Questions: Magnetic Dipole Moment from Current Loops

Question 1

A current loop sits in a uniform external magnetic field B. When is the loop's potential energy U = -μ·B at its minimum?

• A. When the plane of the loop is perpendicular to B (the loop's axis is aligned with the field)
• B. When the plane of the loop is parallel to B (the loop's axis is perpendicular to the field)
• C. When μ is antiparallel to B
• D. Potential energy is the same in all orientations because B is uniform

Answer: (A)

U = -μ·B = -μB cosθ, where θ is the angle between μ and B. This is minimized when cosθ = 1, i.e., when μ is parallel to B. The magnetic moment vector μ points along the loop's axis (perpendicular to the loop's plane), so the loop's axis must align with B. Option B is the maximum-torque orientation (θ = 90°), not minimum energy. Option C (antiparallel) gives U = +μB — maximum energy, an unstable equilibrium.

Question 2

Two circular current loops have identical areas. Loop A carries current I; Loop B carries current 2I. How do their magnetic dipole moments compare?

• A. They are equal — magnetic dipole moment depends only on area, not on current magnitude
• B. Loop B has twice the magnetic dipole moment of Loop A
• C. Loop A has twice the magnetic dipole moment of Loop B
• D. The comparison requires knowing the loops' radii, not just their areas

Answer: (B)

The magnetic dipole moment is μ = IA, so it is proportional to both current and area. Loop B carries 2I with the same area A, giving μ_B = 2IA = 2μ_A. Option A is the common error of confusing area as the sole determinant. Since both loops have the same area, the difference in current is the only variable — and it enters linearly.

Question 3

The torque on a magnetic dipole in a uniform external field tends to push the dipole toward regions of stronger magnetic field.

• T. True
• F. False

Answer: False

Torque and translational force are distinct effects. Torque (τ = μ × B) acts to rotate the dipole — aligning μ with B, it brings the system to lower energy. A net force pushing the dipole toward stronger field regions is a separate phenomenon that requires a non-uniform field: F = ∇(μ·B). In a uniform field, there is torque but zero net translational force. Conflating torque with force is a common error that comes from confusing two different equations.

Question 4

A current loop and a spinning uniformly charged sphere can both possess magnetic dipole moments because both involve charge in circulation.

• T. True
• F. False

Answer: True

Any closed path of moving charge constitutes a current loop and therefore has a magnetic dipole moment μ = IA. A spinning charged sphere has surface charges rotating about the spin axis — each charged element traces a circular orbit, constituting a small current loop. The total dipole moment is the sum over all such elements. This is why spinning particles (electrons, protons, neutrons) have magnetic moments, and why the magnetic dipole concept spans from laboratory coils to intrinsic quantum spin.

Question 5

Why does a compass needle align with Earth's magnetic field? Explain using the magnetic dipole moment and the torque equation.

Model answer: A compass needle is a magnetic dipole with moment μ. In Earth's external field B, it experiences torque τ = μ × B. This torque is nonzero whenever μ is not aligned with B, and it acts to rotate the needle toward alignment. The torque vanishes when μ is parallel to B (minimum energy, U = -μB). Any slight perturbation from alignment restores the torque, making parallel alignment a stable equilibrium. The needle swings to this minimum-energy orientation and (after damping) stays there.

This question tests whether students can connect the abstract equation τ = μ × B to a concrete physical phenomenon. The key steps are: (1) a compass needle is a magnetic dipole; (2) a dipole in an external field experiences a torque that is zero only when μ and B are parallel; (3) parallel alignment is minimum energy (U = -μB), so it is a stable equilibrium; (4) therefore the torque consistently drives the needle to point along the field. The same reasoning explains how MRI machines manipulate nuclear dipole moments and how electric motors produce rotation.