6.5 Torque on current loops

Definition of torque

When a current-carrying loop sits in a magnetic field, the field pushes on different sides of the loop in different directions. The result is a torque that tries to rotate the loop. This is the core idea behind electric motors and many measuring instruments.

Torque on current loops

The torque depends on how much current flows, how large the loop is, how strong the field is, and the loop's orientation relative to the field. The full expression is:

$\tau = NIAB \sin \theta$

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 normal to the loop's plane.

Notice the $\sin \theta$ factor: torque is maximum when the loop's plane is parallel to the field ($\theta = 90°$) and zero when the loop's normal is aligned with the field ($\theta = 0°$). The direction of the torque is determined by the right-hand rule, perpendicular to both the magnetic field and the loop's normal.

Magnetic dipole moment

The magnetic dipole moment bundles the current and geometry of a loop into a single vector quantity:

$\vec{\mu} = NI\vec{A}$

• The magnitude is $\mu = NIA$, measured in ampere-square meters (A·m²)
• The direction points perpendicular to the loop plane, determined by curling your right-hand fingers in the direction of current flow; your thumb gives the direction of $\vec{\mu}$
• This is analogous to the electric dipole moment in electrostatics

Using $\vec{\mu}$, the torque equation becomes the compact cross product $\vec{\tau} = \vec{\mu} \times \vec{B}$, which is easier to work with in vector problems.

Forces on current loops

Beyond torque, magnetic fields can also exert net translational forces on current loops, depending on whether the field is uniform or not. This distinction matters for understanding real devices.

Uniform vs. non-uniform fields

• In a uniform field, the forces on opposite sides of the loop are equal and opposite. They cancel out, producing pure torque with no net translational force. The loop rotates but doesn't move sideways.
• In a non-uniform field, the field is stronger on one side of the loop than the other. The forces no longer cancel, so the loop experiences both torque and a net translational force. The magnitude of that translational force depends on the field gradient.

Uniform fields are approximated inside long solenoids or between large parallel current-carrying plates. Non-uniform fields occur near the ends of solenoids or close to permanent magnets.

Direction of magnetic force

For any small segment of the loop, the force is:

$d\vec{F} = I(d\vec{l} \times \vec{B})$

where $d\vec{l}$ is a small length element pointing in the direction of current flow. You can use Fleming's left-hand rule (or the cross product) to find the direction on each segment.

For a closed loop in a uniform field, adding up all these segment forces gives a net force of zero. In a non-uniform field, the sum is nonzero, and the loop accelerates toward the region of stronger or weaker field depending on its orientation.

Magnetic dipole in a field

A current loop in an external field behaves much like an electric dipole in an electric field. It has both a torque trying to rotate it and a potential energy that depends on its orientation.

Potential energy

The potential energy of a magnetic dipole in an external field is:

$U = -\vec{\mu} \cdot \vec{B} = -\mu B \cos \theta$

• Minimum energy ($U = -\mu B$) occurs when $\vec{\mu}$ is aligned with $\vec{B}$ ($\theta = 0°$). This is the stable configuration.
• Maximum energy ($U = +\mu B$) occurs when $\vec{\mu}$ points opposite to $\vec{B}$ ($\theta = 180°$).
• The total energy difference between aligned and anti-aligned states is $\Delta U = 2\mu B$. This energy splitting is directly relevant to MRI and NMR, where transitions between these states are detected.

Torque on dipole

$\vec{\tau} = \vec{\mu} \times \vec{B}$

The magnitude is $\tau = \mu B \sin \theta$. The torque always acts to rotate the dipole toward alignment with the field.

• At $\theta = 0°$ (aligned) or $\theta = 180°$ (anti-aligned), $\sin \theta = 0$, so the torque is zero.
• Maximum torque occurs at $\theta = 90°$.

A compass needle is a great everyday example: Earth's magnetic field exerts a torque on the magnetized needle, rotating it until it aligns with the field.

Applications of torque

Electric motors

Electric motors convert electrical energy into mechanical rotation using torque on current loops.

1. A current-carrying coil (the rotor) sits inside a magnetic field created by the stator (permanent magnets or electromagnets).
2. The magnetic field exerts torque on the rotor, causing it to rotate.
3. A commutator reverses the current direction every half-turn so the torque always pushes the rotor in the same rotational direction.
4. Since $\tau = NIAB \sin \theta$ varies with angle, real motors use multiple coils offset from each other to produce smoother, more constant torque.

Motors built on this principle appear in electric vehicles, power tools, and household appliances.

Galvanometers

A galvanometer is a sensitive instrument for measuring small currents.

1. A current-carrying coil is suspended in a magnetic field between the poles of a permanent magnet.
2. The magnetic torque rotates the coil by an amount proportional to the current.
3. A spring provides a restoring torque, so the coil deflects until the magnetic torque and spring torque balance. This creates a linear relationship between current and deflection angle.
4. By reading the deflection on a calibrated scale, you measure the current.

Galvanometers can be converted into ammeters (by adding a low-resistance shunt in parallel) or voltmeters (by adding a high-resistance multiplier in series).

Calculation methods

Right-hand rule

To find the direction of the magnetic dipole moment and torque:

1. Curl the fingers of your right hand in the direction of current flow around the loop.
2. Your thumb points in the direction of $\vec{\mu}$ (the magnetic dipole moment).
3. The torque $\vec{\tau} = \vec{\mu} \times \vec{B}$ is then perpendicular to both $\vec{\mu}$ and $\vec{B}$, following the standard right-hand rule for cross products.

This same rule applies when finding the direction of magnetic force on any current-carrying wire segment.

Cross product formulation

The cross product $\vec{\tau} = \vec{\mu} \times \vec{B}$ is the most general way to express the torque.

• Magnitude: $\tau = \mu B \sin \theta$
• Direction: Perpendicular to the plane containing $\vec{\mu}$ and $\vec{B}$, with the sense given by the right-hand rule.

For problems with simple geometry, you can often just use the magnitude formula. For three-dimensional problems or complex loop orientations, writing out the full cross product in component form is more reliable.

Factors affecting torque

The equation $\tau = NIAB \sin \theta$ tells you exactly which variables you can adjust to change the torque. Each factor plays a distinct role in device design.

Current magnitude

Torque is directly proportional to current ($\tau \propto I$). Doubling the current doubles the torque. In practice, you increase current by raising the voltage or lowering the resistance. The limit is heating: too much current causes resistive losses ($P = I^2R$) that can damage the conductor. Motor designers balance higher current for more torque against thermal management.

Loop area

Torque is directly proportional to the loop area ($\tau \propto A$). A larger loop produces more torque for the same current and field. But bigger loops mean bigger, heavier devices. A practical workaround is to use multiple turns (N): winding 100 turns of wire effectively multiplies the torque by 100 without increasing the physical loop size proportionally.

Field strength

Torque is directly proportional to the external field strength ($\tau \propto B$). Stronger magnets produce more torque. The field can come from permanent magnets (neodymium magnets are common in modern motors) or electromagnets. In applications like MRI machines, extremely strong fields (1.5 T to 3 T) are used, and the torque on magnetic dipoles at those field strengths becomes very significant.

Equilibrium conditions

Stable vs. unstable equilibrium

A current loop in a magnetic field has two equilibrium positions where the torque is zero:

• Stable equilibrium ($\theta = 0°$): $\vec{\mu}$ aligned with $\vec{B}$. Potential energy is at a minimum. If you nudge the loop slightly, the torque pushes it back toward alignment.
• Unstable equilibrium ($\theta = 180°$): $\vec{\mu}$ anti-aligned with $\vec{B}$. Potential energy is at a maximum. Any small disturbance causes the loop to swing away toward the stable position.

This is analogous to a pendulum: hanging straight down is stable equilibrium, balanced straight up is unstable.

Rotational dynamics

If displaced from stable equilibrium by a small angle, the loop oscillates. The equation of motion is:

$I\frac{d^2\theta}{dt^2} = -\mu B \sin \theta$

where $I$ is the moment of inertia of the loop (not the current).

For small angles, $\sin \theta \approx \theta$, and this reduces to simple harmonic motion with period:

$T = 2\pi\sqrt{\frac{I}{\mu B}}$

In real systems, damping from friction or induced eddy currents causes the oscillations to decay over time. Critical damping is the condition where the loop returns to equilibrium as quickly as possible without overshooting.

Experimental demonstrations

Magnetic torque balance

This experiment measures the magnetic moment of a sample by balancing magnetic torque against gravitational torque.

1. Suspend a sample between the poles of an electromagnet.
2. Gradually increase the applied field strength.
3. Measure the deflection angle at each field value.
4. From the balance condition, extract the magnetic susceptibility and construct a magnetization curve.

This technique is used in materials science to characterize magnetic properties of new materials.

Rotating coil experiments

A coil rotated in a uniform magnetic field demonstrates both generator and motor principles.

• Generator mode: An external drive rotates the coil. The changing magnetic flux through the coil induces an EMF (Faraday's law), and you can measure how the induced voltage varies with rotation angle and speed.
• Motor mode: Apply a current to the coil instead. The resulting torque causes the coil to rotate, demonstrating motor action.

These experiments also illustrate Lenz's law: the induced current always opposes the change in flux that produced it.

Relationship to angular momentum

The connection between magnetic moments and angular momentum leads to some of the most important phenomena in modern physics.

Precession of magnetic moments

A magnetic moment that isn't aligned with an external field doesn't simply flip into alignment. Instead, it precesses around the field direction, much like a spinning top precesses around the vertical under gravity. The torque from the field continuously changes the direction of the angular momentum vector, tracing out a cone.

This precession is observed in nuclear magnetic resonance (NMR) and electron spin resonance (ESR), and it forms the physical basis for MRI imaging.

Larmor frequency

The precession occurs at a characteristic rate called the Larmor frequency:

$\omega_L = \gamma B$

where $\gamma$ is the gyromagnetic ratio, a constant that depends on the type of particle (electron, proton, specific nucleus). For protons in a 1.5 T MRI field, the Larmor frequency is about 64 MHz.

This frequency is central to NMR spectroscopy, MRI, and atomic clock technology. By measuring $\omega_L$ precisely, you can determine either the field strength or the gyromagnetic ratio.

Quantum mechanical aspects

Classical electromagnetism treats magnetic moments as continuous vectors that can point in any direction. Quantum mechanics reveals that this picture is incomplete.

Spin magnetic moment

Fundamental particles possess an intrinsic magnetic moment associated with their quantum mechanical spin, even though nothing is physically spinning.

• The electron's spin magnetic moment is approximately one Bohr magneton ($\mu_B \approx 9.274 \times 10^{-24}$ J/T).
• Protons and neutrons have much smaller magnetic moments because of their internal quark structure.
• In an external magnetic field, these moments can only take on discrete orientations, leading to quantized energy levels. This quantization is fundamental to atomic structure and spectroscopy.

Stern-Gerlach experiment

The 1922 Stern-Gerlach experiment provided direct evidence for the quantization of angular momentum.

1. A beam of silver atoms was sent through a strongly non-uniform magnetic field.
2. Classically, the beam should have spread into a continuous smear (since the dipole orientations should be random and continuous).
3. Instead, the beam split into two discrete spots on the detector.

This discrete splitting showed that the component of angular momentum along the field direction is quantized. The experiment provided early evidence for electron spin and became one of the foundational results of quantum mechanics.