Torque on a Current-Carrying Coil
In 1882, Arsène d'Arsonval at the Collège de France, Paris, built the first moving-coil galvanometer using a permanent horseshoe magnet and a rotating coil — sensitive to currents as small as 10⁻¹⁰ A at full-scale deflection. Edward Weston made it portable in 1888. The torque τ = nBIA that d'Arsonval exploited is the principle behind every analog ammeter, voltmeter, and multimeter ever manufactured.
Practise this lesson
Four printable worksheets that build from the foundations up to exam-style questions — start at whatever level suits you.
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?
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 plane perpendicular to B at all times (except at the gap)
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
Core Content
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:
$\tau = nBAI\cos\theta$
$\tau$ = 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.
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…
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 shape the field so that at every position (except the vertical gap), the field lines are perpendicular to the plane of the coil. This keeps $\cos\theta \approx 1$ for almost the entire rotation, giving smooth, continuous torque.
Figure 2 — Curved pole pieces create a radial field that maximises torque throughout rotation
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.
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.
Using the Motor Effect Lab: the force on a conductor is MAXIMUM when the angle between current and field is…
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 _____.
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.
Connect the ideas
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 $\theta \approx 0°$ always
- 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.
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Pick your answer, then rate your confidence — that tells the system what to drill next.
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)
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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).
Five timed questions on torque and motor design. Beat the boss to bank a tier — gold (perfect + fast), silver (80%+), or bronze (cleared).
⚔ Enter the arenaAt the start you were asked about d'Arsonval's 1882 galvanometer at the Collège de France: a 50-turn coil (area 2.0 cm², B = 0.15 T) with the plane parallel to the field — maximum, minimum, or zero torque?
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.