the coil is stretched so that there
are gaps between successive elements of the spiral coil, will the
current increase or decrease? Explain.

Prove that the average power over a complete cycle of a.c. through an ideal inductor is zero.

The output voltage of an ideal transformer connected to a 240 V a.c. mains is 24 V. When this transformer is used to light a bulb with rating 24 V, 24 W. calculate current in primary coil of the circuit.

A series LCR circuit with L = 5 H, C = 80 μF, R = 40 Ω is connected to a variable frequency source of 230 V (i) determine resonance frequency of the circuit (ii) obtain impedance of circuit  and amplitude of current at resonance.

An alternating voltage E = 200 sin 300 t is applied across a series combination of R = 10 Ω and an

Inductor of 800 mH. Calculate (i) impedance of circuit (ii) peak value of current in circuit (iii) powerfactor of the circuit.

The electric mains in a house are marked 220 V, 50 Hz. Write down the equation for instantaneous voltage.

Draw a schematic diagram of a step-up transformer. Explain its working principle.  Deduce the expression for the secondary to primary voltage in terms of the number of turns in the two coils. In an ideal transformer, how is this ratio related to the currents in the two coils?

.(a)         State he condition for resonance to occur in a series LCR a.c. circuit and derive an expression for the resonance frequency.

(b)          Draw a plot showing the variation of the peak current (i₀) with frequency of the a.c. source used. Define the quality factor, Q of the circuit.

For given a.c. circuit, distinguish among resistance, reactance and impedance.

Obtain the resonant frequency and Q factor of a series LCR circuit with L= 3H, C= 27μF and

            R= 7.4 Ώ. Write two different ways to improve quality factor of a series LCR circuit

The natural frequency of an LC circuit is 1,25,000 Hz. Then the capacitor C is replaced by another          capacitor with a dielectric medium k, which decreases the frequency by 25 KHz. What is the value of k?

In an AC circuit V and I are given by V=100Sin100t volts and I= 100 Sin(100t+π/3)ma respectively.    What is the power dissipated in the circuit?

When an AC source is connected to a capacitor with a dielectric slab between its plates, will

therms current increase or decrease or remain constant?

The peak value of an AC is 5A and its frequency is 60Hz. Find its rms value. How long will the

current take to reach the peak value starting from zero?

An electric lamp connected in series with a capacitor and an AC source is glowing with certain        brightness. How does the brightness of the lamp change on reducing the capacitance?

.     How do R, XL and XC get affected when the frequency of applied AC is doubled?

Draw a labeled diagram of a step down transformer .  Mention two sources of energy loss in a transformer.

Define mutual inductance.  Write its S.I. unit. Give two factors on which the coefficient of mutual inductance between a pair of coils depends.

Define self inductance. Write its unit. Give expression for self inductance of a long solenoid having N turns.

How are eddy currents produced? Give two applications of eddy currents.

 State Lenz’s law. Show that this law follows the principle of conservation energy.

State faraday's laws of electromagnetic induction (EMI).

Why is a.c. more dangerous than d.c. for same voltage?

A plot of magnetic flux (φ) versus current is shown in figure for two inductors A & B. which of the two has larger value of self inductance? 

If number of turns of a solenoid is doubled, keeping the other factors constant, how does self inductance of the solenoid changes?

What is the power factor of an LCR series circuit at resonance.

 In series LCR circuit, when voltage and current are in same phase?

What is the power dissipation of an a.c. circuit in which voltage and current are given by :

V = 300 sin (ωt –π/2) and I = 10 sin ωt ?

 What is the power consumed in  (i) purely  inductive and (II) purely capacitive  a.c. circuits?

Give the phase difference between applied a.c. voltage and current in a LCR circuit at resonance.

What is the phase difference between the the voltage across an inductor and a capacitor in an a.c. circuit.

Why a transformer cannot be used to step up d.c. voltage?

A choke coil and a bulb are connected in series to a d.c. source. The bulb shines brightly. How does the brightness changes  when an iron core is inserted in the choke coil?

If the speed of rotation of armature is increased twice how would it affect the (a) maximum

e.m.f produced (b) frequency of the e.m.f?

An electric lamp, connected in series with a capacitor and an a.c. source is glowing with certain brightness. How does the brightness of bulb changes on reducing the capacitance?

A line charge λ per unit length is lodged uniformly onto the rim of a wheel of mass M and radius R. The wheel has light non-conducting spokes and is free to rotate without friction about its axis (Fig. 6.22). A uniform magnetic field extends over a circular region within the rim. It is given by,

B = - B0 k (r Ã¢”°Â¤ a; a < R)

= 0 (otherwise)

What is the angular velocity of the wheel after the field is suddenly switched off?

(a) Obtain an expression for the mutual inductance between a long straight wire and a square loop of sidea as shown in Fig. 6.21.

(b) Now assume that the straight wire carries a current of 50 A and the loop is moved to the right with a constant velocity, v = 10 m/s.

Calculate the induced emf in the loop at the instant when x = 0.2 m.

Take a = 0.1 m and assume that the loop has a large resistance.

An air-cored solenoid with length 30 cm, area of cross-section 25 cm² and number of turns 500, carries a current of 2.5 A. The current is suddenly switched off in a brief time of 10^-3 s. How much is the average back emf induced across the ends of the open switch in the circuit? Ignore the variation in magnetic field near the ends of the solenoid.

Figure 6.20 shows a metal rod PQ resting on the smooth rails AB and positioned between the poles of a permanent magnet. The rails, the rod, and the magnetic field are in three mutual perpendicular directions. A galvanometer G connects the rails through a switch K. Length of the rod = 15 cm, B = 0.50 T, resistance of the closed loop containing the rod = 9.0 mΩ. Assume the field to be uniform.

(a) Suppose K is open and the rod is moved with a speed of 12 cm s^-1 in the direction shown. Give the polarity and magnitude of the induced emf.

(b) Is there an excess charge built up at the ends of the rods when

K is open? What if K is closed?

(c) With K open and the rod moving uniformly, there is no net force on the electrons in the rod PQ even though they do experience magnetic force due to the motion of the rod. Explain.

(d) What is the retarding force on the rod when K is closed?

(e) How much power is required (by an external agent) to keep the rod moving at the same speed (=12 cm s^-1) when K is closed? How much power is required when K is open?

(f) How much power is dissipated as heat in the closed circuit?

What is the source of this power?

(g) What is the induced emf in the moving rod if the magnetic field is parallel to the rails instead of being perpendicular?

It is desired to measure the magnitude of field between the poles of a powerful loud speaker magnet. A small flat search coil of area 2 cm² with 25 closely wound turns, is positioned normal to the field direction, and then quickly snatched out of the field region. Equivalently, one can give it a quick 90° turn to bring its plane parallel to the field direction). The total charge flown in the coil (measured by a ballistic galvanometer connected to coil) is 7.5 mC. The combined resistance of the coil and the galvanometer is 0.50 Ω. Estimate the field strength of magnet.

A square loop of side 12 cm with its sides parallel to X and Y axes is moved with a velocity of 8 cm s^-1 in the positive x-direction in an environment containing a magnetic field in the positive z-direction. The field is neither uniform in space nor constant in time. It has a gradient of 10^-3 T cm^-1 along the negative x-direction (that is it increases by 10^{- 3} T cm^-1 as one moves in the negative x-direction), and it is decreasing in time at the rate of 10^-3 T s^-1. Determine the direction and magnitude of the induced current in the loop if its resistance is 4.50 mΩ.

Suppose the loop in Exercise 6.4 is stationary but the current feeding the electromagnet that produces the magnetic field is gradually reduced so that the field decreases from its initial value of 0.3 T at the rate of 0.02 T s^-1. If the cut is joined and the loop has a resistance of 1.6 Ω how much power is dissipated by the loop as heat? What is the source of this power?

A jet plane is travelling towards west at a speed of 1800 km/h. What is the voltage difference developed between the ends of the wing having a span of 25 m, if the Earth's magnetic field at the location has a magnitude of 5 x 10^-4 T and the dip angle is 30°.

A pair of adjacent coils has a mutual inductance of 1.5 H. If the current in one coil changes from 0 to 20 A in 0.5 s, what is the change of flux linkage with the other coil?

Current in a circuit falls from 5.0 A to 0.0 A in 0.1 s. If an average emf of 200 V induced, give an estimate of the self-inductance of the circuit.

A horizontal straight wire 10 m long extending from east to west is falling with a speed of 5.0 m s^-1, at right angles to the horizontal component of the earth's magnetic field, 0.30 x 10^-4 Wb m^-2.

(a) What is the instantaneous value of the emf induced in the wire?

(b) What is the direction of the emf?

(c) Which end of the wire is at the higher electrical potential?

A circular coil of radius 8.0 cm and 20 turns is rotated about its vertical diameter with an angular speed of 50 rad s^-1 in a uniform horizontal magnetic field of magnitude 3.0 x 10^-2 T. Obtain the maximum and average emf induced in the coil. If the coil forms a closed loop of resistance 10Ω, calculate the maximum value of current in the coil. Calculate the average power loss due to Joule heating. Where does this power come from?

 A 1.0 m long metallic rod is rotated with an angular frequency of 400 rad s-1 about an axis normal to the rod passing through its one end. The other end of the rod is in contact with a circular metallic ring. A constant and uniform magnetic field of 0.5 T parallel to the axis exists everywhere. Calculate the emf developed between the centre and the ring.

A rectangular wire loop of sides 8 cm and 2 cm with a small cut is moving out of a region of uniform magnetic field of magnitude 0.3 T directed normal to the loop. What is the emf developed across the cut if the velocity of the loop is 1 cm s^-1 in a direction normal to the (a) longer side, (b) shorter side of the loop? For how long does the induced voltage last in each case?

A long solenoid with 15 turns per cm has a small loop of area 2.0 cm2 placed inside the solenoid normal to its axis. If the current carried by the solenoid changes steadily from 2.0 A to 4.0 A in 0.1 s, what is the induced emf in the loop while the current is changing?

Use Lenz's law to determine the direction of induced current in the situations described by Fig. 6.19:

(a) A wire of irregular shape turning into a circular shape;

(b) A circular loop being deformed into a narrow straight wire.

Predict the direction of induced current in the situations described by the following Figs. 6.18(a) to (f ).

(a)

(b)

(c)

(d)

(e)

(f)