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Faraday's law and Lenz's law - TCE Physics (Tasmania)

Syllabus dot point

“Apply Faraday's law and Lenz's law to determine the magnitude and direction of induced EMF.”

TCEPhysicsUnit 3: Gravity and Electromagnetism9 min read

Quick answer

A changing magnetic flux through a coil induces an EMF. Faraday's law gives its size, ε=−NΔΦΔt\varepsilon = -N\dfrac{\Delta\Phi}{\Delta t}, proportional to the number of turns and the rate of flux change. Lenz's law gives its direction: the induced current always opposes the change that caused it, which is just conservation of energy.

What this dot point is asking

This dot point is the quantitative heart of electromagnetic induction: how to find both the magnitude and the direction of an induced voltage.

Magnetic flux

Magnetic flux measures how much magnetic field passes through an area:

Φ=BAcos⁡θ\Phi = BA\cos\theta

where BB is the field strength, AA is the area of the loop, and θ\theta is the angle between the field and the normal to the loop. Flux is measured in webers. Flux is largest when the field is perpendicular to the loop face and zero when the field lies in the plane of the loop. Induction depends on flux changing, which can happen by changing the field strength, the area, or the angle.

Magnetic flux

Magnetic flux is the product of the magnetic field strength and the area it passes through perpendicularly. It represents the total amount of magnetic field threading a surface and is measured in webers.

Faraday's law

Faraday found that the induced EMF equals the rate of change of flux linkage, where flux linkage is the flux multiplied by the number of turns NN:

ε=−NΔΦΔt\varepsilon = -N\frac{\Delta\Phi}{\Delta t}

So a faster change, a stronger field change, a larger area, or more turns all increase the induced EMF. If the flux is steady, no EMF is induced, which is why a stationary magnet in a coil produces nothing. On a flux-time graph the EMF is proportional to the gradient, so a steep section of the graph corresponds to a large EMF.

Lenz's law and the negative sign

The minus sign in Faraday's law is Lenz's law: the induced current flows in the direction that opposes the change in flux that produced it. If the flux through a coil is increasing, the induced current creates its own field to oppose the increase; if decreasing, it tries to maintain the flux.

This is conservation of energy in action. The opposing current means you must do work to push a magnet into a coil, and that work is what becomes electrical energy. If the induced current aided the change instead, it would create energy from nothing.

Finding the direction step by step

A reliable method for any TCE direction question is: (1) decide whether the flux through the coil is increasing or decreasing and in which direction; (2) state, by Lenz's law, that the induced current must oppose that change; (3) work out which face of the coil must become a north or south pole to provide the opposition; (4) apply the right-hand grip rule to the windings to read off the current direction, and hence the direction through any external load. Eddy currents in a solid conductor follow the same rule, which is why a magnet dropped through a copper pipe falls slowly: the induced currents oppose its motion.

Lenz's law is conservation of energy

The induced current always opposes the change causing it because the alternative would let energy appear from nowhere. The mechanical work done against the opposing force is exactly the electrical energy generated, so the books balance.

Worked example: EMF from a changing field

A coil of 200200 turns and area 0.015 m20.015\ \text{m}^2 sits in a uniform magnetic field perpendicular to the coil. The field drops steadily from 0.40 T0.40\ \text{T} to 0.10 T0.10\ \text{T} in 0.20 s0.20\ \text{s}. Find the average induced EMF.

Step 1: Find the change in flux through one turn

Because the field is perpendicular to the coil face, the angle between the field and the normal to the loop is zero, so cos⁡θ=1\cos\theta = 1 and Φ=BA\Phi = BA. We only need the change in BB, since the area stays constant:

ΔΦ=(ΔB)A=(0.10−0.40)(0.015)=−4.5×10−3 Wb.\Delta\Phi = (\Delta B)A = (0.10 - 0.40)(0.015) = -4.5\times10^{-3}\ \text{Wb}.

The negative sign confirms the flux is decreasing, which is consistent with Lenz's law predicting an induced current that will try to maintain the original flux.

Step 2: Apply Faraday's law across all turns

Faraday's law multiplies the single-turn flux change by the number of turns NN because each turn contributes its own EMF and the turns are connected in series, so the contributions add. Taking the magnitude gives the size of the induced EMF:

∣ε∣=N∣ΔΦ∣Δt=(200)4.5×10−30.20=4.5 V.|\varepsilon| = N\frac{|\Delta\Phi|}{\Delta t} = (200)\frac{4.5\times10^{-3}}{0.20} = 4.5\ \text{V}.

Final answer: The average induced EMF is 4.5 V4.5\ \text{V}.

Forgetting that steady flux gives no EMF

A frequent error is thinking a strong but constant magnetic field induces a voltage. Only a changing flux induces an EMF. A magnet held still inside a coil, however strong, produces nothing; the EMF appears only while the flux is changing, as the magnet moves in or out.

In the exam, first work out the flux and what is changing about it, then apply Faraday's law for the size. Use Lenz's law separately for the direction: state which way the flux is changing, then choose the induced current direction that opposes that change. A clear sentence linking Lenz's law to energy conservation often earns the explanation marks.

Exam-style questions

Questions in the style of TASC exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.

Practice3 marks
In a science museum an aluminium plate is levitated over a large coil carrying 800 A800\ \text{A} of alternating current changing at 800 Hz800\ \text{Hz}. Explain why the aluminium plate levitates, with reference to the relevant laws.
Show worked answer →

The alternating current produces a magnetic field that is constantly changing in both magnitude and direction at 800 Hz800\ \text{Hz}.

Faraday's law: this changing flux through the aluminium plate induces an EMF, which drives circulating eddy currents in the conducting plate.

Lenz's law: the induced eddy currents flow in the direction that opposes the change in flux that created them. The plate therefore behaves like a magnet whose poles oppose the coil's field, so the plate and coil repel each other.

Because the current alternates, this repulsion is sustained on average over each cycle and pushes the plate upward. When the repulsive force balances the plate's weight, it levitates. Markers want: changing flux (Faraday), induced eddy currents, Lenz's law opposition producing a repulsive force that supports the weight.

Practice2 marks
A magneto is a simple generator. A magnet spins anticlockwise between the poles of a laminated iron horseshoe with a coil connected to a load XY. The north pole is entering the horseshoe at A. Justify the direction of the induced current through the load resistor.
Show worked answer →

As the magnet rotates, the magnetic flux threading the coil around the horseshoe changes. With the north pole entering at A, the flux through the coil in that direction is increasing.

Faraday's law says the changing flux induces an EMF; Lenz's law says the induced current flows so as to oppose the change. To oppose the increasing flux from the approaching north pole, the coil's near face must become a north pole to push back against it.

Applying the right-hand grip rule to the coil windings to make that face a north pole fixes the direction the current must flow through the windings, and hence through the load XY (the direction that opposes the rising flux). Markers want the link: increasing flux, Lenz's law opposition, induced field set up to oppose it, then the grip rule giving the current direction through the load.

Practice3 marks
A coil of 150150 turns and area 4.0×10−3 m24.0 \times 10^{-3}\ \text{m}^2 lies with its plane perpendicular to a field that falls steadily from 0.50 T0.50\ \text{T} to zero in 0.25 s0.25\ \text{s}. Calculate the magnitude of the average induced EMF.
Show worked answer →

Find the change in flux per turn. The plane is perpendicular to the field, so θ=0\theta = 0 and Φ=BA\Phi = BA:

ΔΦ=(ΔB)A=(0−0.50)(4.0×10−3)=−2.0×10−3 Wb.\Delta\Phi = (\Delta B)A = (0 - 0.50)(4.0\times10^{-3}) = -2.0 \times 10^{-3}\ \text{Wb}.

Apply Faraday's law (magnitude):

∣ε∣=N∣ΔΦ∣Δt=(150)2.0×10−30.25=1.2 V.|\varepsilon| = N\frac{|\Delta\Phi|}{\Delta t} = (150)\frac{2.0\times10^{-3}}{0.25} = 1.2\ \text{V}.

The average induced EMF is 1.2 V1.2\ \text{V}. Markers want ΔΦ=(ΔB)A\Delta\Phi = (\Delta B)A and ∣ε∣=N∣ΔΦ∣Δt|\varepsilon| = N\dfrac{|\Delta\Phi|}{\Delta t}.

Practise this

Sources & how we know this

Based on the official TASC course document. Not yet individually reviewed by a person; always check TASC and your school's materials for current rules.

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