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Electromagnetic Induction
The phenomenon in which electric current is generated by varying magnetic fields in called Electromagnetic Induction.
The experiments of Faraday and Henry
Experiment 1 Consider a coil C 1 connected to a galvanometer as shown in the figure. When the north pole of a bar magnet is pushed towards the coil, there is a deflection in the galvanometer indicating the presence of electric current in the coil.
- The current flows for as long as the magnet is in motion, there is no deflection when magnet is stationary.
- When the north pole is pulled away from the coil, there is a deflection in the opposite direction i.e. current flows in the opposite direction
- When the south pole of the magnet is pushed/pulled, the deflection is opposite to that when the north pole was being moved.
- The deflection is larger when the magnet is pushed/pulled faster
- Even when the magnet is fixed and the coil C 1 is moved, there is a deflection. Conclusion: Relative motion between the coil and magnet is responsible for generation (induction) of current in the coil. Experiment 2 The bar magnet in experiment 1 is replaced by a coil C 2 connected to a battery. Steady current in the coil C 2 sets up a steady magnetic field. when C 2 is moved towards C 1 , the galvanometer records a deflection. This indicates current is induced in C 1
- When C 2 is moved away, there is a deflection in the opposite direction
- The deflection lasts as long as the coil C 2 is in motion
- When C 2 is fixed and C 1 is moved, the same effects are recorded Conclusion: Relative motion between the two coils is responsible for inducing electric current Experiment 3 Consider the two coils C 1 and C 2 but this time C 2 is connected to the battery through a key. When the key is closed the galvanometer records a momentary deflection. After closing the key for a long time, there is no deflection. When if ' taxi tiniest..
the key is opened, there is again a momentary deflection but in the opposite direction. The deflection increases by many folds when an iron rod is inserted into the coils along their axis Note: Uniform field – field constant w.r.t. position | Steady field- field constant w.r.t. time
Magnetic Flux (φB)
It gives a measure of the number of magnetic field lines intersecting a given area. The magnetic flux through area A due to field B can be written as- Where θ is the angle between B and area vector A SI unit – weber (Wb) or tesla meter squared (Tm^2 ) | scalar quantity
Faraday’s law of Induction
1 st^ Law- If the flux linked with a coil varies with time, an EMF is induced across the coil 2 nd^ Law- The magnitude of induced EMF is directly proportional to the rate of change of flux linked with coil The negative sign indicates the direction of ε and the direction of current in the loop For a coil of N turns- Note: Flux can be varied by varying B, A or θ. Therefore, emf can be induced by-
- Changing B (e.g. time varying field)
- Changing the shape of the coil (area changes)
- Rotating coil in a magnetic field so that angle θ between B and A changes
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Ques: Predict the directions of induced currents in metal rings 1 and 2 lying in the same plane where current I is increasing steadily (PYQ 2012) [1M] Ans: The magnetic field on the sides of the wire is as shown in the figure. As current is increasing, the field and hence flux through the rings will also increase. So, to oppose the increase the change in flux, current will be induced according to Lenz law-
Methods of generating EMF-
1. B as a function of time (t) Consider a solenoid with a time varying current i= I˳sinωt, no of turns per unit length n and radius a. A loop of N turns and radius b (b>a) is placed as shown. Let’s find the emf ε induced in the loop. Case 2: θ as a function of time (PYQ 2011) Consider a coil of radius a and N turns kept in a uniform magnetic field B with its plane perpendicular to the field. It is rotated about an axis perpendicular to the field, in the plane of the loop with an angular velocity ω. The flux through the loop at a time t is- (Not. as field only /ll a) S (^) VI ••^ B N (^) Tt B Fib is
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Motional EMF (PYQ 2020, 2018 , 2011)
The emf generated by moving a conductor in a magnetic field i.e. by changing the flux in the circuit is called motional EMF. In the figure, there is a rectangular conductor PQRS kept in a magnetic field B , in which arm PQ of length L is free to move. The rod PQ is moved towards left with a constant velocity v. Assume that there is no loss of energy due to friction. Explanation 1: Using Faraday’s Law As PQ is moved, the area enclosed by PQRS keeps on changing. At time t let RQ= x, then the flux associated with PQRS – Since x is changing w.r.t. time, the flux through PQRS will also vary with time and hence due to Faraday’s law, an EMF will be induced which can be written as- (Where dx/dt = - v as x is Decreasing with time) Explanation 2: Using Lorentz force Consider a charge q in the conductor PQ. As PQ moves towards the left with velocity v , the charge will also move with same velocity. As the charge is moving in a magnetic field, it will experience a force F = qvB towards Q. All like charges will experience the same force in the same direction. The work done in moving a charge from P to Q- EMF is the work done per unit charge-
- Direction of positive polarity of motional emf is given by direction of v × B.
- The magnitude of motional emf is the product of B, L, v in mutually perpendicular directions. For e.g.- Direc/on of emf e= 0 ( i.e. if any 2 of B,l,v are parallel, e=0) B K
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Method 2: Consider a closed loop OPQ with rod OP rotating with ω. As OP rotates, the area and hence flux linked with the loop changes and an emf is induced across the loop. At time t, let angle between OP and OQ be θ. Area OPQ is- Hence flux linked with loop is-
Energy Conservation
Let resistance of PQ be R and resistance of all other arms is negligible. Therefore, the net resistance is R and doesn’t change as PQ moves. The current I in the circuit can be written as- As current flows through the arm PQ, it will experience a force due to the magnetic field which can be written as- The direction of this force will be towards right, opposite to that of its velocity. To keep the rod moving with constant velocity v, an equal force in the opposite direction is needed. The power required for this will be- The power spent is dissipated as Joule’s heat and is given by- Therefore, the mechanical energy required to move PQ is converted to electrical energy and then thermal energy.
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Eddy Currents
When conductors are subjected to changing magnetic flux induced currents are produced in them. Their flow patterns resemble swirling eddies in water hence they are called eddy currents. Consider a copper a plate which allowed two swing like a pendulum Between the pole pieces of strong magnets. We see that the motion Of the plate is damped and that it comes to rest after a while. This is Because the flux linked with the plate keeps on changing a sit moves In and out of the region between the plates. This induces eddy currents in the plate which opposes the motion of the plate. The direction of eddy current is opposite when the plate swings in and out of the region. Disadvantages of eddy currents Eddy currents are undesirable as they heat up the core and dissipate electrical energy in the form of heat. Methods of reducing eddy currents
- Holes or slits made in the plate reduces the area available for eddy currents to flow
- Laminations – laminations of a metal are used to make a core. The laminations are separated by an insulating layer like lacquer. The plane of the laminations must be parallel to the magnetic field so that they cut across the eddy current paths. This reduces the strength of eddy currents and hence the loss of energy as heat (P=I^2 R) **Advantages/ Uses of Eddy currents
- Magnetic breaking in trains –** Strong electromagnets are situated above the rails. When activated, they produce eddy currents which oppose the motion of the train. As there is no mechanical contact, breaking is smooth 2. Electromagnetic damping- certain galvanometers have a fixed core made of nonmagnetic metallic material. When the coil oscillates, the eddy currents generated in the coil oppose its motion and quickly bring the coil to rest 3. Induction furnace- they are used to produce high temperatures to make alloys by melting the constituent metals. Ahigh frequency A/C is passed through a coil which surrounds the metal to be melted. The eddy currents induced in the metals produce high temperatures sufficient to melt it 4. Electric power meters- The shiny metal discs in the electric power meter rotates due to eddy currents. Electric currents are induced in the disc by magnetic fields produced by sinusoidally varying currents in a coil
Electromagnetic damping (PYQ 2014)
Take two cylindrical pipes of equal diameters, one of aluminum and the other of PVC. Drop a cylindrical magnet through the pipes such that it does not touch the sides of the pipes. We observe that the magnet dropped through the PVC pipe falls faster than the one dropped through the aluminum pipe. This is because as the magnet falls through the aluminum pipe, the flux associated with it changes which generates eddy currents in the pipe that oppose the motion of the magnet. Eddy currents are not generated in the PVC pipe as it is an insulator. (From 1,2)
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We now consider the reverse case. A current I 1 is flown through S 1. The flux linked with S 2 - Where M 21 is mutual inductance/ coefficient of mutual induction of S 2 w.r.t. S 1 Therefore, we see that M 12 = M 21 = M. If there was a medium of relative permeability μR in the solenoids then, Note: The mutual inductance of a pair of solenoids also depends on their separation and their relative orientation If current through one of the coils is varied with time, an emf is set up in the other coil which is calculated as- Important PYQs Ques: A pair of adjacent coils have a mutual inductance of 1.5 H. if the current in one coil changes from 0 to 20 A in 0.5s, what is the change influx linkage of the other coil? (PYQ 2016) [2M] Ans:
Self inductance (PYQ 2019, 2017 , 2013, 2012, 2010)
The phenomenon due to which an emf is induced in a coil due to varying current through the same coil is called self induction. In this case, the flux through a coil of N turns is directly proportion to the current in the coil- Where L is a proportionality constant knows as self inductance/ coefficient of self induction From 3,4. Also, (From 1,2)
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Inductance of solenoid Consider a solenoid of no of turns per unit length n, radius a, current I. we know- For a solenoid with a core of relative permeability μR - if current through the coil is varied with time an emf is induced- Note: Self induced emf is also called back emf as it opposes any change in the current in a circuit. Physically, self inductance plays the role of inertia. It is the electromagnetic analogue of mass in mechanics. Important PYQs Ques: A plot of magnetic flux versus current is shown in the figure for two inductors A and B. which of the two has a larger self-inductance (PYQ 2010) [1M] Ans: We know- Therefore, greater the slope, greater is the self inductance. i.e. A has greater self-inductance than B Ques: If number of turns of solenoid is doubled without changing the length and area of cross-section. The self- inductance of the solenoid will become___________? (PYQ 2020) [1M] Ans: Ques: Ans: Also, From 1, Also, (From 1,2) It will be doubled (Since, (PYQ 2013) [3M] i) (^) ii)
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Construction: - it consists of a coil mounted on a rotor shaft kept in a strong magnetic field. The axis of rotation of the coil is perpendicular to the direction of magnetic field.
- The coil is connected to external circuit with the help of slip rings and brushes Working-
- The coil of area A is rotated in field B, the flux through the loop will be BAcosθ where θ is the angle between field and area vector.
- As the coil rotates, θ and hence the flux through the coil changes which sets up an induced emf in the coil
- Let the coil rotate with a constant angular velocity ω. The angle between A and B at a time t is θ = ωt, assuming θ= 0° at t=0, the flux at any time t can be written as - Since value of sine function varies between +1 and – 1, the polarity of the emf also varies. Hence, direction of current also varies periodically with time therefore, current is called alternating current. Types of commercial generators- 1. Hydro-electric generators- mechanical energy required to rotate armature is provided by water falling from a height 2. Thermal generators- water is heated using coal to produce high pressure steam to rotate the armature 3. Nuclear generators- instead of coal, nuclear fuel is used. Note: Frequency of rotation of armature/ electromagnets – India = 50Hz | USA= 60Hz Important PYQs Ques: For an A/C generator, show a plot of variation of- 1. Magnetic flux 2. Alternating emf versus time generated by a loop of wire rotating in a magnetic field (PYQ 2013) [2M] Ans: For a coil rotaGng in a magneGc field with constant angular velocity t÷E
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