What is Magnetic moment? — Physics

Definition 28.13 University Physics — Year 1 · Chapter 28 — Magnetostatics: Field, Forces, and Dipoles

A plane loop of area $S$ carrying $I$ , oriented by the right-hand rule (normal $\vect n$ along the thumb when the fingers follow $I$ ), has the magnetic moment $\vect m = IS\,\vect n$ ( $\mathrm{A}\,\mathrm{m}^{2}$ ); for $N$ turns, $NIS\,\vect n$ . Seen from distances $r \gg \sqrt S$ the loop is a magnetic dipole: its field has exactly the form of the electric dipole’s with $\vect p/\varepsilon_0 \to \mu_0\vect m$ :

B_r = \frac{\mu_0}{4\pi}\,\frac{2m\cos\theta}{r^3}, \qquad B_\theta = \frac{\mu_0}{4\pi}\,\frac{m\sin\theta}{r^3} .

A bar magnet, an atom, the Earth are magnetic dipoles: the Earth’s moment is $8 \times 10^{22}\,\mathrm{A}\,\mathrm{m}^{2}$ , an electron’s $9.3 \times 10^{-24}\,\mathrm{A}\,\mathrm{m}^{2}$ (the Bohr magneton).

Left: a current loop in a uniform field, seen along its axis of rotation — the Laplace forces on the two sides form a couple that aligns the moment with the field. Right: the field lines of a bar magnet are those of a magnetic dipole, identical in shape to the electric dipole’s, leaving the north face and returning to the south.

Examples

Example 28.15 (The compass and the Earth)

A compass needle is a small magnet, moment $m \sim 0.1\,\mathrm{A}\,\mathrm{m}^{2}$ ; in the horizontal component of the Earth’s field, $B_h \approx 2 \times 10^{-5}\,\mathrm{T}$ , the torque $mB_h\sin\theta$ turns it north, and it oscillates about north with period $2\pi\sqrt{J/mB_h} \sim 1\,\mathrm{s}$ (Exercise 28.9). The Earth’s own dipole, $8 \times 10^{22}\,\mathrm{A}\,\mathrm{m}^{2}$ , is equivalent to a current of a few billion amperes circling in the liquid iron core; the field it makes at the surface is the weakest of the chapter and the first one anybody used.

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Examples

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