IB Physics D.4 Induction — Notes (HL)

What is magnetic flux?

Induction is about change in the amount of magnetic field passing through a circuit. To describe “amount of field through an area” we need a new quantity.

• \(\Phi\) = magnetic flux (Wb, weber; \(1\ \text{Wb}=1\ \text{T m}^2\))
• \(B\) = magnetic field strength (magnetic flux density) (T)
• \(A\) = area enclosed by the loop or coil (m²)
• \(\theta\) = angle between the magnetic field and the normal (perpendicular) to the area

This equation is in the data booklet. Points to note:

• Flux is greatest, \(\Phi=BA\), when the field passes straight through the area (field parallel to the normal, \(\theta=0\)).
• Flux is zero when the plane of the coil is parallel to the field (\(\theta=90^\circ\)): no field lines pass through it.
• \(A\cos\theta\) is the area of the coil projected onto a plane perpendicular to the field — the “effective area” the field lines cross.
• Flux is a scalar, but it has a sign: if the coil turns through 180°, the same field passes through it the other way and \(\Phi\) changes from \(+BA\) to \(-BA\).
• A useful picture: \(B\) is proportional to the density of field lines, so \(\Phi\) is proportional to the number of field lines passing through the area. This is why \(B\) is also called the magnetic flux density.

Flux linkage

• Each turn of the coil has the same flux \(\Phi\) through it, and an emf is induced in every turn. The turns are in series, so their emfs add — hence the factor \(N\).
• The unit of flux linkage is the weber (some books write Wb turns). Flux linkage itself is not in the data booklet, but it appears inside Faraday's law as \(N\Delta\Phi\).

What does Faraday's law of induction say?

• \(\varepsilon\) = induced emf (V)
• \(N\) = number of turns of the coil
• \(\Delta\Phi\) = change in magnetic flux through one turn (Wb)
• \(\Delta t\) = time taken for the change (s)

• This is in the data booklet. The minus sign expresses Lenz's law (see below): the emf acts to oppose the change. When a question asks only for the size of the emf, you may ignore it.
• It is the rate of change of flux that matters. A coil sitting in a strong, steady field has a large flux linkage but no emf.
• \(\dfrac{\Delta\Phi}{\Delta t}\) over a finite interval gives the average emf. The instantaneous emf is \(-N\) times the gradient of the tangent to a graph of \(\Phi\) against \(t\) (or minus the gradient of a graph of \(N\Phi\) against \(t\)).
• Units check: \(1\ \text{Wb s}^{-1}=1\ \text{T m}^2\,\text{s}^{-1}=1\ \text{V}\).
• An emf is induced whether or not the circuit is complete. A current flows only if there is a closed circuit; its size is then given by the emf and the total resistance, as in B.5.

Three ways to change the flux

Since \(\Phi=BA\cos\theta\), the flux linkage \(N\Phi\) changes if any of \(B\), \(A\) or \(\theta\) changes. Examples of induction fall into these groups:

Only relative motion matters: moving a magnet towards a stationary coil induces the same emf as moving the coil towards the stationary magnet at the same speed.

How is an emf induced in a moving straight conductor?

• \(\varepsilon\) = induced emf (V)
• \(B\) = magnetic field strength (T)
• \(v\) = speed of the conductor, perpendicular to the field (m s⁻¹)
• \(L\) = length of the conductor in the field (m), perpendicular to both \(B\) and \(v\)

This is in the data booklet. You can derive it in two ways, and both are acceptable.

Derivation 1 — from the force on the charges (D.3)

• The free electrons in the rod move with the rod at speed \(v\) through the field, so each has a magnetic force \(evB\) along the rod (\(F=qvB\), D.3).
• Electrons pile up at one end, leaving the other end positive. This separation of charge sets up an electric field \(E\) along the rod.
• Charge stops building up when the electric force balances the magnetic force: \(eE=evB\), so \(E=vB\).
• In a uniform field \(E=\dfrac{\varepsilon}{L}\), so \(\varepsilon=EL=BvL\).

Derivation 2 — from Faraday's law

• In a time \(\Delta t\) the rod moves \(v\Delta t\) and sweeps out an area \(\Delta A=Lv\Delta t\).
• The flux through the circuit changes by \(\Delta\Phi=B\Delta A=BLv\Delta t\) (field perpendicular to the area, \(\cos\theta=1\)).
• For a single conductor (\(N=1\)): \(\varepsilon=\dfrac{\Delta\Phi}{\Delta t}=BvL\). The emf is the rate at which the rod “cuts” flux.

The emf exists whether or not the rod is part of a circuit. If the ends are connected through a resistor — for example by sliding the rod along two conducting rails — a current \(I=\dfrac{\varepsilon}{R}\) flows. Then the rod is a current-carrying conductor in a field, and it feels a force \(F=BIL\) (D.3) that opposes its motion.

Direction. In the diagram, positive charges in the rod would be pushed towards the top rail (\(F=qvB\) with \(v\) to the right and \(B\) into the page), so the current flows up through the rod and anticlockwise round the circuit. Lenz's law gives the same answer: the flux into the page is increasing, so the induced current makes its own field out of the page inside the loop. Check with the force: a current flowing up the rod in a field into the page feels a force to the left, against the motion — exactly as Lenz's law requires.

What determines the direction of the induced emf?

• The induced current opposes the change in flux, not the flux itself. If the flux through a coil is increasing, the induced current's field points the opposite way to the external field inside the coil; if the flux is decreasing, the induced field points the same way, trying to keep the flux up.
• In terms of forces: the induced current always produces a force that opposes the relative motion. If the coil is part of a closed circuit, a magnet approaching it is repelled and a magnet moving away is attracted back.
• Lenz's law is the reason for the minus sign in \(\varepsilon=-N\dfrac{\Delta\Phi}{\Delta t}\).

How to find the direction in three steps:

1. Decide whether the flux through the circuit, in the direction of the external field, is increasing or decreasing.
2. The induced field inside the circuit must point against the external field if the flux is increasing, and with it if the flux is decreasing.
3. Use the right-hand grip rule (D.2) to turn the direction of that induced field into a current direction.

Why Lenz's law is a consequence of energy conservation

Suppose the induced current helped the change instead of opposing it. A magnet pushed gently towards a coil would be pulled in faster, increasing the rate of change of flux, which would increase the current, which would pull harder… The magnet would gain kinetic energy and the circuit would gain electrical energy, both from nothing. That breaks the conservation of energy.

Because the induced current opposes the change, an external agent must do work against the opposing force to keep the change going. That work is exactly the electrical energy delivered to the circuit (and dissipated in its resistance). In worked example 3, the 0.060 W supplied by the pulling force is the 0.060 W dissipated in the resistor. Without the opposition there would be no work done, and no energy to supply the current.

A magnet on a spring oscillating over a coil

This example combines time-varying flux, Lenz's law and energy. A magnet hangs on a spring with one pole just above a coil. As it oscillates, the flux through the coil rises and falls, so an alternating emf is induced.

What happens when a coil moves into and out of a field?

A rectangular coil moving at constant speed into a region of uniform field is the IB's standard quantitative example. Treat each stage separately.

• Entering: the area of coil inside the field grows at a rate \(Lv\), where \(L\) is the side perpendicular to the motion. Only the leading side cuts the field. Each turn has emf \(BvL\), so the coil has \(\varepsilon=NBvL\). The flux into the page increases, so the induced current is anticlockwise (seen as drawn).
• Fully inside: the flux linkage is constant, so the emf of the coil is zero. (The two sides perpendicular to the motion do each have an emf \(BvL\), but they act in opposite directions around the loop and cancel.) No current, no opposing force.
• Leaving: the flux decreases at the same rate, so the emf has the same size and opposite sign; the current is clockwise. Only the trailing side is now in the field.
• While there is a current, the side in the field feels a force \(NBIL\) opposing the motion — whether the coil is entering or leaving. To keep the speed constant, an external force must do work, which is dissipated in the coil's resistance.
• If the coil is simply released, this opposing force slows it down on the way in and on the way out. This is electromagnetic braking.

How does a rotating coil produce a sinusoidal emf?

In a simple generator (alternator), a rectangular coil of \(N\) turns and area \(A\) is turned at constant angular velocity \(\omega\) about an axis perpendicular to a uniform field \(B\). The angle between the field and the normal to the coil changes steadily, so the flux linkage changes all the time and an emf is induced. Slip rings and brushes connect the rotating coil to the external circuit.

Take \(t=0\) when the normal to the coil is parallel to the field. The angle at time \(t\) is \(\theta=\omega t\), so

• \(\varepsilon\) = instantaneous emf (V); \(\varepsilon_0\) = peak emf (V)
• \(N\) = number of turns; \(A\) = area of the coil (m²); \(B\) = field strength (T)
• \(\omega\) = angular velocity of the coil (rad s⁻¹); \(f\) = frequency of rotation (Hz), with \(\omega=2\pi f\) and \(T=\dfrac1f\) (both in the data booklet)

The expressions for \(\varepsilon\) and \(\varepsilon_0\) are not in the data booklet. They follow from Faraday's law, because the emf is minus the gradient of the cosine graph of \(N\Phi\), which is a sine graph. Without calculus you can derive the peak value from \(\varepsilon=BvL\):

• Let the sides parallel to the axis have length \(L\) and be a distance \(\tfrac12w\) from it, so \(A=Lw\). Each moves in a circle at speed \(v=\omega\cdot\tfrac12w\).
• Only the component of velocity perpendicular to \(B\) cuts the field: \(v\sin\omega t\). Each side therefore has an emf \(BLv\sin\omega t\).
• The two sides move in opposite directions, so their emfs add round the loop. For \(N\) turns: \(\varepsilon=2NBL\left(\tfrac12\omega w\right)\sin\omega t=NBA\omega\sin\omega t\).
• For the two sides perpendicular to the axis, the magnetic force on the charges acts across the wire rather than along it, so they contribute no emf.

So the flux linkage and the emf are a quarter of a cycle (\(\tfrac{\pi}{2}\) rad) out of phase: each is zero when the other is at a peak.

What happens when the frequency of rotation changes?

Increasing the frequency of rotation \(f\) has two effects on the emf, and you need to state both:

• The period decreases (\(T=1/f\)): the alternating emf completes more cycles per second, because it has the frequency of rotation.
• The peak emf increases in proportion (\(\varepsilon_0=2\pi fNBA\)): the flux linkage swings between the same limits \(\pm NBA\), but in less time, so its rate of change is greater at each corresponding point of the cycle.

The peak flux linkage does not change — only how quickly it changes. On the emf–time graph, doubling \(f\) doubles the height of the curve and halves its period.

How can the power output of a generator be increased?

The peak emf is \(\varepsilon_0=NBA\omega\), so a generator gives a larger emf, and delivers more power to a given load, if it has:

• more turns \(N\) on the coil;
• a stronger magnetic field \(B\) (stronger magnets, or an electromagnet; winding the coil on an iron core concentrates the field);
• a larger coil area \(A\);
• a higher frequency of rotation \(f\).

For a resistive load \(R\), the power at the peak of the cycle is \(\dfrac{\varepsilon_0^2}{R}\), so doubling \(f\) multiplies it by four. That energy is not free: the larger current produces a larger opposing torque on the coil (Lenz's law), so whatever turns the generator — a turbine, an engine, or your hand — must do work faster.

Exam tips for D.4

Common misconceptions

How this topic connects

From fuel to grid: why overall efficiency depends on the energy source

Almost every power station ends in the same device: a turbine turning a generator, which uses electromagnetic induction to transfer kinetic energy of rotation to electrical energy. The generator itself is very efficient — its losses are mainly resistive heating in the coils and friction in the bearings. The big differences come before the turbine. Fossil-fuel and nuclear stations are heat engines: they heat water to drive a steam turbine, and the second law of thermodynamics (B.4) requires a large fraction of the energy to be rejected as thermal energy to the surroundings. Their overall efficiency is therefore limited — typically about a third for older coal and nuclear stations, and up to about 60 % for the best combined-cycle gas stations. Hydroelectric and wind turbines take kinetic energy directly from moving water or air, with no heating stage, so this thermodynamic limit does not apply; hydroelectric stations are among the most efficient of all. A wind turbine cannot extract all the kinetic energy of the wind, because the air must keep moving away from the blades. Photovoltaic cells (not part of D.4) generate electricity with no generator at all.

Faraday's law and the many rates of change in physics

Many laws in physics are statements about how fast something changes: velocity and acceleration are rates of change of displacement and velocity (A.1); force is the rate of change of momentum (A.2); power is the rate of doing work (A.3); torque is the rate of change of angular momentum (A.4); current is the rate of flow of charge (B.5); activity is the rate of radioactive decay (E.3). In each case the same graphical tools apply: the gradient of a graph of the quantity against time gives the rate, a tangent gives the instantaneous value, and the area under the rate–time graph gives the total change — just as the area under an emf–time graph gives the size of the change in flux linkage. Recognising this shared mathematical structure is part of how physicists build general models from particular laws.

Where motors and generators lose energy, and how design reduces it

(This connection also appears in B.4 Thermodynamics.)

A generator transfers mechanical work to electrical energy and a motor does the reverse; both use a coil turning in a magnetic field. In each, some energy is dissipated: resistive heating (\(I^2R\)) in the coil windings, and friction in the bearings and brushes. These losses heat the machine, waste energy and limit how much current the coil can safely carry. Engineers raise efficiency by using thick, low-resistance windings, good bearings and strong fields, so that the required emf or torque is reached with a smaller current. Lenz's law sets the background: the electrical energy delivered by a generator can never exceed the work done turning it.

Key equations

Check your understanding

← Previous: D.3 Motion in electromagnetic fields  ·  Next: E.1 Structure of the atom →