Torque on a Current Loop, and the Magnetic Dipole Moment

Q: Is θ measured to the coil's plane or to its normal?

This page measures it to the normal and uses sinθ, following τ = μ × B. If your book measures to the plane it will write τ = NIAB cosα, and α = 90° − θ. Both are above and they always give the same torque. The quick check that settles it in an exam: the torque must be zero when the coil faces the field square-on, and greatest when the field lies in the coil's plane.

A current loop in a magnetic field feels no net force and a very definite twist. This gives you τ = NIAB sinθ, the magnetic dipole moment μ = NIA that makes the formula short, the energy stored at each angle — and it settles the one thing textbooks disagree about: whether θ is measured to the coil’s plane or to its normal.

Last updated 11 Nov 2026

Torque, dipole moment and orientation energy

τ = NIAB sin θ · μ = NIA

Number of turns on the coilEach turn feels the same couple, so the torque is N times larger. A moving-coil galvanometer and a small DC motor both use a few hundred turns for exactly this reason — it is far cheaper than a bigger magnet.

Current in the coil (A)The torque is directly proportional to the current, which is what makes a moving-coil meter linear: a scale marked in equal divisions reads current in equal steps.

Area enclosed by the coil (cm²)In square centimetres. Only the area matters, not the shape — a square, a circle and a long thin rectangle of the same area all feel the same torque in a uniform field. Remember that 12 cm² = 12 × 10⁻⁴ m².

Flux density of the field (T)The field must be uniform for the net force to vanish and leave a pure couple. In a real galvanometer the pole pieces are curved and the core is soft iron precisely so that the field is radial, which keeps the angle at 90° whatever the deflection and keeps the scale linear.

Angle between the coil’s NORMAL and the field (degrees)Measured to the normal, not to the plane. If your textbook writes τ = NIAB cosα it is measuring α to the plane, and α = 90° − θ; both are below and they always agree. At 0° the loop is square-on to the field and the torque is zero; at 90° the field lies in the plane of the coil and the torque is greatest. The engine behind this page has no degree function, so the expression contains an explicit ×π/180.

The loop seen edge on, tilted to the angle you enter, in a uniform field running left to right. The two red arrows are the forces on the two current-carrying sides, and their length follows BIL on a logarithmic scale so a wide range of currents fits the frame. Those forces do not change direction or size as you turn the coil. What changes is how far apart their lines of action are, and at zero degrees they fall exactly in line and the turning effect vanishes. The green arc is the torque, drawn only while there is one.

7.5000mN·mExample

100 turns, 0.50 A, 12 cm², a 0.25 T field, normal at 30° to the field

The couple, the dipole moment, and which angle is which

τ = NIAB sinθ = μB sinθ · μ = NIA · U = −μB cosθ · force on each side: F = BIL · other convention: τ = NIAB cosα, α = 90° − θ

θ: measured between the field and the coil’s normal — the line sticking out of the loop’s face. That is the convention OpenStax uses with τ = μ × B, and it is the one this page implements. Many Indian school texts measure α to the coil’s plane instead and write cosα. The two are the same statement because α = 90° − θ, and both figures are in the rows above so you can see them agree
μ = NIA: the magnetic dipole moment, in ampere square metres. CBSE Unit III names it directly: “Current loop as a magnetic dipole and its magnetic dipole moment”. Its direction is the coil’s normal, given by curling your right hand the way the current flows. Once you have μ, a current loop and a bar magnet obey identical equations
why there is no net force: in a uniform field the forces on opposite sides of the loop are equal and opposite, so they cancel as a force and survive as a couple. The loop twists on the spot. Put the same loop in a non-uniform field and the cancellation fails and it is dragged as well
U = −μB cos θ: the orientation energy, taken as zero at 90°. It is most negative at 0° (stable, aligned) and most positive at 180° (unstable, backwards). The work to turn the coil between two angles is the difference of these, and two rows above give it
the galvanometer trick: τ depends on sinθ, which would make a meter’s scale hopelessly non-linear. So a real moving-coil meter uses curved pole pieces and a soft iron core to make the field radial, which holds θ at 90° however far the coil turns and makes the deflection proportional to the current

Worked example

100 turns, 0.50 A, 12 cm², a 0.25 T field, normal at 30° to the field
AREA INTO SI. 12 cm² = 12 × 10⁻⁴ m². Squared conversion again, and still the commonest slip on the topic.
FIND THE DIPOLE MOMENT FIRST. μ = NIA = 100 × 0.50 × 12 × 10⁻⁴ = 0.060 A·m². Doing this as a separate step is worth the twenty seconds: everything else on the page is μ multiplied by something.
MULTIPLY BY THE FIELD AND THE SINE. τ = μB sinθ = 0.060 × 0.25 × sin 30° = 0.015 × 0.5 = 7.5 × 10⁻³ N·m = 7.5000 mN·m. Note sin 30° = 0.5 exactly; a calculator in radian mode would give −0.988 and an answer wrong in both size and sign.
CHECK IT THE OTHER CONVENTION'S WAY. The angle to the coil's plane is 90° − 30° = 60°, and NIAB cos 60° = 0.015 × 0.5 = 7.5 mN·m. Same answer. If your textbook and your teacher use different letters for this angle, that row is the reconciliation.
WHERE THE TORQUE COMES FROM. An equivalent square loop has sides of √(12 × 10⁻⁴) = 0.0346 m, so the force on each side is NBIL = 100 × 0.25 × 0.50 × 0.0346 = 0.433 N. That force does not change with the angle. The moment arm does: at 30° it is 0.0346 sin30° = 0.0173 m, and 0.433 × 0.0173 = 7.5 mN·m.
AND THE ENERGY. U = −μB cosθ = −0.015 × cos 30° = −12.99 mJ. To swing the coil round to 90° you must supply +12.99 mJ; to flip it right over to 180°, +27.99 mJ. That second number is the work a motor's commutator saves you by reversing the current instead.

The same coil at every angle — and why a motor needs a commutator

The torque is zero at both ends and the difference between them is the energy column: 0° is the bottom of the well and 180° the top of the hill. A direct-current motor’s coil would stop dead at 0° every half turn; the commutator reverses the current just as it gets there, which flips the dipole moment and turns the stable point into an unstable one, so the coil keeps going. That is the entire idea of a DC motor in one sentence.
Angle to the normal	sin θ	Torque	Energy U	What it does
0°	0.000	0.0000 mN·m	−15.00 mJ	stable: sits still, lowest energy
30°	0.500	7.5000 mN·m	−12.99 mJ	turning towards 0
60°	0.866	12.9904 mN·m	−7.50 mJ	turning hard
90°	1.000	15.0000 mN·m	0.00 mJ	maximum torque
150°	0.500	7.5000 mN·m	+12.99 mJ	still turning towards 0
180°	0.000	0.0000 mN·m	+15.00 mJ	unstable: a nudge and it flips

A couple, not a force — and the angle textbooks argue about

Start with the dipole moment and the rest is one multiplication. μ = NIA bundles the three things about the coil that matter into one vector pointing along its normal, and then τ = μB sinθ is the same equation that governs a compass needle, a bar magnet and an electron’s spin. CBSE Unit III asks for exactly this move — “current loop as a magnetic dipole and its magnetic dipole moment” — and it is worth making it a habit, because a question that gives you μ directly is otherwise confusing.

The angle is the thing to check first in any question. This page measures θ from the field to the coil’s normal and uses sinθ, which is the convention behind τ = μ × B. Plenty of school texts measure the angle from the field to the coil’s plane and write cos. They are the same physics — the two angles add to 90° — but a candidate who reads one convention’s diagram with the other’s formula gets the complementary answer and no warning. Both figures are in the rows above, so you can always check which one your question wants. If in doubt, ask yourself where the torque should be zero: it is zero when the coil faces the field square-on.

There is no net force, and the diagram shows why. The two current-carrying sides feel BIL in opposite directions. In a uniform field those cancel exactly, leaving a pure couple. Turn the angle on this page and watch the red arrows: they never change length, because BIL has no θ in it. What shrinks as the angle falls is the perpendicular distance between their lines of action, and at θ = 0 they lie along the same line, pulling the loop apart and turning it not at all. That is the whole origin of the sinθ.

And this is how a motor works, with one more trick. The coil accelerates towards θ = 0, where the torque dies. Left alone it would oscillate and stop there. A commutator reverses the current just as the coil passes that point, which flips μ end for end and turns the stable position into an unstable one, so the coil carries on round. A galvanometer does the opposite: it wants the coil to stop, so it adds a spring and shapes the pole pieces to make the field radial, holding θ at 90° so that the deflection stays proportional to the current. For the force on a single wire this is built from, see the magnetic force on a charge and a current; for the field the loop itself produces, the field of a coil.

Frequently asked questions

Is θ measured to the coil’s plane or to its normal?

This page measures it to the normal and uses sinθ, following τ = μ × B. If your book measures to the plane it will write τ = NIAB cosα, and α = 90° − θ. Both are above and they always give the same torque. The quick check that settles it in an exam: the torque must be zero when the coil faces the field square-on, and greatest when the field lies in the coil’s plane.

Why is there no net force on the loop?

Because the field is uniform. The forces on opposite sides are equal in size and opposite in direction, so they cancel as a force and survive as a couple. In a non-uniform field they do not cancel and the loop is dragged towards the stronger field as well as twisted — which is why a magnet attracts iron filings rather than merely aligning them.

Does the shape of the loop matter?

Not in a uniform field. Only the enclosed area appears in μ = NIA, so a circle, a square and a long thin rectangle of the same area feel the same torque. That is a genuinely useful shortcut and it is exact, not an approximation.

What does the negative orientation energy mean?

U = −μB cosθ is taken as zero at 90° by convention, so aligned positions come out negative. Negative means bound: you have to put work in to move the coil away from alignment. The most negative value, at 0°, is the stable equilibrium; the most positive, at 180°, is the unstable one. Only differences in U are physical, which is why the zero can be put wherever is convenient.

Why does a moving-coil ammeter have a linear scale if the torque goes as sin θ?

Because the field in one is not uniform by design. Curved pole pieces and a cylindrical soft-iron core make the field radial, so the field is always in the plane of the coil however far it has turned, θ stays at 90° and sinθ stays at 1. The torque is then simply proportional to the current, and a hairspring provides a restoring torque proportional to the deflection. Equal divisions, equal currents.

Related calculators

References

Annual Syllabus, Class XII, Physics, Code 42, Session 2025-26, Central Board of Secondary Education, edustud.nic.in/edu/SYLLABUS_2025_26/12/12_Physics_EM.pdf (read 11 November 2026). The board’s own document, cited and not endorsed by it. Unit I Chapter 1 asks for “Electric flux, statement of Gauss’s theorem and its applications to find field due to infinitely long straight wire, uniformly charged infinite plane sheet and uniformly charged thin spherical shell (field inside and outside)”. Unit III Chapter 4 asks for “force on a moving charge in uniform magnetic and electric fields”, “torque experienced by a current loop in uniform magnetic field” and “Current loop as a magnetic dipole and its magnetic dipole moment”. Unit IV Chapter 6 is one line and covers three of this batch’s records: “Electromagnetic induction; Faraday’s laws, induced EMF and current; Lenz’s Law, Self and mutual induction”. The Hall effect is not in this document.
University Physics Volume 2, chapters 11 and 14, OpenStax, Rice University (openstax.org/books/university-physics-volume-2, key-equations pages read 11 November 2026). An openly licensed university text, cited here because the coaching sites that dominate searches for this topic are not citable and mostly copy one another. Chapter 11 gives the radius of a charged particle’s path as r = mv/(qB), its period as T = 2πm/(qB), the Hall potential as V = IBl/(neA), the drift velocity in crossed fields as v_d = E/B, the magnetic dipole moment as μ = NIA and the torque as τ = μ × B. Chapter 14 gives L_solenoid = μ₀N²A/l, NΦ = LI, ε = −L dI/dt, M = N₂Φ₂₁/I₁ = N₁Φ₁₂/I₂ and U = ½LI². Neither key-equations page lists a cyclotron frequency or a field energy density, so those two are stated here from the definitions and not attributed to this source.
Syllabus: Cambridge International AS & A Level Physics 9702, for examination in 2025, 2026 and 2027, Cambridge Assessment International Education, cambridgeinternational.org/Images/664565-2025-2027-syllabus.pdf (read 11 November 2026). Cited for outcomes a Cambridge candidate is examined on that the Indian board words differently or omits. 20.3.3: “understand the origin of the Hall voltage and derive and use the expression V_H = BI /(ntq), where t = thickness”. 20.3.4: “understand the use of a Hall probe to measure magnetic flux density”. 20.5.2: “recall and use Φ = BA”. 20.5.3: “understand and use the concept of magnetic flux linkage”. 20.5.4 asks candidates to explain experiments demonstrating “that the induced e.m.f. is in such a direction as to oppose the change producing it”. 20.5.5: “recall and use Faraday’s and Lenz’s laws of electromagnetic induction”. Note that induction is topic 20.5 here and not a topic of its own; topic 21 is alternating currents. The copy read was retrieved in two parts and the last 19,825 characters were not read, so nothing is claimed about what is on the data-and-formulae sheet.
Fundamental Physical Constants — Frequently used constants (2022 CODATA wallet card), NIST Committee on Data for Science and Technology, physics.nist.gov/cuu/pdf/wallet_2022.pdf (read 11 November 2026). Every constant on this page comes from here, and the thing worth noticing is which numbers have a bracket after them. The elementary charge is printed 1.602 176 634 × 10⁻¹⁹ C and the speed of light 299 792 458 m s⁻¹, both with no bracket at all, because since 2019 they are exact by definition — the coulomb and the metre are now defined from them. But the vacuum electric permittivity is printed 8.854 187 8188(14) × 10⁻¹² F m⁻¹ and the vacuum magnetic permeability 1.256 637 061 27(20) × 10⁻⁶ N A⁻²: both now carry an uncertainty of about 1.6 parts in 10¹⁰, where before 2019 μ₀ was exactly 4π × 10⁻⁷ and ε₀ exactly 1/μ₀c². Almost every school book still prints the old exact forms. At three significant figures nothing changes; which constants are definitions changed completely.