Year 12 Physics Module 6 IQ4 ⏱ ~20 min 5 MC · 2 Short Answer Lesson 10 of 21

Torque on a Current-Carrying Coil

Use force pairs and one declared angle convention to explain how a current-carrying coil turns, then analyse how radial fields support useful motor torque.

Today's hook: A 50-turn coil of area $2.0$ cm² carries $1.0\times10^{-10}$ A in a $0.15$ T field. When the coil plane is parallel to the field, is its torque maximum or zero, and which orientation gives zero torque?

0/5TASKS

Learn 0 of 6 steps complete

Retrieve the motor effect and declare the plane-angle convention used throughout the lesson.

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 rectangular loop of wire carrying current is placed in a uniform magnetic field. The loop is free to rotate about a horizontal axis through its centre.

When the plane of the loop is parallel to the magnetic field, do you think the torque is maximum, minimum, or zero?
What about when the plane is perpendicular to the field?
If you had to design a motor, would you want the torque to stay constant as the coil spins, or is a pulsing torque acceptable?

Warm-up, when does a current-carrying conductor experience maximum force in a magnetic field?

Learning Intentions

goals

Know, Torque Law

The torque on a current-carrying coil is $\tau = nBAI \cos\theta$
$\theta$ is the angle between the plane of the coil and the magnetic field
Torque is maximum when the plane is parallel to B ($\theta = 0°$)

Understand, Why Torque Varies

The force on each side of the coil is $F = BIl$, but the lever arm changes with angle
In a uniform field, torque drops to zero when the coil is perpendicular to B
A radial magnetic field keeps the local field approximately parallel to the coil plane through most of the rotation

Can Do, Calculate and Predict

Calculate torque given $n$, $B$, $A$, $I$, and $\theta$
Predict torque at any orientation of the coil
Explain why radial magnets improve motor performance

Scan these before reading

vocab

Torque ($\tau$)A turning force. Measured in N m. On a coil: $\tau = nBAI\cos\theta$.

Number of turns ($n$)The number of loops in the coil. More turns = more torque.

Angle $\theta$The angle between the plane of the coil and the magnetic field direction.

Radial magnetic fieldCurved pole pieces keep the local field approximately parallel to the coil plane and perpendicular to its area normal through most of the rotation.

Split-ring commutatorReverses current direction every half-turn so torque always acts in the same rotational direction.

Misconceptions to fix

Wrong: "The torque is maximum when the coil is perpendicular to the field, because that's when the most field lines pass through the coil."

Right: Maximum flux (perpendicular plane) corresponds to zero torque, not maximum. Torque depends on the lever arm, not flux. When the plane is parallel to B, the lever arm is greatest and torque is maximum.

Wrong: "The angle $\theta$ in $\tau = nBAI\cos\theta$ is measured from the normal to the coil."

Right: $\theta$ is the angle between the plane of the coil and the field. This is the complement of the angle between the normal and the field. Confusing these two angles is a common HSC error.

Cross-lesson links: L08 found forces between wires. L09 applies the same force to a current loop in a field, the torque on a rectangular coil (τ = nBIA) is the fundamental mechanism of the galvanometer and the DC motor. This is the bridge between static force calculations and the rotating machines in L10.

Core Content

Set the geometry

Sketch the coil plane, its area normal, and $\vec B$ before selecting sine or cosine.

Connect the motor-effect force pair to coil torque and calculate with the plane-angle convention.

Torque on a Single Coil

+5 XP

From force on a wire to turning force on a loop

Hold a flat rectangular loop of wire horizontally between the poles of a horseshoe magnet and switch on a current. The two sides of the loop parallel to the magnet poles experience upward and downward forces simultaneously, an equal-and-opposite pair that tries to twist the loop around its vertical axis. This is the couple that drives every galvanometer needle and every DC motor armature. For a coil of $n$ turns, area $A$, carrying current $I$ in field $B$, the forces from F = BIl combine to give a torque of:

Torque on a Coil

$\tau = nBAI\cos\theta$

$\tau$ = torque (N m) · $n$ = number of turns · $B$ = magnetic field strength (T)

$A$ = area of coil (m²) · $I$ = current (A) · $\theta$ = angle between plane of coil and B

Critical point: $\theta$ is measured between the plane of the coil and the magnetic field, not between the normal to the plane and the field. This is a common HSC error.

$\theta = 0°$ (plane parallel to B): $\cos 0° = 1$ → maximum torque
$\theta = 90°$ (plane perpendicular to B): $\cos 90° = 0$ → zero torque

Figure 1, Torque on a coil at three angles. Maximum when plane is parallel to B; zero when perpendicular.

Text description

The magnetic field B is uniform and points down. Each panel shows the coil end-on: a line for the coil plane, a dot at the side carrying current out of the page, a cross at the side carrying current into the page, and the axis at the centre.
The force on the out-of-page side points right and the force on the into-page side points left, at every angle.
θ = 0°: the coil plane is parallel to B, the forces act furthest from the axis and the torque is maximum.
θ = 45°: the lever arm is shorter and τ = 0.71 τmax.
θ = 90°: the coil plane is perpendicular to B, both forces act along the plane through the axis and the torque is zero.
θ is the angle between the coil plane and B, so τ = nBAI cos θ.

Worked Example, Torque on a Coil

A rectangular coil has 50 turns, each of dimensions 4.0 cm × 3.0 cm. It carries a current of 3.0 A in a uniform magnetic field of 0.20 T. (a) Calculate the maximum torque. (b) Calculate the torque at $\theta = 30°$. (c) At what orientation is the torque zero?

Part (a), Maximum torque ($\theta = 0°$). $A = (4.0\times10^{-2})(3.0\times10^{-2}) = 1.2\times10^{-3}$ m². $\tau_{max} = nBAI = (50)(0.20)(1.2\times10^{-3})(3.0) = 3.6\times10^{-2}$ N m.
Part (b), Torque at $\theta = 30°$. $\tau = (3.6\times10^{-2})\cos 30° = 3.1\times10^{-2}$ N m.
Part (c), Zero torque. Torque is zero when $\cos\theta = 0$, i.e. $\theta = 90°$. This is when the plane of the coil is perpendicular to B. Forces on the sides act through the axis of rotation, producing no turning effect.

Torque on a coil: $\tau = nBAI\cos\theta$ (N m). $\theta$ = angle between the plane of the coil and $\vec{B}$ (NOT the normal). Maximum when $\theta = 0°$ (plane parallel to B); zero when $\theta = 90°$ (plane perpendicular to B).

Pause, copy the highlighted torque formula and angle convention into your book before moving on.

A coil has maximum torque $\tau_{max}$. When the plane of the coil makes an angle of 60° with the field, the torque is…

Check the angle

Explain why a plane-field angle of $60^\circ$ gives half the maximum torque.

Correct the radial-field geometry and separate smooth torque from current reversal.

Radial Magnetic Fields and DC Motors

+5 XP

How motor designers keep torque pointing the right way

We just saw that torque varies as $\cos\theta$ in a uniform field, dropping to zero every half-turn. That raises a question: how do real DC motors spin smoothly if the torque keeps collapsing? This card answers it → a radial field keeps $\theta \approx 0°$ throughout rotation, and a split-ring commutator reverses the current to maintain the same turning direction.

In a uniform magnetic field, the torque on a coil varies as $\cos\theta$. This means the torque pulses: maximum, then zero, then maximum in the opposite direction. A motor built this way would judder, not spin smoothly.

A radial magnetic field solves this. The curved pole pieces keep the field approximately parallel to the plane of the coil, and therefore perpendicular to the coil's area normal, through most of the rotation. With this lesson's plane-angle convention, this keeps $\cos\theta \approx 1$ and produces smooth, continuous torque.

Figure 2, Curved pole pieces keep the local field approximately parallel to the coil plane (perpendicular to its area normal), maintaining large torque.

Text description

A curved north pole piece on the left and a curved south pole piece on the right surround a cylindrical soft-iron core, leaving a narrow curved gap.
In the gap the field is radial: on the left it points from the N face in towards the core, and on the right it points out from the core to the S face.
The coil is wound on the core. Its two sides (a dot for current out of the page, a cross for current into the page) sit in the gap, joined by the coil plane through the axis. The area normal is perpendicular to the plane.
At each side B points along the coil plane, perpendicular to the area normal, whatever angle the coil has turned to. So θ, the angle between the plane and B, stays near 0° and the torque stays near maximum.

The split-ring commutator reverses the current direction every half-turn. Without it, the coil would oscillate around the equilibrium position instead of rotating continuously. With it, the torque always pushes the coil in the same direction.

HSC Tip

When asked about DC motor design, always mention two features: (1) radial magnetic field for smooth torque, and (2) split-ring commutator for continuous rotation in one direction.

Radial field: curved pole pieces keep $\theta \approx 0°$ → $\cos\theta \approx 1$ → nearly constant torque throughout rotation. Split-ring commutator: reverses current every half-turn so torque always acts in the same rotational direction. Both are essential for smooth, continuous DC motor operation.

Add the highlighted DC motor design features to your notes before the check below.

In a DC motor with a radial field, the torque varies from maximum to zero and back each half-turn.

The split-ring commutator ensures the coil always experiences torque in the same rotational direction.

Torque is maximum when the plane of the coil is perpendicular to the magnetic field.

Name both design roles

State what the radial field changes and what the split-ring commutator changes.

Investigate the conductor force and translate between plane and normal angle conventions.

Interactive Tool, Motor Effect Lab Open fullscreen ↗

Using the Motor Effect Lab: the force on a conductor is MAXIMUM when the angle between current and field is…

Equivalent angle conventions

This page uses $\theta$ between the coil plane and $\vec B$, giving $\tau=nBIA\cos\theta$. If a question instead defines $\alpha$ between the area normal and $\vec B$, use $\tau=nBIA\sin\alpha$. Because $\alpha+\theta=90°$, the two forms are equivalent. Sketch the plane, its normal and $\vec B$ before selecting sine or cosine.

Translate conventions

If the plane angle is $30^\circ$, state the normal angle and verify that the sine and cosine forms agree.

Apply area conversion, units, proportional reasoning, and the declared angle convention.

Current and resistance condition.

For an ideal coil connected to a fixed voltage before rotational back emf is considered, $I=V/R$. Increasing winding resistance therefore reduces current and, because $\tau\propto I$, reduces torque. Lesson 17 owns the deeper running-motor back-emf and power model.

Activity 1, Torque Calculations

ApplyBand 3–4

Practise applying $\tau = nBAI\cos\theta$ at different orientations

A coil with 20 turns, area $2.0\times10^{-3}$ m², carries 4.0 A in a 0.15 T field. Calculate (a) the maximum torque; (b) the torque when the plane makes 45° with the field.
The torque on a coil is 0.10 N m at $\theta = 0°$. Calculate the torque at $\theta = 60°$.
Explain in one sentence why the torque formula uses $\cos\theta$ rather than $\sin\theta$.
A student says increasing the resistance of the coil wire will increase the maximum torque. Is this correct? Explain.

Fill the gap. A coil with 10 turns, area $5.0\times10^{-3}$ m², carries 2.0 A in a 0.10 T field. The maximum torque (in mN m, to one decimal place) is _____.

Complete the calculation set

Record units and the chosen angle definition in every calculation.

Explain motor-design choices and consolidate the L09 bridge to motor operation.

Activity 2, Motor Design Analysis

UnderstandBand 5

Apply your understanding of radial fields and commutators to motor design

A student builds a simple DC motor using a rectangular coil in a uniform magnetic field (no radial field, no commutator). Describe two specific problems this motor would have, and for each problem, explain which design feature solves it and how.

Wrap-up, Synthesis and Summary

Connect the ideas

In a uniform field, torque varies as $\cos\theta$, pulsing between maximum and zero. This causes smooth rotation?

No. Pulsing torque causes juddering. A radial field keeps $\cos\theta \approx 1$ throughout rotation, solving this problem. The split-ring commutator solves the separate problem of reversing direction.

Copy into your books

Torque Formula

$\tau = nBAI\cos\theta$
$\theta$ = angle between plane and B
Max torque: $\theta = 0°$
Zero torque: $\theta = 90°$

Key Components

$n$ = turns (↑n → ↑torque)
$B$ = field strength
$A$ = coil area
$I$ = current

Radial Field

Keeps $\theta \approx 0°$ through most of each turn
Produces nearly constant torque
Curved pole pieces needed

Commutator

Reverses current every half-turn
Maintains one direction of rotation
Without it: oscillation not rotation

Three of these statements about the torque on a current-carrying coil are correct. Pick the odd one out.

Respond

Complete Activity 2 before moving to independent Practice.

Analyse6 marks

A rectangular coil of $n$ turns carrying current $I$ is free to rotate in a uniform magnetic field. Analyse how the torque on the coil varies as it rotates through one half turn. (6 marks)

Unpack the verb

This dot point names analyse in its own text — "to analyse the functions of its components, the production of torque" — so the directive is quoted, not substituted.

Plan: identify which sides produce useful force, build the torque expression, then track it through the rotation and name the two critical orientations.

Starting with which sides contribute prevents the commonest error, which is treating all four sides of the coil as equivalent.

Account5 marks

A simple DC motor built with a uniform magnetic field and a split-ring commutator rotates, but its torque is noticeably uneven through each revolution. Account for this unevenness, and for the fact that a motor with a radial field does not show it. (5 marks)

Unpack the verb

Account for is a cause chain ending at a given observation. Here there are two observations, so there are two chains sharing one starting point.

Plan: start from the torque relationship, follow it through a revolution in a uniform field, then change one thing — the field geometry — and follow it again.

Sharing the starting point makes the second chain short, because only one variable has changed between the two cases.

Retrieve the angle convention, calculate torque, and evaluate a motor claim.

Quick recall, torque on a coil

+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, 8 marks

+5 XP

ApplyBand 4(4 marks) 1. A rectangular coil has 100 turns and dimensions 5.0 cm × 4.0 cm. It carries a current of 2.0 A in a uniform magnetic field of 0.30 T. (a) Calculate the maximum torque on the coil. (2 marks) (b) Explain why a DC motor uses a radial magnetic field rather than a uniform field. (2 marks)

2 marks: correct area conversion and $\tau = nBAI$ calculation · 2 marks: radial field keeps $\theta \approx 0°$ throughout → $\cos\theta \approx 1$ → nearly constant torque (accept: avoids pulsing torque)

AnalyseBand 5(4 marks) 2. A student claims: "Increasing the resistance of a DC motor's coil winding will increase the torque, because more resistance means more power." Evaluate this claim. In your answer, use $\tau = nBAI\cos\theta$ and relevant circuit principles.

1 mark: identifies $\tau \propto I$ · 1 mark: higher resistance reduces I (for constant V) · 1 mark: therefore torque decreases, not increases · 1 mark: student confused resistance with another torque factor (e.g. n, B, A)

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 (4 marks): (a) $A = (5.0\times10^{-2})(4.0\times10^{-2}) = 2.0\times10^{-3}$ m². $\tau_{max} = nBAI = (100)(0.30)(2.0\times10^{-3})(2.0) = 0.12$ N m (2 marks). (b) A radial field ensures the plane of the coil is always approximately parallel to the field lines ($\theta \approx 0°$), so $\cos\theta \approx 1$ throughout most of the rotation. This gives nearly constant torque, producing smooth rotation rather than a pulsing force (2 marks).

Q2 (4 marks): The torque formula $\tau = nBAI\cos\theta$ shows that torque is directly proportional to current $I$ (1 mark). By Ohm's law, for a fixed voltage $V$, $I = V/R$, so increasing resistance $R$ decreases the current (1 mark). Decreased current means decreased torque (1 mark). The student's claim is incorrect; they have confused resistance (which affects current and therefore torque negatively) with one of the direct factors like turns $n$, field $B$, or area $A$ (1 mark).

Check retention and revisit the relationship between torque, plane angle, and area-normal angle.

Check what actually stuck

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quiz

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How did your thinking change?

At the start you were asked about a 50-turn coil of area $2.0$ cm² carrying $1.0\times10^{-10}$ A in a $0.15$ T field. With its plane parallel to the field, is the torque maximum or zero?

The answer is maximum torque. When the plane is parallel to B, the forces on the coil sides act at the greatest perpendicular distance from the axis, the lever arm is maximised. Using $\tau = nBIA\cos\theta$ with $\theta = 0°$: $\tau = 50 \times 0.15 \times 10^{-10} \times 2.0 \times 10^{-4} \times \cos 0° = 1.5 \times 10^{-13}$ N m. Zero torque occurs when the plane is perpendicular to B ($\theta = 90°$, $\cos 90° = 0$).

The common misconception is to confuse magnetic flux (maximum when perpendicular) with torque (maximum when parallel). Flux and torque are related to different geometries of the same system.

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