Spin-Only Magnetic Moment
Compute the spin-only magnetic moment µ = √(n(n+2)) in Bohr magnetons.
µ (µB)
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Magnetic moment: µ = I·A
A loop carrying current acts like a magnetic dipole, and its moment is µ = I · A. Here I is the current (A) and A the area enclosed by the loop (m²). The vector points perpendicular to the plane; the right-hand rule tells you which way. Wind the wire into N turns and you get µ = N · I · A. Down at the atomic scale people work in Bohr magnetons, µ_B = 9.274·10⁻²⁴ J/T (set by the electron), while nuclei are measured in nuclear magnetons, µ_N = 5.051·10⁻²⁷ J/T. Put the dipole in an external field B and it stores energy U = −µ·B·cos θ while feeling a torque τ = µ·B·sin θ. Example: a single-turn loop carrying 2 A over 0.01 m² has µ = 0.02 A·m²; in B = 0.5 T at 90° the torque is τ = 0.01 N·m.
Applications: motors, MRI and compasses
You meet magnetic moments all over physics. Electric motors spin because a coil in the stator field feels a torque. NMR and MRI rely on nuclear spins lining up with B and precessing at the Larmor frequency, which is what produces the imaging signal. A compass needle is just a tiny permanent moment swinging into Earth's 25–65 µT field. And the electron g-factor stands as one of the sharpest tests of QED ever made (Schwinger's prediction, Nobel 1965).
FAQ
Why is µ perpendicular to the loop? A circulating current produces a field that looks like a small bar magnet, and that magnet's axis runs normal to the plane of the loop. The right-hand rule pins down which of the two perpendicular directions it points.
What's the difference between µ_B and µ_N? One is built from the electron mass, the other from the proton mass. Since µ ∝ 1/m, the nuclear magneton ends up roughly 1836× smaller than the Bohr magneton.
Does the loop's shape matter? Not at all. Only the enclosed area and the current count, so a square, a circle and a triangle of equal area all give the same µ.
What happens when µ is parallel to B? The torque drops to zero and the energy bottoms out at U = −µB. That's the stable equilibrium, the orientation the dipole "wants" to settle into along the field.
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