3. Magnetic Measurements
Magnetic Moment.—The moment M of a magnet may be determined in many ways,[17] the most accurate being that of C. F. Gauss, which gives the value not only of M, but also that of H, the horizontal component of the earth’s force. The product MH is first determined by suspending the magnet horizontally, and causing it to vibrate in small arcs. If A is the moment of inertia of the magnet, and t the time of a complete vibration, MH = 4π²A / t² (torsion being neglected). The ratio M/H is then found by one of the magnetometric methods which in their simplest forms are described below. Equation (44) shows that as a first approximation.
M / H = (d² − l²) tan θ/2d,
where l is half the length of the magnet, which is placed in the “broadside-on” position as regards a small suspended magnetic needle, d the distance between the centre of the magnet and the needle, and θ the angle through which the needle is deflected by the magnet. We get therefore
M² = MH × M/H = 2π²A (d² − l²)² tan θ/t²d
(42)
H² = MH × H/M = 8π²Ad / {t² (d² − l²)² tan θ}.
(43)
When a high degree of accuracy is required, the experiments and calculations are less simple, and various corrections are applied. The moment of a magnet may also be deduced from a measurement of the couple exerted on the magnet by a uniform field H. Thus if the magnet is suspended horizontally by a fine wire, which, when the magnetic axis points north and south, is free from torsion, and if θ is the angle through which the upper end of the wire must be twisted to make the magnet point east and west, then MH = Cθ, or M = Cθ/H, where C is the torsional couple for 1°. A bifilar suspension is sometimes used instead of a single wire. If P is the weight of the magnet, l the length of each of the two threads, 2a the distance between their upper points of attachment, and 2b that between the lower points, then, approximately, MH = P(ab/l) sin θ. It is often sufficient to find the ratio of the moment of one magnet to that of another. If two magnets having moments M, M′ are arranged at right angles to each other upon a horizontal support which is free to rotate, their resultant R will set itself in the magnetic meridian. Let θ be the angle which the standard magnet M makes with the meridian, then M′/R = sin θ, and M/R = cos θ, whence M′ = M tan θ.
| Fig. 5. |