7.9 Electromagnetic Induction

 A metal detector is moved horizontally at a constant speed just above the Earth’s surface to search for buried metal objects 

Figure 1 shows the coil P of a metal detector moving over a circular ring made from a single band of metal. The planes of the coil and the ring are both horizontal. 

Figure 1

In this metal detector, P carries a direct current so that the magnetic flux produced by P does not vary. The ring is just below the surface, so the flux is perpendicular to the plane of the ring. The field is negligible outside the shaded region of P. 

Figure 2 shows how the magnetic flux through the ring varies with time when P is moving at a constant velocity.

Figure 2

Sketch a graph on the grid to show how the e.m.f. induced in the ring varies with time as P moves across the ring. 

Use the same scale on the time axis as in Figure 2.

Use the laws of Faraday and Lenz to explain the shape of your graph.

P moves at a velocity of 0.3 m s⁻¹. 

Show that the diameter of the ring is about 3 cm.

Determine: 

A guitarist and a physicist collaborate on a project to understand what happens when a guitar string is plucked.

The physicist wants to model a guitar string using a metal conductor PQ of length L = 1.5 cm. It is placed in a uniform magnetic field of flux density B, and they consider a scenario in which the conductor is displaced by a small distance Δx perpendicularly to the field at a constant velocity v. This is shown in Figure 1:

Figure 1

Show that the magnitude of the induced emf  is given by the expression: 

                 = BLv

The physicist then connects PQ to an extended piece of metal string, such that it can be connected to a cathode ray oscilloscope (c.r.o.) as shown in Figure 2. 

Figure 2

The guitarist plucks the string such that it vibrates at its fundamental frequency f in the magnetic field, with a displacement x at time t given by: 

            x =  cos ωt 

where  is the amplitude and ω is the angular frequency of the vibration respectively. 

Deduce an equation for the magnitude of the maximum emf induced in PQ.

The trace on the c.r.o. is shown in Figure 3. 

It is set up such that the y-sensitivity is 1.5 mV cm^–1 and the time base is 5.0 ms cm^–1. 

Figure 3

Given the amplitude of vibration is 2.8 cm, and using Figure 3, calculate the magnitude of the magnetic flux density B.

A resistor is now connected across RS in Figure 2. 

Discuss how the trace on the c.r.o. would be affected as a result of a resistor being connected across RS.

As part of an introductory lesson to electromagnetic induction, a physics teacher plans a series of experiments for her students to carry out. 

In experiment 1, two metal rings A and B are dropped from the same height between two magnets, as shown in Figure 1. They are identical, apart from a small slit cut in B as shown. 

Figure 1

The teacher asks her students to observe the motion of the two metal rings as they fall between the magnets. 

Describe and explain the motion of A and B as they fall between the magnets.           

You may wish to include sketches in your answer. 

Experiment 2 is designed as a thought experiment. The physics teacher displays a sketch of the variation of induced emf  in a coil with time. This is shown in Figure 2a. 

Figure 2a

The students are asked to use discuss the properties of the graph shown in Figure 2 to determine how it might be reproduced. 

Use the axes provided in Figure 2b to sketch a graph of the magnetic flux linkage Nϕ through the coil between t = 0 to t = .

Figure 2b

The physics teacher reveals that the graph shown in Figure 2a was produced by a coil moving at a constant speed into and out of a uniform magnetic field. They draw a sketch of this motion in three stages, as shown in Figure 3: 

Figure 3

A student disagrees, saying that while the coil is moving across the magnetic field between  to  as shown in Figure 3, magnetic field lines are being cut, so there must be an induced emf in the coil. 

Suggest how the teacher should explain what happens within the coil during time   to to improve the student’s understanding.

Experiment 3 gets students to model an airplane flying parallel to the Earth’s surface at a constant speed v. 

The wingspan of the plane is modelled by a thin metallic rod of length L = 75.0 cm which is connected to a voltmeter. 

Given that the Earth’s magnetic field is at 70° to the vertical and has a flux density of 1.8 × 10^–4 T, calculate the speed a student would have to run at in order to generate an emf of 0.15 mV across the ends of the metallic rod.

Figure 1 shows a simple induction motor. An alternating current  is supplied to a stationary coil (stator), which is wrapped around an iron magnet. 

A rotating metal square of length 5.5 cm (rotor) is shown end on in Figure 1. 

Figure 1

Explain how current is induced in the rotor coil.

Sometimes, induction motors are used to rotate the turntable on record decks. 

The induction motor in Figure 1 can generate a maximum emf of 6 V when the magnetic flux density is 2.4 T. 

Show that approximately 130 revolutions per second is required to generate the maximum emf.

In more sophisticated analyses of induction motors, the ‘back emf’ generated in the rotor must be considered. 

With reference to the laws of electromagnetic induction, suggest how a back emf is generated in the rotor and describe its implications on the operation of the turntable.

A coil is connected to a zero-centred ammeter, as shown in Figure 1. A student drops a magnet so that it falls vertically and completely through the coil. 

Figure 1

Describe what the student would observe on the ammeter as the magnet falls through the coil, and explain how this demonstrates Lenz’s law.

Next, the student wants to investigate the effects of a magnet falling through a brass tube. 

Figure 2 shows two small, solid metal cylinders, Y and Z. Y is made from silver. Z is made from a steel alloy and is a strong permanent magnet. 

Figure 2

Y and Z are released separately from the top of a long, vertical brass tube so that they pass down the centre of the tube, as shown in Figure 3. 

Figure 3

Explain why you would expect an e.m.f. to be induced in the tube as Z passes through it.

The time taken for Z to pass through the tube is much longer than that taken by Y. 

State the consequences of this induced e.m.f., and hence explain why Ztakes longer than Y to pass through the tube.

The brass tube is replaced by a tube of the same dimensions made from copper. The resistivity of copper is much lower than that of brass. 

Describe and explain how, if at all, the times taken by Yand Z to pass through the tube would be affected.

A third solid metal cylinder X is used in the experiment. X is found to be a much stronger magnet than Z.

Using Faraday’s law, explain why a larger e.m.f. is induced when X passes through the brass tube.

A metal detector is moved horizontally at a constant speed just above the Earth’s surface to search for buried metal objects 

Figure 1 shows the coil P of a metal detector moving over a circular ring of diameter 3.0 cm and made from a single band of metal. The planes of the coil and the ring are both horizontal. 

Figure 1

In this metal detector, P carries a direct current so that the magnetic flux produced by P does not vary. The ring is just below the surface, so the flux is perpendicular to the plane of the ring. The field is negligible outside the shaded region of P. 

Figure 2 shows how the induced emf in the ring varies with time when P is moving at a constant velocity.

Figure 2

State the value of the total area between the line and the axis in Figure 2 and explain what it represents. 

Calculate the magnitude of the flux through the ring when the field strength due to P is 1.3 T. 

Give an appropriate unit with your answer.

Describe and explain the effect on the emf induced in the ring if the velocity at which P moves increases. 

The metal detector P is replaced with a rectangular metal detector Q and the process repeated as shown in Figure 1. The planes of the coil and the ring remain horizontal. 

Figure 3 shows how the induced emf in the ring varies with time when Q is moving at a constant velocity. 

Figure 3

Show that the magnitude of the change in flux through the ring in the first 1.1 × 10^–1 s is equal to 0.03 Wb

Figure 1 shows a dynamo torch, a device which is operated by successive squeezes of the handle. 

Figure 1

Squeezing the handle causes a permanent magnet to rotate within a fixed coil of wires. The harder the handle is squeezed, the faster the magnet rotates. 

Using Faraday’s law:

• Explain how the torch works
• Discuss factors which affect the brightness of the bulb 

Sketch on the diagram below to help you in your answer.

The circular coil in the dynamo torch has a diameter of 150 mm and contains 50 turns. It is placed so that its plane is perpendicular to a horizontal magnetic field of uniform flux density 25 mT. 

Calculate the magnetic flux passing through the coil when in this position.

The magnet is rotated through 90° about a vertical axis in a time of 15 ms. 

Calculate the change of magnetic flux linkage produced by this rotation.

Calculate the average e.m.f. induced in the coil when the magnet is rotated.

 Figure 1 illustrates the main components of one type of electromagnetic braking system. A metal disc is attached to the rotating axle of a vehicle. An electromagnet is mounted with its pole pieces placed either side of the rotating disc, but not touching it. 

When the brakes are applied, a direct current is passed through the coil of the electromagnet and the disc slows down. 

Figure 1

Explain, using the laws of electromagnetic induction, how the device in Figure 1 acts as an electromagnetic brake.

A conventional braking system has friction pads that are brought into contact with a moving metal surface when the vehicle is to be slowed down. 

State one advantage and one disadvantage of an electromagnetic brake compared to a conventional brake.

Figure 2 shows an arrangement of the coils in a different device which uses electromagnetic induction. 

Figure 2

When the switch is closed there is a current in the coil in circuit P. The current is in a clockwise direction as viewed from position O. 

Circuit Q is viewed from position O. 

Explain how Lenz’s law predicts the direction of the induced current when the switch is opened and again when it is closed.

An Earth inductor is a device used to measure the horizontal component of the Earth’s magnetic field. 

Figure 3 shows the setup of the device. When the coil is rotated an e.m.f. is induced. 

Figure 3

The Earth inductor consists of a 750 turn coil and the mean diameter of the turns on the coil is 55 cm. Figure 4 shows the output recorded for the variation of potential difference V with time t when the coil is rotated at 1.5 revolutions per second.

Figure 4

Determine the flux density, , of the horizontal component of the Earth’s magnetic field.