Torque on a Current Loop Motors and Meters Study Pack

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Last updated May 21, 2026

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Torque on a Current Loop Motors and Meters Study Guide

Master the physics behind rotating current loops by working through torque equations, magnetic dipole moments, and the τ = NIAB sin θ relationship. This pack covers how DC motors use commutators to sustain rotation and how galvanometers are modified into voltmeters and ammeters using series and shunt resistors — everything you need for motors and meters.

Key Takeaways

  • •A current-carrying loop placed in an external magnetic field experiences a net torque because the magnetic forces on opposite sides of the loop form a couple — equal, opposite, and offset forces that produce rotation rather than translation.
  • •The torque on a rectangular current loop is given by τ = NIAB sin θ, where N is the number of turns, I is the current, A is the loop area, B is the magnetic field strength, and θ is the angle between the magnetic field and the plane of the loop's normal vector.
  • •Torque is maximized when the plane of the loop is parallel to the magnetic field (θ = 90°) and drops to zero when the loop's normal vector aligns with the field (θ = 0°).
  • •The magnetic dipole moment μ = NIA quantifies how strongly a current loop responds to an external field, and torque can be written compactly as τ = μB sin θ.
  • •DC electric motors convert this magnetic torque into continuous rotation by using a commutator and brushes to reverse current direction each half-turn, maintaining torque in the same rotational sense.
  • •Galvanometers detect small currents by measuring the deflection of a current loop against a calibrated restoring spring; the equilibrium deflection angle is directly proportional to the current.
  • •Both voltmeters and ammeters are built from galvanometers modified with precision resistors — a large series resistor for voltmeters and a small parallel shunt resistor for ammeters.

Magnetic Force on a Current-Carrying Conductor

Understanding torque on a current loop begins with the fundamental interaction between a magnetic field and a straight current-carrying wire, because the loop is simply four such wire segments arranged in a closed rectangle.

Force on a Straight Current-Carrying Wire

  • •A wire carrying current I in a magnetic field B experiences a force F = BIL sin α, where L is the wire's length and α is the angle between the current direction and the field.
  • •The force direction is found using the right-hand rule: point fingers along the current direction, curl them toward B, and the thumb points in the direction of the force on positive charge carriers.
  • •When the wire is parallel to the field (α = 0°), the force is zero; the force is maximum when the wire is perpendicular to the field (α = 90°).

Why a Loop Produces Torque Instead of Net Translation

  • •In a rectangular loop, opposite sides carry current in opposite directions relative to the field, so the forces on those sides point in opposite directions.
  • •Because these two equal-and-opposite forces act on spatially separated sides of the loop, they form a force couple — a pair that produces pure rotation with no net translational acceleration.
  • •The two sides of the loop parallel to the rotation axis (the sides through which the axis passes) experience forces that cancel along the axis and therefore do not contribute to torque.

Quantifying Torque on a Rectangular Current Loop

The net torque acting on a multi-turn rectangular loop in a uniform external magnetic field can be expressed with a single equation that links geometry, current strength, and field orientation.

Derivation of the Torque Equation

  • •Each of the two active sides of a rectangular loop (length a, perpendicular to the field) experiences force F = BIb, where b is the side length parallel to the field.
  • •The moment arm for each force depends on how the loop is tilted: when the loop's plane makes angle θ with the field, each moment arm is (a/2) sin θ.
  • •Combining both forces and their moment arms gives τ = BIab sin θ = BIA sin θ for a single-turn loop, where A = ab is the enclosed area.

Effect of Multiple Turns and the Full Torque Formula

  • •Stacking N turns of wire multiplies the torque by N because each turn contributes independently: τ = NIAB sin θ.
  • •θ is defined as the angle between the magnetic field vector and the normal to the plane of the loop — equivalently, the complement of the angle between the field and the loop's face.
  • •Torque is greatest (τ_max = NIAB) when θ = 90°, meaning the loop's plane is aligned with the field and the normal is perpendicular to it.

About this Study Pack

Created by Kibin to help students review key concepts, prepare for exams, and study more effectively. This Study Pack was checked for accuracy and curriculum alignment using authoritative educational sources. See sources below.

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