D.4 Induction (HL)

A square coil of 50 turns and side 4.0 cm is in a uniform magnetic field of flux density 0.20 T. The normal to the plane of the coil makes an angle of 60° with the field. The coil is turned until its plane is parallel to the field.

What is the magnitude of the change in flux linkage NΦ?

A coil on a laboratory carriage is moved into, through and out of a region of uniform magnetic field. The graph shows how the flux linkage NΦ of the coil varies with time t.

Which row gives the magnitude of the average emf induced in the coil between 0 and 0.20 s and between 0.30 s and 0.40 s?

A flat coil rotates at a constant frequency in a uniform magnetic field. The maximum magnetic flux linkage of the coil is NΦ_max and the peak emf induced in it is ε₀.

The frequency of rotation is doubled. Which row gives the new maximum flux linkage and the new peak emf?

The antenna of a contactless payment card is a flat rectangular loop of area 3.0 × 10⁻³ m². Near a card reader there is a uniform magnetic field of flux density 2.0 mT. The field makes an angle of 60° with the plane of the loop. What is the magnetic flux through the loop?

Which combination of units is equivalent to the weber, the unit of magnetic flux?

A flat coil of N turns, each of area A, lies in a uniform magnetic field of flux density B. At first the normal to the coil is parallel to the field. The coil is then rotated through 180° about an axis perpendicular to the field. What is the magnitude of the change in flux linkage?

The rectangular coil of a wind-turbine generator rotates at constant angular speed in a uniform magnetic field. At the instant when the magnitude of the induced emf is greatest, the plane of the coil is

A rectangular coil of a hand-cranked generator in a library display rotates in a uniform magnetic field. The crank is turned twice as fast. All other quantities are unchanged. The peak emf induced in the coil

A rectangular coil of 50 turns and area 0.020 m² is placed in a uniform magnetic field of 0.30 T. The normal to the coil makes an angle of 60° with the field. The magnetic flux linkage of the coil is

The south pole of a bar magnet is moved towards the face of a flat circular coil that is connected to a resistor. Which row gives the direction of the induced current, as seen by an observer looking at the coil from the side of the magnet, and the force that the coil exerts on the magnet?

A flat coil of area 200 cm² is in a uniform magnetic field of magnitude 0.50 T. The normal to the coil makes an angle of 60° with the field. What is the magnetic flux through the coil?

A straight metal rod of length 0.40 m moves at a constant speed of 3.0 m s⁻¹ in a uniform magnetic field of flux density 0.25 T. The rod, its velocity and the field are mutually perpendicular. What is the emf induced between the ends of the rod?

A bar magnet is released from rest above a coil and falls vertically along the axis of the coil, passing completely through it. The coil is connected to a data logger that records the induced emf. The magnet is moving faster as it leaves the coil than when it entered. Compared with the emf peak recorded as the magnet enters, the emf peak recorded as it leaves is

A conducting rod is pulled at constant speed v along two parallel horizontal rails that are joined by a resistor of resistance R. A uniform vertical magnetic field acts on the rod. The resistances of the rod and the rails are negligible.

The speed of the rod is doubled and the resistor is replaced by one of resistance 2R. What is (new electrical power dissipated)/(original electrical power dissipated)?

A rectangular coil rotates at constant angular speed in a uniform magnetic field. The graph shows how the magnetic flux linkage NΦ of the coil varies with time t.

What is the peak emf induced in the coil?

A simple generator coil rotates in a uniform magnetic field and produces a sinusoidal emf of peak value ε₀.

The magnetic field strength is doubled and the angular speed of rotation is halved. What is the new peak emf?

A bar magnet hangs from a spring and oscillates vertically with frequency f above a fixed coil. The coil is connected in series with a resistor and a switch.

Which statement is correct?

A yacht has a vertical aluminium mast 16 m tall. In the harbour the Earth's magnetic field has a horizontal component of 1.8 × 10⁻⁵ T directed due north and a vertical component of 4.5 × 10⁻⁵ T directed downwards. The yacht sails at a constant speed of 5.0 m s⁻¹.

Which row gives the magnitude of the emf induced between the top and the bottom of the mast when the yacht sails due east and when it sails due north?

Two identical lamps P and Q are connected in parallel branches across a battery through a single switch. P is in series with a resistor; Q is in series with a coil of many turns wound on an iron core. The resistor and the coil have equal resistance. The switch is closed.

Which statement describes and explains what is observed?

The bob of a pendulum is a flat copper plate that swings in its own vertical plane. The plate swings through the narrow gap between the poles of a strong magnet, with the magnetic field perpendicular to the plate. The field occupies only a small region around the lowest point of the swing. The oscillations die away quickly.

I. The kinetic energy lost by the pendulum is transferred to internal energy of the plate.

II. If the uniform magnetic field extended over the whole region through which the plate swings, the oscillations would die away just as quickly.

III. With a stronger magnet, the oscillations would die away more quickly.

Which statements are correct?

An electric bicycle uses regenerative braking. When the rider brakes, the hub motor acts as a generator and charges the battery. Which statement best explains why the bicycle slows down?

A flat square coil rotates at a constant rate in a uniform magnetic field and produces a sinusoidal emf. Which single change doubles the peak emf but leaves the period of the emf unchanged?

A circuit contains a battery, a switch and a coil with many turns wound on an iron core. When the switch is opened, a spark jumps across the switch contacts. What is the best explanation?

A coil rotates at a constant rate in a uniform magnetic field. At one instant the plane of the coil is parallel to the magnetic field lines. What are the flux linkage and the induced emf at this instant?

An aircraft with a wingspan of 40 m flies horizontally, with its wings level, at 250 m s⁻¹. The Earth's magnetic field there has magnitude 5.0 × 10⁻⁵ T and points at 53° below the horizontal (sin 53° ≈ 0.80, cos 53° ≈ 0.60). What is the magnitude of the emf induced between the wing tips?

A small, strong cylindrical magnet is released from rest at the top of a long vertical copper tube. It soon falls at a constant terminal speed. Air resistance is negligible. Which statement is correct while the magnet falls at terminal speed?

A maglev test vehicle travels horizontally at 120 m s⁻¹. A straight metal antenna rod of length 2.0 m is mounted horizontally on the vehicle, perpendicular to the direction of travel. The vertical component of the Earth's magnetic field is 5.0 × 10⁻⁵ T. What is the emf between the ends of the rod?

A bar magnet hangs from a spring with its north pole pointing down. It oscillates vertically with frequency f above a fixed flat coil whose axis is vertical. The magnet never enters the coil. Which statement about the emf induced in the coil is correct?

A simple ac generator is connected to a fixed resistor. The internal resistance of the generator is negligible. The rotation frequency of the coil is doubled while the magnetic field is unchanged. By what factors do the peak emf and the mean power in the resistor change?

A metal axle of length 0.80 m belongs to a library book trolley rolling at 1.5 m s⁻¹ along a level floor. The axle is horizontal and perpendicular to the trolley's direction of motion. The vertical component of the Earth's magnetic field is 50 μT. The potential difference induced between the ends of the axle is

A strong magnet is dropped down a long vertical aluminium tube and takes much longer to emerge than an identical magnet dropped through a plastic tube of the same size. The best explanation is that

A coil in a wind-turbine generator produces a sinusoidal emf with a peak value of 200 V. Its resistance is negligible. It is connected across a 40 Ω resistor. The mean power dissipated in the resistor is

A generator coil starts with its plane perpendicular to a uniform magnetic field and rotates at constant speed with period \(T\). The emf has peak value \(\varepsilon_0\). What is the first time at which the emf equals \(\frac12\varepsilon_0\)?

An electromagnetic flowmeter measures the speed of blood in a vessel of internal diameter 4.0 mm. A uniform magnetic field of 0.080 T is applied perpendicular to the vessel, and electrodes on opposite sides of the vessel record a potential difference of 0.24 mV. What is the speed of the blood?

A wind-turbine generator contains a coil rotating at a constant frequency of 40 Hz in a uniform magnetic field, producing a peak emf of 15 V. The turbine speeds up so that the coil rotates at 60 Hz. The coil and field are unchanged. What are the new peak emf and the new period of the alternating emf?

A metal rod of length 0.50 m and negligible resistance slides at a constant speed of 2.0 m s⁻¹ along horizontal conducting rails connected through a resistor, so that the total resistance of the circuit is 0.80 Ω. A uniform vertical magnetic field of 0.40 T acts on the rod. Ignoring friction, what horizontal force must be applied to keep the rod at constant speed?

Two adjacent regions of uniform magnetic field of the same flux density B are separated by a straight boundary. The field is directed into the page on the left of the boundary and out of the page on the right. A flat square coil of N turns and side L moves to the right at constant speed v, in the plane of the page, with two of its sides parallel to the boundary. Each region is wider than the coil.

What is the magnitude of the emf induced in the coil while it is crossing the boundary?

A bar magnet is dropped along the vertical axis of a horizontal, complete copper ring and falls through it.

I. The magnet takes longer to fall through the complete ring than through an identical ring that has a narrow cut in it.

II. While the magnet approaches the ring, the magnetic force on the magnet is directed upwards.

III. While the magnet moves away below the ring, the magnetic force on the magnet is directed upwards.

Which statements are correct?

A flat coil of N turns and area A rotates with constant angular velocity ω in a uniform magnetic field of flux density B. The peak emf induced in the coil is ε₀ = NBAω.

The coil turns through 90° from the position in which its plane is perpendicular to the field. What is (magnitude of the average emf induced during this quarter turn)/ε₀?

A square conducting loop of side 0.10 m moves at constant velocity through a region of uniform magnetic field 0.30 m wide. The field is perpendicular to the plane of the loop. The loop enters the region, passes through it, and leaves completely. Which describes the emf induced in the loop?

On a linear test track, a metal bar slides without friction along two parallel conducting rails that are joined by a fixed resistor. A uniform magnetic field is perpendicular to the plane of the rails. An external force F keeps the bar moving at constant speed v, and the power dissipated in the resistor is P. The speed is changed to 2v. What are the new force and the new power?

A coil connected to a fixed resistor is flipped through 180° in a uniform magnetic field, taking time t. The experiment is repeated with the flip taking time t/2. Assume the emf changes with time in the same way in both trials. Compared with the first trial, what happens to the total charge that flows and to the total thermal energy produced in the second trial?

A square conducting loop of side L and resistance R lies fully inside a region of uniform magnetic field directed perpendicular to its plane. The loop is pulled out of the field at constant speed v. The experiment is repeated at speed 2v. By what factor does the total energy dissipated in the loop change?

A conducting tether of length \(L\) and resistance \(R\), with \(R\) proportional to \(L\), is dragged at constant speed perpendicular to a uniform magnetic field and to its own length, as part of a circuit whose resistance is entirely that of the tether. It is replaced by a tether of length \(2L\) made of the same wire. The speed and field are unchanged. Compared with the original tether, the current \(I\) in the new tether and the magnetic force \(F\) on it are

A metal rod of length 0.60 m is pivoted at one end and rotates at 5.0 revolutions per second in a plane perpendicular to a uniform magnetic field of 0.20 T. What is the emf between the ends of the rod?

A flat rectangular coil of 200 turns measures 5.0 cm × 4.0 cm. It is placed in a uniform magnetic field of flux density B with the normal to its plane at an angle θ to the field, and connected to a charge meter; the total resistance of the circuit is 25 Ω.

The coil is quickly turned until its plane is parallel to the field, and the total charge Q that flows through the meter is recorded. This is repeated for different starting angles θ.

A student investigates how the peak emf ε₀ of a small generator depends on the frequency of rotation f of its coil. The coil has 250 turns and an area of 1.20 × 10⁻³ m², and it rotates in a uniform magnetic field of flux density B. The peak emf is read from a data-logger trace, and each value has an absolute uncertainty of ±0.10 V. The frequency is measured with a stroboscope; its uncertainty is negligible.

Theory gives ε₀ = 2πfNBA, where N is the number of turns and A the area of the coil. The graph shows the data with their error bars.

A small generator consists of a rectangular coil of 500 turns and area 2.0 × 10⁻³ m², rotating at a constant rate in a uniform magnetic field of flux density 0.050 T. The coil has a resistance r and is connected to a 30 Ω resistor. A data logger connected across the resistor records the potential difference V across it; the trace is shown.

The peak emf of a coil of N turns and area A rotating with frequency f in a field B is ε₀ = 2πfNBA.

A bar magnet is released from rest above a vertical coil that is connected to a data logger, which records the induced emf. The height h of the lower end of the magnet above the top of the coil is measured with a metre rule. The first peak of the emf, ε_peak, occurs when the magnet has fallen a further, unknown distance h₀ into the coil.

The student assumes that ε_peak = kv, where v is the speed of the magnet at that instant and k is a constant, and that the magnet falls freely until the first peak. The graph shows five of the six values of ε_peak² against h.

A strong bar magnet hangs from a spring of spring constant 12 N m⁻¹ so that one pole is just above a flat coil fixed to the bench. The magnet oscillates vertically. With the ends of the coil unconnected the amplitude decreases slowly; the ends are then connected to a resistor and the amplitude A is measured every 5 s.

With the coil unconnected, a graph of ln(A/mm) against t has a gradient of −0.017 s⁻¹ (dashed line). Assume that the damping forces from the air and from the induced current are each proportional to the speed of the magnet, so that their effects on the decay constant λ of the amplitude add.

An electric toothbrush is recharged without any electrical contacts. A coil in the charging base produces a magnetic field that passes through a 200-turn coil inside the toothbrush handle. The handle coil has a cross-sectional area of 3.0 × 10⁻⁴ m². In one part of a cycle, the magnetic flux density through the handle coil rises uniformly from 0 to 12 mT in 2.5 ms.

A hand-cranked emergency radio contains a small a.c. generator. Its rectangular coil has 150 turns and an area of 4.0 × 10⁻³ m². The coil rotates at 5.0 revolutions per second in a uniform magnetic field of flux density 0.080 T.

A free-fall ride at a theme park is stopped by magnetic braking. Copper fins attached to the car pass at high speed between rows of strong permanent magnets fixed to the tower.

At a science exhibit in a library, a student pulls a straight metal bar of length 0.45 m at a constant speed of 2.4 m s⁻¹ along two horizontal parallel conducting rails. The rails are connected by a resistor of resistance 0.60 Ω, and the bar is in contact with the rails and perpendicular to them. A uniform vertical magnetic field of 0.35 T acts throughout. The resistances of the bar and rails are negligible.

In a demonstration of energy harvesting, the north pole of a bar magnet is withdrawn from one end of a coil. The coil is connected in a complete circuit to a sensitive ammeter.

A contactless card reader in a shop contains a coil of 150 turns and cross-sectional area 4.0 cm². A magnetic field, perpendicular to the coil, increases uniformly from 0 to 0.060 T in 0.020 s and then remains steady.

A small wind-turbine generator contains a rectangular coil of 150 turns measuring 6.0 cm × 4.0 cm. The coil rotates at a constant rate in a uniform magnetic field about an axis through its centre that is parallel to its 6.0 cm sides and perpendicular to the field. The graph shows how the emf ε across the ends of the coil varies with time t.

A bar magnet of mass 0.20 kg hangs from a spring with its lower pole just above a flat coil fixed to the bench. The coil is connected to a data logger of very high resistance, so that no current flows in the coil. The magnet is pulled up, then released at t = 0. The graph shows how the displacement x of the magnet from its equilibrium position varies with time t. Air resistance is negligible.

A bar magnet is released from rest, north pole downwards, with its lower end 18 cm above the top of a vertical coil of 500 turns. The coil is connected to a data logger. The graph shows how the induced emf varies with time as the magnet falls through the coil. Air resistance is negligible.

A speed sensor on a roller-coaster car consists of a flat rectangular coil of 40 turns, 6.0 cm long in the direction of motion and 3.0 cm wide. As the car passes a checkpoint the coil moves horizontally at a constant 12 m s⁻¹ through a region of uniform magnetic field of flux density 0.25 T, directed into the page. The field region is 4.0 cm long in the direction of motion — shorter than the coil — and wider than the coil, as shown.

In transcranial magnetic stimulation (TMS), a coil held against the scalp produces a brief, intense magnetic field in the brain. In a simple model, a circular path of conducting brain tissue of radius 1.0 cm lies perpendicular to the field. The magnetic flux density through this path falls uniformly from 1.5 T to zero in 0.10 ms.

A vehicle sensor buried under a traffic-light stop line contains a coil. As a car's steel body passes over, the magnetic flux linkage in the coil changes. The table gives a simplified model of the flux linkage NΦ during one event.

t / ms: 0, 20, 40, 60, 80
NΦ / mWb: 0, 6.0, 12.0, 12.0, 0

Between the tabulated values, NΦ changes linearly with time.

A gym exercise bike uses an eddy-current brake. An aluminium flywheel spins between the poles of strong permanent magnets, and no part of the brake touches the flywheel. Tests show that the braking torque is proportional to the angular speed of the flywheel. At an angular speed of 40 rad s⁻¹ the braking torque is 3.0 N m.

A small search coil of 20 turns, each of area 4.0 cm², is used to monitor the field inside a medical MRI scanner. The uniform magnetic flux density is 1.5 T, and the normal to the plane of the coil makes an angle of 30° with the field. The coil is connected to a meter, and the total resistance of the circuit is 2.5 Ω.

An engineer measures the field between the poles of an industrial electromagnet with a 'flip coil'. The coil has 150 turns, each of area 2.0 cm², and is connected to a charge-measuring meter. The total circuit resistance is 40 Ω. The coil starts with its plane perpendicular to the field and is quickly rotated through 180°. The meter records a charge of 3.6 × 10⁻⁵ C.

A prototype generator for a small wind turbine has a coil of 200 turns, each of area 1.5 × 10⁻³ m², rotating in a uniform magnetic field. The peak emf ε₀ is measured at several rotation frequencies f:

f / Hz: 5.0, 10.0, 15.0, 20.0, 25.0
ε₀ / V: 0.94, 1.88, 2.83, 3.77, 4.71

Each metal blade of a helicopter rotor has a length of 5.0 m from the hub to the tip. The rotor turns in a horizontal plane with a frequency of 6.0 Hz. The vertical component of the Earth's magnetic field is 5.0 × 10⁻⁵ T.

A library security gate contains a rectangular coil of 240 turns, 0.10 m by 0.15 m, with its plane perpendicular to a uniform magnetic field. The field strength decreases uniformly from 0.080 T to 0.020 T in 0.150 s.

An electromagnetic flowmeter is used to measure the speed of blood in an artery. A uniform magnetic field of magnitude 0.10 T is applied perpendicular to the artery, whose internal diameter is 5.0 mm. Two electrodes touch opposite sides of the artery wall, along a line perpendicular to both the field and the flow. The blood contains positive and negative ions.

A bar magnet of mass 0.045 kg is released from rest 0.80 m above the top of a coil of 120 turns and falls with its axis vertical along the axis of the coil. Air resistance is negligible before the magnet reaches the coil. The coil is connected to a data logger.

A conducting rod of length 0.35 m slides at a constant speed of 1.8 m s⁻¹ along two horizontal parallel rails connected by a resistor of resistance 2.5 Ω. The rails and rod have negligible resistance. A uniform vertical magnetic field of magnitude 0.60 T acts over the whole circuit. Use unrounded values in later parts.

A coil of 250 turns and area 0.040 m² rotates at a constant frequency of 2.5 Hz in a uniform magnetic field of magnitude 0.30 T, as part of a small wind-turbine generator. At time t = 0 the plane of the coil is perpendicular to the field. Use unrounded values in later parts.

A rectangular loop of copper wire has a mass of 8.0 g and a total resistance of 0.020 Ω. Its horizontal sides are 0.10 m long and its vertical sides 0.25 m long. The loop is released from rest, with its plane vertical, above a deep region of uniform horizontal magnetic field that is perpendicular to the plane of the loop. The upper boundary of the field region is horizontal. The graph shows how the speed v of the loop varies with time t after its release. Air resistance is negligible.

A model electromagnetic launcher consists of two long horizontal rails 0.25 m apart, joined at one end through a switch to a battery of emf 6.0 V. A copper rod of mass 0.050 kg rests across the rails in a uniform vertical magnetic field of flux density 0.80 T. The total resistance of the circuit is 1.5 Ω and remains constant. A constant frictional force of 0.12 N acts on the rod whenever it moves. The switch is closed and the rod accelerates along the rails, as shown.

A vertical solenoid has an iron core that extends well above its top. An aluminium ring of mass 5.0 g rests on the top of the solenoid, around the core. The solenoid is connected through a switch to a d.c. supply. When the switch is closed the ring jumps vertically upwards to a height of 0.45 m, then falls back and comes to rest on the solenoid, although the current in the solenoid remains on. The graph shows how the current I in the solenoid varies with time t after the switch is closed.

An electromagnetic flowmeter measures the flow of seawater along a horizontal plastic pipe of internal diameter d = 0.050 m. The water flows due east. Coils outside the pipe produce a uniform magnetic field of flux density B, directed vertically downwards, across the pipe. Two small metal electrodes are set in the wall of the pipe at the ends of a horizontal diameter, one on the north side and one on the south side.

Seawater contains positive and negative ions. Assume that the water moves with the same speed v across the whole cross-section of the pipe.