Question 6.1:

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

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The direction of the induced current in a closed loop is given by Lenz’s law. The given pairs of figures show the direction of the induced current when the North pole of a bar magnet is moved towards and away from a closed loop respectively.

Using Lenz’s rule, the direction of the induced current in the given situations can be predicted as follows:

The direction of the induced current is along **qrpq***.*

* *

The direction of the induced current is along **prqp**.

The direction of the induced current is along ** yzxy**.

The direction of the induced current is along ** zyxz**.

The direction of the induced current is along ** xryx**.

No current is induced since the field lines are lying in the plane of the closed loop.

Question 6.2:

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.

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Length of the rod, *l *= 1 m Angular frequency,*ω *= 400 rad/s

Magnetic field strength, *B *= 0.5 T

One end of the rod has zero linear velocity, while the other end has a linear velocity of *lω.*

* *

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Average linear velocity of the rod,

Emf developed between the centre and the ring,

Hence, the emf developed between the centre and the ring is 100 V.

Question 6.3:

A long solenoid with 15 turns per cm has a small loop of area 2.0 cm^{2} 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?

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Number of turns on the solenoid = 15 turns/cm = 1500 turns/m

Number of turns per unit length,
*n *= 1500 turns

The solenoid has a small loop of area, *A *= 2.0 cm^{2} = 2 × 10^{−4} m^{2}

Current carried by the solenoid changes from 2 A to 4 A.

Change in current in the solenoid,
*di *= 4 − 2 = 2 A

Change in time, *dt *= 0.1 s

Induced *emf *in the solenoid is given by Faraday’s law as:

Where,

= Induced flux through the small loop

= *BA *... (*ii*)

*B *= Magnetic field

=

μ_{0} = Permeability of free space

*= *4π×10^{−7} H/m

Hence, equation (*i*) reduces to:

Hence, the induced voltage in the loop is

Question 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?

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Length of the rectangular wire,
*l *= 8 cm = 0.08 m Width of the rectangular wire, *b *= 2 cm = 0.02 m Hence,
area of the rectangular loop,

*A *= *lb*

* *

= 0.08 × 0.02

= 16 × 10^{−4} m^{2}

Magnetic field
strength, *B *= 0.3 T Velocity of the loop, *v *= 1 cm/s = 0.01 m/s Emf developed in the loop is given as:

*e *= *Blv*

* *

= 0.3 × 0.08 × 0.01 = 2.4 × 10^{−4} V

Hence, the induced voltage
is 2.4 × 10^{−4} V which lasts
for 2 s. Emf developed, *e *= *Bbv*

= 0.3 × 0.02 × 0.01 = 0.6 × 10^{−4} V

Hence, the induced voltage
is 0.6 × 10^{−4} V which lasts for 8 s.

Question 6.5:

A 1.0 m long metallic rod is rotated
with an angular frequency of 400 rads^{-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 0.5T
parallel to the axis exist everywhere. Calculate
the emf developed between the centre
and the ring.

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l = 1.0 cm ω = 400 rad/s

B = 0.5T

ε =

= 100V

Question 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 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?

Max induced
*emf *= 0.603 V Average
induced *emf *= 0 V

Max current in the coil = 0.0603 A Average power loss = 0.018 W (Power comes from the external rotor)

Radius of the circular
coil, *r *= 8 cm = 0.08 m Area of the coil, *A *= π*r*^{2} = π × (0.08)^{2} m^{2} Number of turns
on the coil, *N *=
20

Angular speed,
*ω *= 50 rad/s

Magnetic field strength, *B *= 3 × 10^{−2} T Resistance
of the loop, *R *= 10 Ω Maximum induced
*emf *is given as:

*e
*= *Nω AB*

* *

= 20 × 50 × π
× (0.08)^{2} × 3 × 10^{−2}

= 0.603 V

The maximum *emf *induced in the coil is 0.603 V.

Over a full cycle, the average

Average power loss due to joule heating:

The current induced in the coil produces a torque opposing the rotation of the coil. The rotor is an external agent. It must supply a torque to counter this torque in order to keep the coil rotating uniformly. Hence, dissipated power comes from the external rotor.

Question 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}.

What is the instantaneous value of the emf induced in the wire?

What is the direction of the emf?

Which end of the wire is at the higher electrical potential?

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Length of the wire, *l *= 10 m

Falling speed of the wire, *v *= 5.0 m/s

Magnetic field strength, *B *= 0.3 × 10^{−4} Wb m^{−2}

Emf induced in the wire,

*e
*= *Blv*

* *

Using Fleming’s right hand rule, it can be inferred that the direction of the induced emf is from West to East.

The eastern end of the wire is at a higher potential.

Question 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.

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Initial current,
*I*_{1} = 5.0 A Final
current, *I*_{2} = 0.0 A

Change in current,
Time taken for the change,
*t *= 0.1 s

Average emf, *e *= 200 V

For self-inductance (*L) *of the coil, we have the relation
for average emf as:

*e *= *L*

* *

* *

Hence, the self induction of the coil is 4 H.

Question 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?

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Mutual inductance of a pair of coils, *µ *= 1.5 H

Initial current,
*I*_{1} = 0 A Final
current *I*_{2} = 20 A

Change in current, Time taken for the change, *t *= 0.5 s

Induced emf, Where is the change in the flux linkage with the coil.

Emf is related with mutual inductance as:

Equating equations (1) and (2), we get

Hence, the change in the flux linkage is 30 Wb.

Question 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°.

Speed of the jet plane, *v *= 1800 km/h = 500 m/s

Wing spanof jet plane, *l *= 25 m

Earth’s magnetic
field strength, *B *= 5.0 × 10^{−4} T Angle of dip,

Vertical component of Earth’s magnetic field,

*B _{V} *=

= 5 × 10^{−4} sin 30°

= 2.5 × 10^{−4} T

Voltage difference between the ends of the wing can be calculated as:

*e *= (*B _{V}*) ×

* *

= 2.5 × 10^{−4} × 25 × 500

= 3.125 V

Hence, the voltage difference developed between the ends of the wings is

3.125 V.

Question 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?

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Sides of the rectangular loop are 8 cm and 2 cm.

Hence, area of the rectangular wire loop,

*A *= length × width

= 8 × 2 = 16 cm^{2}

= 16 × 10^{−4} m^{2}

Initial value of the magnetic field,

Rate of decrease of the magnetic field,

*Emf *developed in the loop is given as:

Where,

= Change in flux through the loop area

= *AB*

* *

Resistance of the loop, *R *= 1.6 Ω

The current induced in the loop is given as:

Power dissipated in the loop in the form of heat is given as:

The source of this heat loss is an external agent, which is responsible for changing the magnetic field with time.

Question 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 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.

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Side of the square
loop, *s *= 12 cm = 0.12 m

Area of the square loop, *A *= 0.12 × 0.12 = 0.0144 m^{2} Velocity of the loop, *v *= 8 cm/s = 0.08 m/s

Gradient of the magnetic
field along negative
*x*-direction,

And, rate of decrease of the magnetic field,

Resistance of the loop,

Rate of change of the magnetic flux due to the motion of the loop in a non-uniform magnetic field is given as:

Rate of change of the flux due to explicit time variation in field *B *is given as:

Since the rate of change of the flux is the induced *emf*, the total induced
*emf *in the loop can be calculated as:

∴Induced current,

Hence, the direction of the induced current is such that there is an increase in the flux through the loop along positive z-direction.

Question 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 cm^{2} 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.

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Area of the small flat search coil, *A *= 2 cm^{2} = 2 × 10^{−4} m^{2}

Number of turns on the coil, *N *= 25

Total charge flowing in the coil, *Q *=
7.5 mC = 7.5 × 10^{−3} C

Total resistance of the coil and galvanometer, *R *= 0.50 Ω

Induced current in the coil,

Induced emf is given as:

Where,

= Charge in flux

Combining equations (1) and (2), we get

Initial flux through the coil, = *BA*

* *

Where,

*B *= Magnetic field strength

Final flux through the coil,

Integrating equation (3) on both sides, we have

But total charge,

Hence, the field strength of the magnet is 0.75 T.

Question 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.

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.

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

K is open? What if K is closed?

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.

What is the retarding force on the rod when K is closed?

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?

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

What is the source of this power?

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

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Length of the rod, *l *= 15 cm = 0.15 m Magnetic
field strength, *B *= 0.50 T

Resistance of the closed loop, *R *= 9 mΩ = 9 × 10^{−3} Ω

Induced emf = 9 mV; polarity
of the induced emf is such that end *P *shows positive
while end *Q *shows negative
ends.

Speed of the rod, *v *= 12 cm/s = 0.12 m/s Induced
emf is given as:

*e *= *Bvl*

* *

*= *0.5 × 0.12 × 0.15

= 9 × 10^{−3} v

= 9 mV

The polarity
of the induced emf is such that end *P *shows positive
while end *Q *shows negative ends.

Yes; when key K is closed, excess charge is maintained by the continuous flow of current.

When key K is open, there is excess charge built up at both ends of the rods.

When key K is closed, excess charge is maintained by the continuous flow of current.

Magnetic force is cancelled by the electric force set-up due to the excess charge of opposite nature at both ends of the rod.

There is no net force on the electrons in rod PQ when key K is open and the rod is moving uniformly. This is because magnetic force is cancelled by the electric force set-up due to the excess charge of opposite nature at both ends of the rods.

Retarding force exerted on the rod, *F *= *IBl*

Where,

*I *= Current flowing through the rod

9 mW; no power is expended when key K is open.

Speed of the rod, *v *= 12 cm/s = 0.12 m/s

Hence, power is given as:

When key K is open, no power is expended.

9 mW; power is provided by an external
agent. Power dissipated as heat = *I*^{2} *R*

= (1)^{2} × 9 × 10^{−3}

= 9 mW

The source of this power is an external agent.

Zero

In this case, no emf is induced in the coil because the motion of the rod does not cut across the field lines.

Question 6.15:

An air-cored
solenoid with length 30 cm, area of cross-section 25 cm^{2} 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.

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Length of the solenoid,
*l *= 30 cm = 0.3 m

Area of cross-section, *A *= 25 cm^{2} = 25 × 10^{−4} m^{2}

Number of turns on the solenoid,
*N *= 500

Current in the solenoid,
*I *= 2.5 A

Current flows for time, *t *= 1

Average back emf, Where,

= Change in flux

= *NAB *… (2)

Where,

*B *= Magnetic field strength

Where,

−3 s

= Permeability of free space = 4π
× 10^{−7} T m A^{−1}

Using equations (2) and (3) in equation (1), we get

Hence, the average back emf induced in the solenoid is 6.5 V.

Question 6.16:

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.

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.

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Take a small element
*dy *in the loop at a distance
*y *from the long straight
wire (as shown in the given figure).

Magnetic flux associated with element Where,

* *

*I *= Current in the wire

= Permeability of free space = 4π
× 10^{−7} T m A^{−1}

*y *tends
from *x *to .

Emf induced in the loop, *e *= *B’av
*

* *

Given,

*I *= 50 A

Question 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
**= − B_{0} **k **(*r *≤ *a*; *a *<
*R*)

= 0 (otherwise)

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

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Line charge per unit length Where,

*r *= Distance of the point
within the wheel
Mass of the wheel = *M*

Radius of the wheel = *R*

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Magnetic field,

At distance *r*,themagnetic force is balanced
by the centripetal force i.e.,

∴Angular velocity,