Torque on a Current Loop Motors and Meters Study Pack

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Last updated May 28, 2026

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Torque on a Current Loop Motors and Meters Study Guide

Master the mechanics of torque on current loops, from the τ = NIAB sin θ equation and magnetic dipole moment to how DC motors use split-ring commutators and galvanometers balance spring torque to measure current.

Key Takeaways

  • •A current-carrying loop in a uniform magnetic field experiences a net torque because equal and opposite forces on opposite sides of the loop create a rotational tendency rather than linear acceleration.
  • •The magnitude of torque on a current loop is given by τ = NIAB sin θ, where N is the number of turns, I is current, A is loop area, B is 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 plane is perpendicular to the field (θ = 0°).
  • •A DC electric motor converts electrical energy into continuous mechanical rotation by using a split-ring commutator to reverse current direction each half-turn, maintaining torque in a consistent rotational direction.
  • •A galvanometer measures small electric currents by balancing magnetic torque on a current loop against a restoring spring torque, producing a deflection proportional to current.
  • •The magnetic dipole moment (μ = NIA) of a current loop determines how strongly the loop responds to an external magnetic field, and the torque equation can be written compactly as τ = μB sin θ.

Magnetic Force on a Current-Carrying Conductor

Before analyzing what happens to a complete loop, it is essential to understand how a magnetic field exerts force on a straight wire carrying current, since loop behavior emerges directly from this interaction.

Origin of Force on a Current-Carrying Wire

  • •Moving charges experience a magnetic force described by F = qv × B; in a conductor, the net drift of electrons constitutes a current, so the wire as a whole feels a collective force.
  • •The force on a straight wire segment is F = ILB sin α, where I is the current, L is the length of wire in the field, B is the magnetic field magnitude, and α is the angle between the current direction and the field.
  • •Maximum force occurs when the current is perpendicular to the field (α = 90°); zero force results when current runs parallel to the field (α = 0°).

Direction of Force: The Right-Hand Rule

  • •Point the fingers of the right hand in the direction of conventional current (positive charge flow), curl them toward the magnetic field vector B, and the extended thumb points in the direction of the magnetic force on the wire.
  • •Reversing either the current direction or the field direction reverses the force, a fact that becomes critical when analyzing the two sides of a current loop.

How a Current Loop Experiences Net Torque

A rectangular current loop placed in a uniform magnetic field does not simply translate; instead, the forces on its sides produce a net rotational effect — a torque — that tends to align the loop with the field.

Force Distribution Around a Rectangular Loop

  • •Consider a rectangular loop with sides of length a (parallel to the field axis) and b (perpendicular to it), carrying current I in a uniform field B.
  • •The two sides of length b that run perpendicular to B each experience a force of magnitude F = IbB, but in opposite directions — one side is pushed upward, the other downward.
  • •The two sides of length a run parallel to B and therefore experience zero net magnetic force, because the angle α between current and field is 0° in that orientation.

Torque Calculation for a Single Loop

  • •The two opposing forces (each IbB) are separated by a moment arm equal to (a/2) sin θ from the loop's central axis, where θ is the angle between the magnetic field B and the normal to the loop's plane.
  • •The total torque is the sum of contributions from both force pairs: τ = IbB · a sin θ = IAB sin θ, where A = ab is the area of the loop.
  • •For a coil of N turns, each turn contributes equally, giving the general torque equation: τ = NIAB sin θ.

Equilibrium and Maximum Torque Positions

  • •Torque reaches its maximum value (τ = NIAB) when θ = 90°, meaning the plane of the loop is parallel to the magnetic field.
  • •Torque falls to zero when θ = 0°, meaning the loop's normal vector aligns with B — this is the equilibrium position the loop naturally seeks.
  • •If the current direction is not reversed at the zero-torque position, the loop oscillates rather than rotating continuously.

Magnetic Dipole Moment and the General Torque Expression

Physicists describe the rotational response of any current loop to a magnetic field using a single quantity called the magnetic dipole moment, which consolidates the loop's geometry and current into one compact descriptor.

Defining the Magnetic Dipole Moment

  • •The magnetic dipole moment is defined as μ = NIA, where N is the number of turns, I is the current, and A is the enclosed area of the loop.
  • •The direction of μ is perpendicular to the plane of the loop, determined by the right-hand rule: curl the fingers in the direction of current flow and the thumb points along μ.
  • •Units of magnetic dipole moment are ampere·meters squared (A·m²).

Compact Form of the Torque Equation

  • •The torque equation τ = NIAB sin θ can be rewritten as τ = μB sin θ, or in vector form as τ = μ × B.
  • •This form reveals that the torque acts to align μ with B, just as an electric dipole in an electric field aligns with that field.
  • •The analogy between electric and magnetic dipoles is useful: both store potential energy in a field and both experience restoring torques when displaced from alignment.

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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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Torque on a Current Loop Motors and Meters Study Pack | Kibin