**12. Electromagnetic Induction**

**Multiple Choice Questions**

**1. Henry is equivalent to …………………..**(A) ampere / second

(B) ampere second

(C)

(D) ohm second

**Ans.** (D) ohm second

**2. A transformer converts **** AC to ****. The secondary has 75 turns. The number of turns in primary are…………………..**

(A) 600

(B) 500

(C) 400

(D) 300

**Ans.** (D) 300

**3. A metre gauge train is heading north with speed **** in earth’s magnetic field ****. The e.m.f. induced across the axle joining the wheels is…………………..**(A)

(B)

(C)

(D)

**Ans.** (B)

**4. In a step-up transformer, ratio of the turns is**(A) greater than 1

(B) less than 1

(C) equal to 1

(D) never equal to 1

**Ans.** (A) greater than 1

**5. If a coil of metal wire is kept stationary in a uniform magnetic field, then…………………..**

(A) an e.m.f. is induced in the coil

(B) a current is induced in the coil

(C) neither e.m.f. nor current is induced

(D) both current and e.m.f. are induced

**Ans.** (C) neither e.m.f. nor current is induced

**6. A small piece of metal wire is dragged across the gap between the pole pieces of magnet in 0.5 second. The magnetic flux between the pole pieces is **** weber. The emf induced in the wire is…………………..**

(A) 1.6 millivolt

(B) 16 millivolt

(C) 1.6 volt

(D) 16 volt

**Ans.** (A) 1.6 millivolt

**7. If a circular coil of 100 turns with a crosssectional area of **** is kept with its plane perpendicular to the magnetic field of ****, the magnetic flux linked with the coil will be…………………..**(A)

(B)

(C) .

(D)

**Ans.** (C) .

**Theory Questions**

**Faraday’s Laws of Electromagnetic Induction**

**1. State Faraday’s laws of electromagnetic induction.**

**Ans:**

i. First law:

Whenever there is a change of magnetic flux in a closed circuit, an induced e.m.f is produced in the circuit. Also, if a conductor cuts the lines of magnetic field, an e.m.f. is induced between its ends.

ii. Second law:

The magnitude of induced e.m.f produced in the circuit is directly proportional to the rate of change of magnetic flux linked with the circuit.

**What is electromagnetic induction? [July 16]**

**Ans:** The phenomenon of producing an induced e.m.f in a conductor or conducting coil due to changing magnetic ‘flux is called electromagnetic induction.

**12.3 Lenz’s Law**

**State Faraday’s laws of electromagnetic induction and Lenz’s law.**

**Ans:**

i. Faraday’s laws of electromagnetic induction Refer Subtopic 12.2: Q. No. 1 .

ii. Lenz’s law:

The direction of induced current in a circuit is such that the magnetic field produced by the induced current opposes the change in the magnetic flux that induces the current. The direction of induced emf is same as that of induced current.

**What does the negative sign indicate in Lenz’s law?**

**Ans:** The negative sign used in Lenz’s law indicates that the emf induced in the coil is in such direction that it opposes the change in magnetic flux linked with the coil.

**12.5 Motional Electromotive Force**

**Derive an expression for****generated in a conductor of length****moving in uniform magnetic field (B) with uniform velocity (v) along****-axis.**

**Ans:**

**A frame of wire **** in magnetic field **** and **

**wire **** is moving with velocity **** along ****-axis**

i. Consider a shaped frame of wires of area situated in a constant magnetic field . Here, QR is a conducting wire that slides on the frame parallel to PS.

ii. As the wire QR of length is moved out with velocity to increase , the area of the loop PQRS increases. Thus the flux of through the loop increases with time.

iii. According to the flux rule, the induced emf will be equal to the rate at which the magnetic flux through a conducting circuit changes.

iv. The induced emf will cause a current in the loop. It is assumed that there is enough resistance in the wire so that the induced currents are very small producing negligible magnetic field.

v. As the flux through the frame is , magnitude of the induced emf can be written as

where is the velocity of wire increasing the length of wires and SR. vi. Now, a charge which is carried along by the moving wire , experiences Lorentz force ; which is perpendicular to both and and hence is parallel to wire QR.

vii. The force is constant along the length of the wire (as and are constant) and zero elsewhere ( for stationary part RSPQ of wire frame).

viii. When the charge moves a distance along the wire, the work done by the Lorentz force is

where angle between and .

ix. The emf generated is,

x. For maximum induced emf,

xi. Thus, from equation (1) and (2), for any circuit whose parts move in a fixed magnetic field, the induced emf is the time derivative of flux regardless of the shape of the circuit.

**12.10 Eddy Currents**

**What are eddy currents? State any two applications of eddy currents. [Mar 08, Oct 09]**

**Ans:**

i. The circulating currents induced in a metal block, when it is placed or moved in a changing magnetic field are called eddy currents.

ii. Applications of eddy currents:

a. Dead beat galvanometer: When a galvanometer is used for measuring current, the coil is wound on a light aluminium frame to make it dead beat i.e., to bring the coil quickly to rest. This is because the motion of the metal frame in the magnetic field gives rise to eddy current in the frame. The eddy current opposes the motion and brings the coil to rest quickly.

b. Induction motor (speedometer):

Eddy currents are used to know the speed of any vehicle. A pointer shows the speed on a calibrated scale. Speedometer consists of a strong magnet, kept rotating according to the speed of the vehicle. A magnet is rotated in an aluminium drum, pivoted by means of spring. Eddy currents are produced in the drum. The drum turns in the direction of the rotating magnet. A pointer attached to the drum indicates the speed of the vehicle on a calibrated scale.

c. Electric brake (Induction brake): When the train is to be stopped, the power supplied to rotate the axle is switched off. At the same time, a stationary magnetic field is applied to the rotating drum giving rise to strong eddy currents in the drum. These eddy currents, produce a

torque which opposes the rotation of the drum and hence the axle. Thus train is brought to rest quickly and smoothly.

d. Induction furnace: The metal which is to be melted is placed in a huge crucible. High frequency alternating current is allowed to flow through the coil. As a result a rapidly variable magnetic field is produced which provide very strong eddy currents. Heat which is produced in this process is enough to melt the entire block of metal in short time. This method is generally used to make alloys of different metals in vacuum.

(Any two applications)

**12.12 Energy Stored in a Magnetic Field**

**Derive an expression for energy stored in the magnetic field in terms of induced current.**

**Ans:**

i. Changing magnetic flux in a coil causes an induced emf.

ii. The induced emf so produced opposes the change and hence the energy has to be spent to overcome it to build up the magnetic field.

iii. This energy may be recovered as heat in a resistance of the circuit.

iv. The induced emf is given as,

v. The work done in moving a charge dq against this emf is,

Therefore total work,

vi. Equation (1) gives the energy stored in magnetic field and is analogous to the energy stored in the electric field in a capacitor.

**12.14 Mutual Inductance (M)**

**1. Explain self induction and mutual induction.**

**Ans:**

i. Self-induction:

a. Consider a circuit (coil) in which the current is changing.

b. The changing current will vary the magnitude of magnetic flux linked with the coil (circuit) itself and consequently an emf will be induced in the circuit.

c. The production of induced emf, in the circuit (coil) itself, on account of a change in the current in it, is termed as the phenomenon of self-inductance.

d. Let at any instant, the value of magnetic flux linked with the circuit itself be corresponding to current in it.

i.e.,

or

where is a constant of proportionality and is termed as the self-inductance (or coefficient of self induction) of the coil.

e. For a closely wound coil of turns, the same magnetic flux will be linked with all the turns.

When the flux through the coil changes, each turn of the coil contributes towards the induced emf.

f. The flux linkage for coil with turns corresponding to current I will be written as,

g. The inductance (L) depends only on the geometry and material properties of the coil.

h. According to Faraday’s law, induced emf e is given by,

Using equation (1),

ii. Mutual induction:

a. Consider a case of two coils placed side by side as shown in figure.

b. Suppose a fixed current is flowing through coil 1 . Due to this current a magnetic field will be produced in the nearby region surrounding the coil 1.

c. Magnetic flux linked with the surface area of the coil 2 due to magnetic field is given as,

d. If the positions of the coils are fixed in space,

Then

where, = constant of proportionality termed as mutual inductance or coefficient of mutual induction of coil 2 (or circuit ) with respect to coil 1 (or circuit ).

e. Suppose changes slowly with time then magnetic field in the vicinity of coil 2 is related to current in coil 1 in the same way as it would be related for a steady current. The magnetic flux will change in proportion as changes.

f. The induced emf in coil 2 is given as,

g. When current flows through coil 2, magnetic flux linked with coil 1 is given as,

Where, mutual inductance of coil 1 with respect to coil 2.

h. The induced emf in coil 1 will be,

**Explain, why the equivalent inductance of two coils connected in parallel is less than the inductance of either of the coils.**

**Ans:**

i. For parallel combination of two coils, the current through each parallel inductor is a fraction of the total current and the voltage across each parallel inductor is same.

ii. As a result, a change in total current will result in less voltage dropped across the parallel array than for any one of the individual inductors.

iii. There will be less voltage drop across parallel inductors for a given rate of change in current than for any of the individual inductors. iv. Less voltage for the same rate of change in current results in less inductance.

v. Thus, the total inductance of two coils is less than the inductance of either coil.

**12.15 Transformer**

**State the principle on which a transformer works. With neat diagram, explain the construction of a step-up transformer.**

**Ans: Principle:**

Working of a transformer is based on the principle of mutual induction i.e., whenever the magnetic flux linked with a coil changes, an e.m.f is induced in the neighbouring coil.

**Construction:**

i. A step- up transformer consists of two sets of coils primary and secondary insulated from each other. The number of turns in secondary coil is greater than the number of turns in primary coil The coil is called the input coil and coil is called the output coil.

ii. The two coils are wound separately on a laminated soft iron core.

**State the principle of a transformer. Explain its construction and working. Derive an expression for the ratio of e.m.f.s in terms of number of turns in primary and secondary coil.**

**OR**

**State the principle on which transformer works. Explain its working with construction. Derive an expression for ratio of e.m.f.s and currents in terms of number of turns in primary and secondary coil. **

**OR**

**State the principle of working of transformer. Explain the construction and working of **** transformer. Derive an expression for e.m.f. and current in terms of turns ratio.**

**Ans:** Refer Subtopic 12.15: Q. No. 1 for Principle.

i. Construction:

a. A transformer consists of two sets of coils primary and secondary insulated from each other. The coil is called the input coil and coil is called the output coil.

b. The two coils are wound separately on a laminated soft iron core.

**ii. Working:**

a. When an alternating voltage is applied to the primary coil the current through the coil goes on changing. Hence, the magnetic flux through the core also changes.

b. As this changing magnetic flux is linked with both the coils, an e.m.f is induced in each coil.

c. The amount of the magnetic flux linked with the coil depends upon the number of turns of the coil.

d. Let, ‘ ‘ be the magnetic flux linked per turn with both the coils at certain instant ‘ ‘.

e. Let ‘ and ‘ ‘ be the number of turns of primary and secondary coil,

magnetic flux linked with the primary coil at certain instant ‘ ‘

magnetic flux linked with the secondary coil at certain instant ‘ ‘

f. Induced e.m.f produced in the primary and secondary coil is given by,

g. Dividing equation (2) by (1),

Equation (3) represents equation of transformer.

The ratio is called turns ratio (transformer ratio) of the transformer.

h. For an ideal transformer, Input power Output power

i. From equation (3) and (4),

**3. Distinguish between step-up and step-down transformer.**

**Ans:**

i. | The number of turnsin its secondary is morethan that in its primary. | The number of turnsin primary is greaterthan secondary. |

ii. | Alternating voltageacross the ends of itssecondary is morethan that across itsprimary i.e., | Alternating voltageacross the ends of theprimary is more thanthat across itssecondary i.e., |

iii. | Transformer ratio | Transformer ratio |

iv. | Primary coil made ofthick wire. | Secondary coil madeof thick wire. |

v. | Secondary coil ismade of thin wire. | Primary coil is madeof thin wire. |

vi. | Current throughsecondary is less thanprimary. | Current throughprimary is less thansecondary. |

**What is a transformer? With the help of a suitable diagram describe working of transformer.**

**Ans: Transformer:**

Transformer is an electrical device which converts low alternating voltage at high current to high alternating voltage at low current and vice-versa.

Working: Refer Subtopic 12.15: Q. No. 2

(Diagram and working only)

**Numericals**

**12.5 Motional Electromotive Force**

**A metal****long rotates about one of its ends perpendicular to a plane whose magnetic induction is****. Calculate the number of revolutions made by the rod per second if the e.m.f. induced between the ends of the rod is****.**

**Solution:**

Given: ,

To find: Number of revolutions ( )

Formula:

Calculation: From formula,

Ans: The number of revolutions made by rod per second is 4.

**An aircraft of wing span of****flies horizontally in earth’s magnetic field of****T. Calculate the velocity required to generate an e.m.f. of****between the tips of the wings of the aircraft.**

**Solution:**

[July 22]

Given:

To find: Velocity (v)

Formula:

Calculation: From formula

Ans: The velocity required is .

**A wire****long is supported horizontally at a height of****along east-west direction. When it is about to hit the ground, calculate the average e.m.f. induced in it.**

**Solution:**

Induced e.m.f. is given by formula,

In given condition, the wire initially supported in east-west direction, when falls towards the ground travels in north-south direction. Hence, velocity is acting in the north – south direction. Earth’s magnetism also acts from the north pole to the south pole. This makes

Ans: The average induced e.m.f. in the wire will be zero.

**12.6 Induced emf in a Stationary Coil in a Changing Magnetic Field**

**A retangular coil of length****and breadth****has resistance of****. The coil is placed in a magnetic induction of****and its direction is perpendicular to the plane of the coil. If the magnetic induction is uniformly reduced to zero in 5 milli seconds, find the e.m.f. and current induced in the coil.**

**Solution:**

Given:

To find:

ii. Induced current (I)

Formulae: i.

ii.

Calculation: Since,

Also,

From formula (i),

From formula (ii),

Ans: i. The e.m.f induced in the coil is .

ii. Current induced in the coil is .

**The magnetic flux through a loop varies according to the relation****, where ‘****‘ is constant, ‘****‘ is in milliweber and ‘****‘ is in second. What is the magnitude of induced e.m.f. in the loop at****seconds?**

**Solution:**

Given: (in milliweber)

To find: Magnitude of induced e.m.f. (e)

Formula: (in magnitude)

Calculation: Using formula,

Ans: The magnitude of induced e.m.f. is .

**The magnetic flux through a loop is varying according to a relation****where****is in milliweber and****is in second. What is the e.m.f. induced in the loop at****second?**

**Solution:**

Given:

To find:

(in milliweber),

Formula: (in magnitude)

Calculation: Using formula,

Ans: The magnitude of induced e.m.f. is .

**12.7 Generators**

**A coil of****turns, each of area****is kept in a uniform field of induction****. If the coil rotates with a speed of 6000 r.p.m. about an axis in the plane of the coil and perpendicular to the magnetic induction, calculate peak value of e.m.f. induced in the coil.**

**Solution:**

Given:

To find: Peak e.m.f

Formula:

Calculation: From formula,

Ans: The peak value of e.m.f. induced in the coil is .

**12.11 Self Inductance**

**Two inductor coils with inductance****and****are connected in series. What is the resultant inductance of the combination of the two coils?**

**Ans:** Resultant inductance

**12.14 Mutual Inductance (M)**

**The co-efficient of mutual induction between primary and secondary coil is****. Calculate induced e.m.f. if current of****is cut off in****seconds.**

**Solution:**

Given:

To find: Induced e.m.f.(e)

Formula:

Calculation: From formula,

Ans: The induced e.m.f. in the coil is .

**2. An emf of **** is induced in the windings of a coil, when the current in a nearby coil is increasing at the rate of ****, what is the mutual inductance **** of the two coils in mH?**

**Solution:**

Given:

To find: Mutual Inductance (M)

Formula:

Calculation: From formula

Ans: Mutual Inductance of the two coils is .

**An emf of****is induced in the windings of a coil when the current in a nearby coil is increasing at the rate of****. What is the mutual inductance (M) of the coils? [July 22]**

**Solution:**

Given:

To find: Mutual Inductance (M)

Formula:

Calculation: From formula

Ans: Mutual Inductance of the two coils is .