Page images
PDF
EPUB

BOOK VI.

MAGNETISM AND ELECTRICITY.

SECTION XLIX.

THE MEASUREMENT OF MAGNETIC FIELDS.

Apparatus required: Mirror magnetometer, lamp and scale, bar magnet, and vibration box.

The intensity of the magnetic field at a point is measured numerically by the force on a unit pole placed at that point. It can therefore be represented by a straight line, the direction of the line representing the direction, and the length representing the intensity, of the force. If two fields are superposed on each other the intensity of the combined field may be obtained in the same way as the resultant of two forces. If OP, Fig. 101 a, represents the direction and magnitude of the magnetic force on a unit pole placed at O, due to one of the fields, and OQ that due to the other, the combined effect will be represented by the diagonal OR of the parallelogram OPQR. The magnitude and direction of the resultant field may be calculated like the corresponding quantities in the case of a resultant force, by the following equations:

OR2 = OP2 + OQ2 + 20P. OQ cos POQ.........(1),

[blocks in formation]

Two special cases are of frequent occurrence. If OP and OQ are at right angles to each other (Fig. 101 b), the magnitude of the resultant is given by

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

The second case (Fig. 101 c) is that in which the resultant OR is at right angles to one of the forces OP. Then

[blocks in formation]

Exercise. The magnetic force on unit pole at a point distant r cms. from a straight wire of infinite length through which a current of strength i units is passing, is equal to 2i/r, and the lines of force are circles having the wire as axis. If the wire is vertical and carries a current of one unit, find by calculation or geometrical construction the direction in which a small magnetic needle will point when placed respectively east, west, north, south, and north-east of the wire at distances of 4, 11, or 15 centimetres from the wire. The earth's horizontal magnetic force may be taken as 17.

If the magnetic field is due to a magnet having its north pole of strength μ at B and its south pole of strength – μ at A, the magnetic force on unit pole at a point P due to the north pole at B is μ/BP2, that due to the south pole at A is - μ/AP2. If the point P is on the straight line through the poles A and B, the two forces act in the same straight line, and have a

resultant in the direction of the stronger, of intensity I such that

[blocks in formation]

If we call the magnetic moment μAB of the magnet M, the distance of P from the centre of AB d, and if I is half the distance between the poles, i.e. about

magnet, the resultant intensity

2 Md

the length of the

2d

[merged small][ocr errors]
[blocks in formation]

M.

(d2 — 12)2 °

If the point P is so far from the magnet that is small

compared to d, we have simply I = M.

2

[ocr errors]

If the axis of the magnet is at right angles to the magnetic meridian, and H is the earth's horizontal force at the point P, the angle which the resultant force makes with the magnetic meridian is such that

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]

A small magnetic needle placed at the point at which the resultant has been calculated will set itself in a direction forming an angle with the magnetic meridian, and therefore enable to be found.

The resultant magnetic force at a point P due to a magnet AB, which in the above case has been calculated when P is on AB produced, may also be easily determined when P lies on a line through the centre C of the magnet perpendicular to AB. Since AP BP the magnetic forces due to the two poles are equal, and their resultant bisects the angle between their directions, ie. is parallel to AB. Each force having an inμAC AP3

=

tensity has a component

μ AP2

[blocks in formation]

parallel to AB, hence the

Writing again M for the magnetic moment of the magnet,

I for half the distance between the poles, and d for the distance

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors]

Hence the field at a distance d from the centre of a short magnet along the axis of the magnet, is twice the field at the same distance along a line through the centre of the magnet perpendicular to the axis. If the law of force had been the inverse nth power instead of the inverse square, the former force would have been found to be n times the latter. Hence an experimental determination of the ratio of these forces will furnish a proof of the law of action of magnetic poles on each other.

If the axis of the magnet is at right angles to the magnetic meridian, and H is again the earth's horizontal force, the resultant of the two fields will make an angle meridian where

[merged small][merged small][merged small][merged small][ocr errors]

with the

[blocks in formation]

(5).

The angle may as before be determined by placing a small magnetic needle at the point P, and observing the - deflection produced when the magnet is placed in position. The apparatus provided Fig. 101d. consists of a small

[ocr errors][merged small]

magnetic needle to which a mirror is attached, suspended by a fine fibre, the torsion of which may be neglected. The centre of the needle is situated 3 mms. above the middle point of the upper of the two horizontal graduated scales placed at right angles to each other. One of the scales is placed in the magnetic meridian, i.e. parallel to the axis of a magnetic needle brought near it. For convenience this scale should be about 2 mms. below the other. The rotation of the mirror is determined in the usual manner by the motion of the image of a cross wire formed after reflection by the mirror, on a scale placed parallel to the mirror when in its central position. The cross wire is attached to the scale and illuminated by a lamp. (For the method of obtaining the angles of rotation see Section XXXIV.)

The position of the image of the cross wire on the scale is first read.

The deflecting magnet provided is then placed on the scale running east and west, say to the west of the needle, with its north pole towards the needle, the positions of the ends of the magnet are observed, and the scale reading of the cross wire is taken. The magnet is then reversed so that its south pole now points towards the needle, and the deflection again observed. It is then transferred to the east of the needle and the two deflections again observed.

To make the corresponding observations with the magnet north and south of the needle, the magnet is placed on a frame which slides along the scale in the magnetic meridian, the magnet itself being at right angles to the meridian. The height of the frame is such that the magnet itself will be at the same level as when placed on the other scale. The frame is moved along the scale till the centre of the magnetic axis is at the same distance from the centre of the needle as previously, the north pole of the magnet being say to the east. The deflection of the cross wire is observed, the magnet is reversed so that its north pole points to the west, and the observation repeated. The frame and magnet are then transferred to the other side of the needle and the observations repeated.

A set of observations should be taken for each of three

« PreviousContinue »