Current loop as a magnetic dipole

Current loop, torque and the galvanometer

A current loop behaves like a small bar magnet. Its magnetic dipole moment is m = NIA, where N is the number of turns, I the current and A the area. Its direction is given by the right-hand rule (perpendicular to the plane of the loop). Unit: A m².

In a uniform field B, the loop feels no net force but a torque τ = mB sin θ = NIAB sin θ, where θ is the angle between m and B. Torque is maximum when the plane of the coil is parallel to B (θ = 90°).

An electron orbiting a nucleus is also a tiny current loop and has a magnetic moment.

Moving coil galvanometer (MCG): a coil in a radial magnetic field (made using concave pole pieces and a soft-iron core) turns when current flows. The magnetic torque NIAB is balanced by the restoring torque kφ of a spring, so NIAB = kφ, giving φ ∝ I. A radial field keeps sin θ = 1 at all positions so the scale is linear.

• Current sensitivity = φ/I = NAB/k.
• Voltage sensitivity = φ/V = NAB/(kG), where G is the coil resistance.

Conversions:

• Ammeter: connect a small shunt S in parallel: S = I_g G / (I – I_g). An ideal ammeter has zero resistance.
• Voltmeter: connect a large resistance R in series: R = V/I_g – G. An ideal voltmeter has infinite resistance.

Worked example: G = 20 Ω, full-scale I_g = 5 mA. For a 5 A ammeter, S = (0.005 × 20)/4.995 ≈ 0.02 Ω. For a 10 V voltmeter, R = 10/0.005 – 20 = 1980 Ω.

Key points

• m = NIA; τ = mB sin θ.
• MCG uses a radial field so deflection ∝ current.
• Ammeter: low shunt in parallel. Voltmeter: high resistance in series.

Exam tips

• Shunt and series resistance calculations appear often; write the formula first.
• Increasing N increases current sensitivity but also G, so voltage sensitivity may not rise.

Common mistakes

• Putting the shunt in series or the voltmeter resistance in parallel.
• Using the angle between B and the plane of the coil in τ = mB sin θ without converting.
• Forgetting to change mA to A.