Year 12 Physics Module 6 ⏱ ~40 min 5 MC · 2 Short Answer Lesson 13 of 21

Magnetic Flux and Changing Flux

Use the area vector to calculate signed magnetic flux, distinguish flux from flux linkage, and explain how changing field strength, area or orientation prepares the way for electromagnetic induction.

Today's hook: A loop can have a large magnetic flux yet zero induced emf. What must change before a galvanometer responds, and why does the answer depend on the loop's area normal?
0/5TASKS
Learn 0 of 6 steps complete
1
Retrieve the flux unit, choose an area normal and predict what can change flux.
Warm up first

Three quick questions from earlier lessons. Pulling old material back to mind before you learn something new makes the new material stick better, so this is not busywork.

Worksheets

Practise this lesson

Use the available foundations worksheet for guided practice on this lesson.

Build Foundations & guided practice
Before you read, predict

A circular loop of wire sits in a uniform magnetic field pointing straight through it.

  1. 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?
  2. If you keep the loop still but strengthen the magnetic field, what happens to the flux?
  3. In which case would you expect an electric current to be induced?

Warm-up, what is the SI unit of magnetic flux?

Learning Intentions
goals

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

  • Constant flux linkage does not induce an emf; changing flux linkage does
  • Flux can change by changing B, changing A, or changing angle
  • Faraday's law relates induced emf to the rate of change of flux linkage

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
Scan these before reading
vocab
Magnetic flux (Φ)The amount of magnetic field passing through a given area. $\Phi = BA\cos\theta$. Measured in webers (Wb).
Weber (Wb)The SI unit of magnetic flux. 1 Wb = 1 T m².
Rate of change of fluxHow quickly flux changes with time: $\Delta\Phi/\Delta t$. Units: Wb/s.
Flux linkage ($N\Phi$)The sum of the single-turn fluxes for a coil. For $N$ identical turns in the same field, the linkage is $N\Phi$.
Induced emfA potential difference produced when flux linkage changes. A current follows only if a conducting path is closed.
Lesson boundary: This lesson defines single-turn flux, flux linkage and ways they change. L13 owns Faraday-law emf calculations; L14 owns induced-current direction; L19 owns eddy-current applications.

Core Content

2
Define signed single-turn flux and represent the field-to-normal angle.
1
What Is Magnetic Flux?
+5 XP

Counting magnetic field lines through a loop

Magnetic flux measures the component of a magnetic field passing through an oriented area. Choose one of the two possible area normals first. The sign of the flux then records whether $\vec B$ points generally with or against that chosen normal.

Magnetic flux

$\Phi = \vec B\cdot\vec A = BA\cos\theta$

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.

Magnetic field and area-normal angle for three loop orientations Three panels show a loop edge-on as an ellipse. In each panel the magnetic field points upward. The chosen area-normal arrow is parallel to the field for maximum positive flux, at sixty degrees for half the maximum flux, and perpendicular for zero flux. Bnormal θ = 0°, Φ = BA 60° θ = 60°, Φ = 0.5BA θ = 90°, Φ = 0

Figure 1, $\theta$ is always measured between $\vec B$ and the chosen area normal. The diagram remains meaningful without colour because every vector is labelled.

Worked example, Calculating flux at different angles

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.

  1. 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
  2. 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
  3. 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?

3
Identify changes in field strength, enclosed area and orientation.
2
Changing Flux Induces Emf
+5 XP

Preparing the quantity used by Faraday's law

We just saw that $\Phi = BA\cos\theta$ describes one oriented turn. That raises a question: what physical changes alter $\Phi$? This card answers it: change $B$, change the area within the field, or change the angle between $\vec B$ and the area normal.

A stationary loop wholly inside a constant uniform field has constant flux. Rotating the loop, changing the area enclosed by it, changing the field, or moving it across the boundary of a field region can change flux. L13 uses the rate of change of flux linkage to calculate induced emf.

Key insight

Constant flux linkage gives zero induced emf. Changing flux linkage can produce an induced emf. If the conducting path is closed, that emf can drive a current. L13 quantifies this relationship.

There are three ways to change flux:

  1. Change B vary the magnetic field strength
  2. Change A alter the area of the loop (e.g., expand or compress it)
  3. 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.

Common misconception
✗ Wrong: A strong magnetic field always induces a current in a nearby wire.
✓ Right: A constant field gives no induced emf in a stationary loop of fixed area and orientation. A change in flux linkage induces an emf; current additionally requires a closed conducting path.

For a fixed chosen normal, change $\Phi=BA\cos\theta$ by changing (1) $B$, (2) the area $A$ within the field, or (3) $\theta$. For a coil, Faraday's law responds to changing linkage $N\Phi$, not merely to a large constant value.

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.

Optional L13-L14 preview

The interactive can show induced-emf magnitude and direction. In this lesson, use it only to compare which changes alter flux; the Faraday and Lenz calculations belong to the next two lessons.

Interactive Tool, Electromagnetic Induction Open fullscreen ↗

According to the induction tool, electromagnetic induction occurs when…

4
Convert a plane angle to the required area-normal angle.
3
The Angle Convention
+5 XP

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.

Angle relationship

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…

5
Distinguish flux from flux linkage and read a flux-time graph.
Signed flux, linkage and graphs

Choose an area normal before assigning a sign: $\Phi=\vec B\cdot\vec A=BA\cos\theta$ is positive when $\vec B$ has a component along that normal and negative when it points oppositely. $\Phi$ is flux through one turn in webers; $N\Phi$ is flux linkage for $N$ identical turns. On a $\Phi$–$t$ graph, the vertical value is flux and the gradient is $d\Phi/dt$. L13 introduces induced emf from the negative gradient of linkage, $-d(N\Phi)/dt$.

Single-turn flux$\Phi=\vec B\cdot\vec A$, measured in Wb. Its sign depends on the chosen area normal.
Flux linkage$N\Phi$ for $N$ identical turns sharing the same flux. It is not the same quantity as $\Phi$.
Magnetic flux versus time graph Flux is constant and positive from zero to two seconds, falls linearly through zero to a negative value by four seconds, then remains constant. The first and last sections have zero gradient while the middle section has a constant negative gradient. Φt constant Φ, zero gradient constant negative gradient constant Φ, zero gradient 2 s4 s

Figure 2, The graph value is flux; its gradient is the rate of flux change. A flat segment can be non-zero flux but has zero rate of change.

A 50-turn coil has $2.0\times10^{-4}$ Wb through each turn. Its flux linkage is…

6
Apply angle conversion, compare flux changes and state the model conditions.
Activity 1, Flux Calculations
ApplyBand 3

Practise calculating magnetic flux with different angles

  1. Set B = 0.50 T, A = 50 cm², θ = 0°. Calculate the flux.
  2. Keep B and A constant. Rotate to θ = 60°. Calculate the new flux. By what factor did it decrease?
  3. Return to θ = 0°. Halve the area to 25 cm². What happens to the flux? Explain.
  4. Set θ = 90° (zero flux). Now increase B to 1.0 T. What is the flux? Explain why.
Activity 2, Three Ways to Change Flux
UnderstandBand 4

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:

  1. The coil is stretched sideways so its area increases.
  2. The coil is rotated from perpendicular to parallel alignment with the field.
  3. An electromagnet near the coil is switched on, increasing the field.
  4. The coil is slid sideways while remaining wholly inside the same uniform field, with its area and orientation unchanged.
Synthesis, connect the ideas

Magnetic flux is the bridge between magnetism and electricity:

  • $\Phi = BA\cos\theta$, measures how much field passes through a loop
  • A constant flux linkage produces zero induced emf; a change in linkage is required
  • Three ways to change flux: change B, change A, or change the angle θ
  • Faraday's law in L13 relates emf to the rate of change of flux linkage

Which statement correctly uses the area-normal convention?

7
Calculate flux and compare rates of flux change independently.
Quick recall, magnetic flux
+5 XP

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.

Short Answer, 7 marks
+5 XP

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): Choose the initial area normal so Φi = Φ0 = BA. Case A: Φf = BA/2, so ΔΦ = Φf − Φi = −Φ0/2 and $|\Delta\Phi|/\Delta t=\Phi_0/(4.0\text{ s})$. Case B: Φf = 0, so ΔΦ = −Φ0 and $|\Delta\Phi|/\Delta t=\Phi_0/(2.0\text{ s})$. Case B has twice the average magnitude of flux-change rate. For identical single-turn loops, L13 therefore predicts twice the average emf magnitude.

8
Retrieve flux conventions and revisit why change, not size alone, matters.
Check what actually stuck
How did your thinking change?

At the start, a loop had a large flux but zero induced emf. You were asked what must change before a galvanometer responds and why the area normal matters.

The answer: the flux value can be non-zero while its time gradient is zero. Changing $B$, $A$ or $\theta$ changes $\Phi$; for a coil, L13 uses the changing linkage $N\Phi$. The chosen normal fixes the sign of $\Phi$ and therefore the sign convention used for the later emf calculation.

Bridge: Switching a nearby electromagnet changes $B$. State what happens to $\Phi$ for one turn and to $N\Phi$ for an $N$-turn coil, without yet calculating the induced emf.

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