Poles & Bar Magnets

A small current loop and an electric dipole obey equations of the same shape. That suggests a handy picture: magnetic “poles” that behave like magnetic charges. It explains bar magnets beautifully.

The magnetic pole model#

A small current loop makes a field (μ₀/4π)(2μ⃗/r³) on its axis and feels a torque μ⃗ × B⃗; an electric dipole makes (1/4πε₀)(2p⃗/r³) and feels p⃗ × E⃗. Swap p⃗ for μ⃗ and 1/4πε₀ for μ₀/4π and they match. So we may imagine:

• Magnetic charges (poles) of two kinds. A pole of strength m in a field B⃗ feels F⃗ = mB⃗: along B⃗ for a positive pole, against it for a negative one.
• A pole m produces a radial field B = (μ₀/4π)(m/r²).
• A magnetic dipole is −m and +m a small distance d apart, with moment md pointing from −m to +m.
• A current loop of area A carrying i is equivalent to a dipole with md = iA, along the loop’s area vector.

A solenoid as two poles#

Replace each turn of a solenoid by a tiny dipole, choosing the separation so the N of one touches the S of the next. Inside they cancel in pairs, leaving one S pole and one N pole of strength m at the two ends, a solenoid’s length apart. This gives the field outside the solenoid correctly. (Inside, the real field runs from S to N, opposite to the pole picture: magnetic field lines are closed loops, never starting or ending like electric ones.)

Why a bar magnet is a magnet#

Atoms contain circulating electrons: tiny current loops. In a magnet these loops are lined up, all circulating the same way. Inside the bar, neighbouring loops cancel; at the surface nothing cancels them, leaving a net surface current. So a magnet is equivalent to a tightly wound solenoid. If the surface current is I per unit length and the cross-section is A, the pole strength is m = IA.

End effects put the poles slightly inside the bar. The distance between them is the magnetic length; magnetic length ÷ geometric length ≈ 0.84. Writing the magnetic length as 2l:

Torque and potential energy#

In a uniform field B at angle θ, the forces ±mB on the poles form a couple of moment 2ml B sin θ:

Turning it slowly from θ₀ to θ takes work ∫MB sin θ dθ = MB(cos θ₀ − cos θ), stored as potential energy. Taking U = 0 at θ = 90°: