Magnetic Flux and Changing Flux
In April 1820, Hans Christian Oersted at the University of Copenhagen deflected a compass needle 90° by placing it parallel to a current-carrying wire — showing for the first time that electric current produces a magnetic field. Faraday spent 11 years pursuing the reverse effect. His breakthrough required the concept of magnetic flux Φ = BA cos θ: only a changing flux induces an emf, which is why Oersted's steady current produced no induction — but a switching current did.
Practise this lesson
Four printable worksheets that build from the foundations up to exam-style questions — start at whatever level suits you.
A circular loop of wire sits in a uniform magnetic field pointing straight through it.
- If you rotate the loop so it faces edge-on to the field, does the amount of field passing through it increase, decrease, or stay the same?
- If you keep the loop still but strengthen the magnetic field, what happens to the flux?
- In which case would you expect an electric current to be induced?
Warm-up — what is the SI unit of magnetic flux?
Know — Flux Definition
- Magnetic flux is $\Phi = BA\cos\theta$ where $\theta$ is the angle between B and the normal to the area
- Flux is measured in webers (Wb), where 1 Wb = 1 T m²
- Flux depends on field strength, area, and orientation
Understand — Why Change Matters
- Stationary flux does not induce an emf — only changing flux does
- Flux can change by changing B, changing A, or changing angle
- The rate of change of flux determines the magnitude of the induced emf
Can Do — Calculate and Predict
- Calculate flux given B, A, and angle
- Determine how flux changes in different scenarios
- Predict whether an emf will be induced in a given situation
Core Content
Counting magnetic field lines through a loop
Magnetic flux is a measure of how much magnetic field passes through a given area. Imagine holding a hula hoop in a stream of water — the flux is like the amount of water passing through the hoop each second. If you turn the hoop edge-on, less water flows through. If you make the stream stronger, more water flows through.
$\Phi = BA\cos\theta$
Φ = magnetic flux (Wb) · B = magnetic field strength (T) · A = area of loop (m²) · θ = angle between B and the normal to the area
When the field is perpendicular to the loop (parallel to the normal), θ = 0° and cos 0° = 1, giving maximum flux Φ = BA. When the field is parallel to the loop (perpendicular to the normal), θ = 90° and cos 90° = 0, giving zero flux.
Figure 1 — Flux through a loop depends on the angle between B and the normal to the area
A circular coil of radius 4.0 cm has 80 turns. It is placed in a uniform magnetic field of 0.30 T such that the field is perpendicular to the plane of the coil.
- Part (a) — Flux perpendicular to plane.
Area $A = \pi r^2 = \pi(0.040)^2 = 5.03 \times 10^{-3}$ m²
Since B is perpendicular to the plane, $\theta = 0°$.
$\Phi = BA\cos 0° = (0.30)(5.03 \times 10^{-3})(1) = 1.5 \times 10^{-3}$ Wb - Part (b) — Flux when plane makes 40° with field.
The angle between the plane and B is 40°, so the angle between the normal and B is 90° − 40° = 50°.
$\Phi = BA\cos 50° = (1.5 \times 10^{-3})(0.643) = 9.6 \times 10^{-4}$ Wb - Part (c) — No induced emf when stationary.
Emf is induced only when magnetic flux changes. When the coil is held stationary, flux is constant — B, A, and θ do not change. With no change in flux, there is no induced emf.
Magnetic flux: $\Phi = BA\cos\theta$ (Wb = T m²). $\theta$ = angle between $\vec{B}$ and the normal to the loop. Maximum flux ($\Phi = BA$) when B perpendicular to plane ($\theta = 0°$); zero flux when B parallel to plane ($\theta = 90°$).
Pause — copy the highlighted flux formula and angle rule into your book before moving on.
A square loop of side 5.0 cm is perpendicular to a 0.40 T field. What is the magnetic flux?
The key principle behind generators and transformers
We just saw that magnetic flux $\Phi = BA\cos\theta$ measures how much field passes through a loop. That raises a question: when does that flux actually do anything electrical — when does it drive a current? This card answers it → only when flux changes; constant flux, no matter how large, induces no emf.
An emf is induced in a conductor only when the magnetic flux through it changes. A stationary loop in a constant field has constant flux — and no induced emf. But change anything — rotate the loop, move it into or out of the field, strengthen or weaken the field — and an emf appears.
Stationary flux = no emf. Changing flux = induced emf. The rate of change of flux determines how large the induced emf is — Faraday's Law (next lesson) will quantify this relationship.
There are three ways to change flux:
- Change B — vary the magnetic field strength
- Change A — alter the area of the loop (e.g., expand or compress it)
- Change θ — rotate the loop so the angle between B and the normal changes
In most practical devices, flux is changed by rotation (generators) or by changing B via another coil (transformers). The rate at which flux changes determines how large the induced emf is.
Emf is induced only when flux changes — constant flux → zero emf. Three ways: (1) change $B$, (2) change $A$, (3) change $\theta$ (rotate). Generators use rotation; transformers use changing $B$. Larger rate of flux change → larger induced emf.
Add the highlighted three methods and misconception correction to your notes before the check below.
A loop sitting stationary inside a very strong uniform magnetic field will have a large current induced in it.
Rotating a loop inside a uniform magnetic field will change the flux and can induce an emf.
Increasing the area of a loop inside a uniform field changes the magnetic flux.
According to the induction tool, electromagnetic induction occurs when…
A common source of error in flux calculations
We just saw that three things can change flux. That raises a question: when a problem gives us the angle between B and the plane (not the normal), which formula do we use? This card answers it → if plane makes angle $\alpha$ with B, the normal makes $90°-\alpha$, so $\Phi = BA\cos(90°-\alpha) = BA\sin\alpha$.
The angle θ in $\Phi = BA\cos\theta$ is the angle between the magnetic field B and the normal to the loop's surface — not the angle between B and the plane of the loop. This distinction causes many exam errors.
If a question states the plane makes angle α with the field, then the normal makes angle (90° − α) with the field.
So: $\Phi = BA\cos(90° - \alpha) = BA\sin\alpha$
Example: Plane at 30° to field → normal at 60° to field → $\Phi = BA\cos 60° = 0.5\,BA$
Always ask: "What angle does B make with the normal?" If B is perpendicular to the plane, it is parallel to the normal, so θ = 0°. If B is parallel to the plane, it is perpendicular to the normal, so θ = 90°.
$\theta$ in $\Phi = BA\cos\theta$ = angle between $\vec{B}$ and the normal (NOT the plane). B perp to plane → $\theta = 0°$, $\Phi = BA$. B parallel to plane → $\theta = 90°$, $\Phi = 0$. Plane at angle $\alpha$ to B → $\Phi = BA\sin\alpha$.
Pause — write the highlighted angle convention and the two special cases into your book before moving on.
A loop has flux 2.0 × 10−3 Wb when perpendicular to a field. When rotated so the plane makes 60° with the field, the new flux is…
Practise calculating magnetic flux with different angles
- Set B = 0.50 T, A = 50 cm², θ = 0°. Calculate the flux.
- Keep B and A constant. Rotate to θ = 60°. Calculate the new flux. By what factor did it decrease?
- Return to θ = 0°. Halve the area to 25 cm². What happens to the flux? Explain.
- Set θ = 90° (zero flux). Now increase B to 1.0 T. What is the flux? Explain why.
Identify and explain the three mechanisms
A rectangular coil is placed in a uniform magnetic field. For each scenario below, state whether flux changes and identify which quantity (B, A, or θ) changes:
- The coil is stretched sideways so its area increases.
- The coil is rotated from perpendicular to parallel alignment with the field.
- An electromagnet near the coil is switched on, increasing the field.
- The coil is slid sideways while remaining in the same uniform field.
Magnetic flux is the bridge between magnetism and electricity:
- $\Phi = BA\cos\theta$ — measures how much field passes through a loop
- Only changing flux induces an emf — stationary flux has no electrical effect
- Three ways to change flux: change B, change A, or change the angle θ
- The rate of change of flux determines the magnitude of the induced emf (Faraday's Law, next lesson)
A fresh five-question set drawn from this lesson's bank — feedback shown immediately. +5 XP per correct · +25 XP all correct
Pick your answer, then rate your confidence — that tells the system what to drill next.
ApplyBand 4(3 marks) 1. A rectangular loop of dimensions 6.0 cm × 4.0 cm is placed in a uniform magnetic field of 0.25 T. (a) Calculate the flux when the plane of the loop is perpendicular to the field. (b) The loop is rotated so that the plane makes 30° with the field. Calculate the new flux. (c) Explain whether an emf is induced if the loop remains stationary at this 30° angle.
1 mark: (a) correct flux using Φ = BA · 1 mark: (b) correct angle conversion and flux · 1 mark: (c) states constant flux → no emf
AnalyseBand 5(4 marks) 2. A loop sits in a constant magnetic field. In Case A, the loop is squeezed so its area halves in 2.0 s. In Case B, an identical loop is rotated from perpendicular to parallel to the field in 2.0 s. In which case is the average rate of change of flux larger? Justify your answer with calculations.
1 mark each: correct initial Φ for each case · 1 mark each: correct ΔΦ/Δt and comparison with reasoning
Show all answers
Multiple choice
MC answers and full explanations are shown inline as you complete each question. Use the retry button to attempt a fresh set drawn from the lesson bank.
Short Answer — Model Answers
Q1 (3 marks): (a) A = 0.060 × 0.040 = 2.4 × 10−3 m². Φ = BA = 0.25 × 2.4 × 10−3 = 6.0 × 10−4 Wb (1 mark). (b) Plane at 30° to B means normal at 60° to B. Φ = BA cos 60° = 6.0 × 10−4 × 0.500 = 3.0 × 10−4 Wb (1 mark). (c) No emf is induced. When the loop remains stationary, B, A, and θ are all constant, so flux is constant. Emf requires a change in flux (1 mark).
Q2 (4 marks): Let initial flux = Φ0 = BA. Case A: area halves, ΔΦ = BA − B(A/2) = BA/2 = Φ0/2, so ΔΦ/Δt = Φ0/4 (per second). Case B: rotated to edge-on, ΔΦ = BA − 0 = Φ0, so ΔΦ/Δt = Φ0/2 (per second). Case B has the larger average rate of change of flux (twice that of Case A), so a larger average induced emf would result.
At the start you were asked about Oersted's 1820 Copenhagen experiment: why did a steady 5 A current produce no induction in Faraday's coil — and what change would move the needle?
The answer: a steady current creates a steady magnetic flux. Faraday's law requires a changing flux to induce an emf — and the rate of change ($\Delta\Phi/\Delta t$) determines the magnitude. Oersted's constant current held $\Delta\Phi/\Delta t = 0$. Switching the current on or off — or using AC — creates $\Delta\Phi/\Delta t \neq 0$, which moves the galvanometer needle.
Extend: An induction cooktop uses a rapidly oscillating magnetic field to induce currents in the pot. Which of the three methods of changing flux does it use? Why does the pot heat up but the cooktop surface does not?
Five timed questions on magnetic flux. Beat the boss to bank a tier — gold (perfect + fast), silver (80%+), or bronze (cleared).
⚔ Enter the arena