Magnetic Dipole Moment of a Current Loop
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Direct answer
Hang a current loop in a magnetic field and it tries to line up like a compass needle: the loop carries a magnetic dipole moment M = NIA (turns × current × area), pointing along the right-hand-curl normal to the plane. The field exerts torque τ = MB sinθ that rotates it toward alignment, and the orientation stores potential energy U = −MB cosθ — stable at θ = 0, unstable at 180°, with 2MB of work needed to flip it through half a turn. Far away, the loop's field mimics a short bar magnet's: B_axial = μ0(2M)/(4πr³) along the axis and B_equatorial = μ0M/(4πr³) opposite to M on the perpendicular bisector — the axial field is twice the equatorial at equal distance. The moving-coil galvanometer is this physics wearing a spring: coil torque NIAB balances spring torque kθ, so current reads directly as needle swing.
What you must remember
- Moment definition: M = NIA for N turns, current I, loop area A; direction by the right-hand curl rule around the circulation — the loop's north-seeking face.
- Torque and energy: τ = MB sinθ (maximum at 90°, zero at 0° and 180°); U = −MB cosθ; work to rotate from 0° to 180° is 2MB.
- Field of the loop: centre B = μ0NI/2R; on the axis at distance x, B = μ0NIR²/(2(R² + x²)^(3/2)), which far away becomes μ0(2M)/(4πx³).
- Dipole pair rule: axial field = 2 × equatorial field at the same distance, in directions parallel and antiparallel to M respectively — the same 2:1 an electric dipole shows.
- Orbiting charge: an electron in a circular orbit is a current loop with magnetic moment M = (e/2m)L; the Bohr magneton μ = eh/4πm = 9.27 × 10⁻²⁴ J/T is atomic magnetism's natural unit.
- Galvanometer physics: NIAB = kθ at equilibrium, so deflection is proportional to current; current sensitivity θ/I = NAB/k, and increasing N, A or B — or softening the spring — raises it.
- Net force on a loop: zero in a uniform field (torque only); a net force appears only where the field varies in space, which is why non-uniform fields attract or repel dipoles.
The galvanometer as a magnetic dipole
The moving-coil galvanometer gathers the chapter into one device. A coil of N = 40 turns, area 5 cm² (5 × 10⁻⁴ m²), sits in a radial field B = 0.25 T arranged so that the plane's normal stays parallel to the field through the whole swing — that radial-field design trick keeps torque NIAB constant rather than oscillating with angle. Torque per ampere: NAB = 5 × 10⁻³ N m/A, so a spring of constant k = 2.6 × 10⁻³ N m/rad shows 1 rad of deflection per 0.52 A — a few-degree swing already resolves milliamperes. Sensitivity levers read straight off the formula: more turns, bigger area, stronger field, softer spring.
Now the dipole mechanics around it. Give the coil a moment M = 0.2 A m² in B = 0.5 T: maximum torque MB = 0.1 N m at 90°, aligned energy −MB = −0.1 J, and a flip to anti-alignment costs 2MB = 0.2 J of work. The far-field formulas close the loop: on the axis at r = 10 cm, B = μ0(2M)/(4πr³) = 4 × 10⁻⁵ T, comparable to the Earth's field — a concrete scale for how weak a small loop's distant field is.
Dipole questions in JEE
The direction of M via the right-hand rule is the first filter: a clockwise loop viewed from above carries M into the table. Second, the energy formula's sign: U = −MB cosθ means alignment is the energy minimum, and "work done in rotating a dipole from 0° to 90°" is MB (from −MB to 0), while 0° to 180° is 2MB — options include MB, 2MB and MB(1 − cosθ) forms to be matched to the stated angles. Third, the axial-versus-equatorial 2:1 pattern transfers from electric dipoles by analogy. Fourth, the galvanometer's radial field is a named design feature, existing to keep torque constant at every deflection and the scale linear. Main tests formula plug-ins; Advanced rotates the loop in a field (induced EMF meets dipole torque) or invokes the Bohr magneton link.
Frequently asked questions
What is the magnetic dipole moment of a current loop?
M = NIA for N turns carrying current I around area A, directed perpendicular to the loop's plane by the right-hand curl rule.
How much work is needed to turn a magnetic dipole from alignment to anti-alignment?
2MB, because the potential energy U = −MB cosθ changes from −MB to +MB as θ goes from 0° to 180°.
Why is the axial field of a dipole twice the equatorial field?
At equal distances the on-axis field is μ0(2M)/(4πr³) while the bisector field is μ0M/(4πr³) — the same 2:1 an electric dipole shows.
How does a moving-coil galvanometer measure current?
Coil torque NIAB is balanced by spring torque kθ; with a radial field keeping torque angle-independent, θ = (NAB/k)I is linear in current.
What magnetic moment does an orbiting electron carry?
M = (e/2m)L, tying magnetic moment to angular momentum; for the ground-state Bohr orbit this equals one Bohr magneton, 9.27 × 10⁻²⁴ J/T.