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PURPOSE

To calibrate an oscillating magnet in the field of a current carrying coil and to measure the Earth's field.

EQUIPMENT  Helmholtz pair, 5 amp power supply, ammeter, compass, reversing switch, 22 Ω (5 A) rheostat, jar magnet, stopwatch, half-meter stick.

RELEVANT EQUATIONS

Torque on a magnet
Period of oscillating magnet
Field of Helmholtz Pair

DISCUSSION

The opposite ends or poles of a magnet are designated North-seeking (N) and South-seeking (S), just as the two kinds of electric charge are designated positive (+) and negative (−). As with charge, opposite poles attract each other and like poles repel each other. Furthermore, the effect that a magnet or group of magnets has on a pole is conveniently described in terms of the magnetic field B, which is analogous to the description of electric forces in terms of the electric field E.

The magnetic field at a point P produced by a magnetic pole is equal to the force exerted by the magnet on a unit North-seeking magnetic pole located at P. Thus, the magnetic field is a vector quantity that can be associated with every point in the vicinity of the magnet. It can be graphically represented by drawing the field lines associated with the magnet, as shown in Fig. 10-1. At any point P, the direction of B is tangent to the field line at P and the magnitude of B is indicated by the spacing of the lines: the closer the lines are spaced, the greater the magnitude of B. The unit of measure for B in the SI system is the tesla (T).

In the case of a C-magnet, the direction of B is from the North-seeking to the South-seeking pole and the lines are more or less equally spaced, which indicates that the magnitude of B is constant throughout the region between the poles. In the case of a bar magnet, the direction and magnitude of B vary throughout the surrounding space in a complicated way. Obviously the magnitude of B is greatest near the poles of the magnet.

The field lines produced by a particular source can be determined by using a small bar magnet, such as a compass needle, mounted on a pivot so that it is free to rotate. A magnet with magnetization M responds to a magnetic field at its location by lining up with the field lines. The reason for this is that an external magnetic field produces a torque τ on the bar magnet that is proportional both to the field strength and to the sine of the angle θ between the bar magnet and the direction of B.

(1)

To be more precise, let us draw an arrow from the South-seeking to the North-seeking pole of the bar magnet. This arrow represents the magnetization vector M of the bar magnet. As shown in Fig. 10-2 the magnetic torque on the bar magnet always acts to align M with B. This kind of torque is called a restoring torque because it tends to restore the magnet to the aligned position.

If the bar magnet is already lined up with B, the torque is zero and the magnet will remain in that orientation. If the bar magnet is not initially lined up with B however, the restoring torque will cause the bar magnet to oscillate about the direction of B.

The period of oscillation T of a magnet that has a moment of inertia I can be shown to be inversely proportional to the square root of the field strength by solving the differential equation obtained by writing down the angular form of Newton's 2nd law:

using the small angle approximation,

and

(2)

Therefore the stronger the field is, the shorter the period is (i.e., the faster the magnet oscillates). These oscillations eventually die out because of friction.

This is essentially the way a compass works. The compass needle is simply a bar magnet and its North-seeking pole is usually designated by a colored or other type of mark. The North-seeking pole of the needle always points North, because the Earth itself produces a magnetic field to which the compass needle responds. In fact, the Earth can be likened to a giant bar magnet, which is approximately aligned with its rotational axis. It is apparent that the way we distinguish between the poles of a bar magnet is to see to which magnetic pole of the Earth they point. Thus, the colored end of a compass needle and the end of a bar magnet marked "N" are properly called North-seeking poles because they point to the North magnetic pole of the Earth.