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


even if the charges are stationary [and the q (v × B) term of the Lorentz
force is not operative], an emf is nevertheless induced in the presence of a
time-varying magnetic field. Thus, moving charges in static field and static
charges in a time-varying field seem to be symmetric situation for
Faraday’s law. This gives a tantalising hint on the relevance of the principle
of relativity for Faraday’s law.

The motion of a copper plate is damped when it is allowed to oscillate
between the magnetic pole-pieces. How is the damping force, produced by
the eddy currents?

    6.2 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.

    FIGURE 6.19

    6.3 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?

    Answer:


    6.4 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?


    6.5 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.


    6.6 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 × 10–2 T. Obtain the
    maximum and average emf induced in the coil. If the coil forms a
    closed loop of resistance 10 Ω, calculate the maximum value of current
    in the coil. Calculate the average power loss due to Joule heating.
    Where does this power come from?

    Answer:


    6.7 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 × 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?


    6.8 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.


    6.9 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?


    6.10 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 × 10–4 T and the dip angle is 30°.

    ADDITIONAL EXERCISES

    6.11 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?


    6.12 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 increasesby 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.


    6.13 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.


    6.14 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.

    FIGURE 6.20

    (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?


    6.15 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.


    6.16 (a) Obtain an expression for the mutual inductance between a long
    straight wire and a square loop of side a 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.

    FIGURE 6.21

    6.17 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 ≤ a; a < R)
    = 0 (otherwise)
    What is the angular velocity of the wheel after the field is suddenly
    switched off?

    Answer: