CHAPTER II.

MECHANICAL AND MAGNETIC UNITS.

§ 1. Introductory.-For the study of magnetism and electricity it is necessary that the student should have a knowledge of the principles of elementary mechanics, and a clear understanding of what is meant by a system of units. He will then be in a position to understand in their exact sense the terms employed in the discussion of magnetic and electrical measurements. For reasons of space it is impossible to give here any satisfactory account of the mechanical principles involved in our present subject. We shall merely indicate briefly the nature and extent of the knowledge required.

§ 2. Fundamental and Derived Units.-The three fundamental units to which all other mechanical quantities can be ultimately referred, are the units of length, time, and mass. We shall in what follows employ the centimètre as the unit of length, the second as the unit of time, and the gramme as the unit of The system of units built up upon this foundation is called the 'centimètre-gramme-second system,' or more usually the 'C.G.S. system.'

mass.

Examples of derived units will occur in §§ 3 and 4 and later on. Very simple cases of derived units are the square centimètre as the unit of area, and the cubic centimètre as the unit of volume.

The exact way in which any derived unit involves the fundamental units constitutes what is called the dimensions of the quantity measured by that derived unit. For the purpose of exhibiting the dimensions of any derived unit it is convenient to represent the three fundamentals by the symbols [L], [M], and [T] respectively. Thus the C.G.S. unit of volume has dimensions represented by [L]3.

§ 3. Velocity and Acceleration.—In the same system the unit of velocity will be the velocity of one centimètre per second. If we represent this unit by the symbol [V], we may express the dimen

The rate at which velocity changes is called acceleration; and in our system unit acceleration will be the adding of unit velocity during unit time. Hence the C.G.S. unit of acceleration is one centimètre per second, per second.

We may express the 'dimensions' of acceleration by the relation

§ 4. Force.-The C.G.S. unit of force is called the dyne. It is that which acting on unit mass for unit time gives to it unit velocity.

Thus we may express symbolically the 'dimensions' of force by the relation—

×

[F]=[M] × [A] = [M] x [V] _ [M] x[L]

[T]

[T]2

The student will have learned, in his study of mechanics, to distinguish carefully between mass and weight. In virtue of gravitation, a mass of one gramme is urged downwards with a certain force; this force being called the weight of one gramme. Near the earth's surface this force is such that acting for one second it will give to the gramme mass a velocity of about 981 centimètres per second. Hence, near the earth's surface, the weight of our unit mass is a force measured by about 981 dynes. A gramme then is a mass; and we take it for our unit of mass. But the weight of a gramme is a force, and is very far indeed from being unit force in our system.

Note.-In the English system of units we take the foot, second, and pouna as our three fundamental units.

The unit of force in this system is called the poundal, and is that which acting for one second on one pound mass will give it a velocity of one foot per second.

§ 5. Parallelogram of Forces, &c.-We shall make frequent use of the principle commonly called the 'Parallelogram of Forces.'

It will be assumed that the student is thoroughly conversant with the formulæ for the composition of two component forces into one resultant, and for the converse resolution of a single force into two components, in the case in which the directions of the two components are at right angles to one another.

The case of oblique resolution' will not occur.

§ 6. Moments and Couples.-Let us suppose that we have a body (such, e.g., as a magnetic needle) capable of rctation about an axis. If one or more forces act on the body we shall in general have, as a result, rotation of the body about the axis.

In our present subject the only cases usually occurring are very simple; the forces acting in one plane which is perpendicular to the axis. We shall have to consider two cases.

6

(i.) A single force acting.-Let a single force of F dynes act upon the body, and let the perpendicular between the axis and the direction of the force bep centimètres. Then the turning power' or moment of the force about the axis is, as we know, measured by Fx p. This is the moment urging the body to rotate about the axis; and there is also a force of F dynes tending to move the axis bodily in the direction of the force.

If, for example, the body be a horizontal bar, suspended from its middle point by a thread, and we push one end of the bar, we shall find that the bar tends to rotate and also to move bodily in the direction of our push.

(ii.) A couple acting.-Now consider the case where two equal, parallel, and opposite forces act upon the body, each force being F dynes, and the distance between them being p centimètres. The moment of this couple is, as we know, measured by F × p. In this case we have simply a tendency to rotation, and no pressure on the axis.

When the product Fx is numerically equal to unity-(F being measured in dynes and p in centimètres)—then we have what we call unit moment in the C.G.S. system.

§ 7. Unit Magnetic Pole.-Returning now to magnets and magnetic poles, we observe that similar poles of various pairs of magnets repel one another with forces of various magnitudes. Thus we naturally say that poles may be of different strengths.

We should then endeavour to express the 'strengths' of mag

netic poles in terms of some convenient unit-pole; and further we should choose our unit strength of pole such as to give the simplest relation to force of repulsion expressed in dynes, and distance apart expressed in centimètres.

We have seen that a single pole never occurs alone. But in an indefinitely long straight thin magnet, perfectly magnetised, we have the poles exactly at the ends and indefinitely far apart; or, in other words, we can easily conceive of, and approximately obtain, a pole isolated from another pole. We can then in our definitions speak of 'a pole.' We choose to call the northseeking pole +, and the south-seeking pole -.

It is natural to define unit-pole as that which at a distance of 1 cm. from a similar pole repels it with a force of 1 dyne.

So a pole of two units would be defined as that which at a distance of 1 cm. from unit-pole repelled it with 2 dynes; or at a distance of i cm. from a similar pole repelled it with 2 × 2 = 4 dynes. We shall designate the numerical value of the strength of a pole by μ.

§ 8. Magnetic Fields, and Unit Field. Any region where a pole (e.g. a + unit-pole, which we may conveniently employ as a test-pole) would be urged by magnetic force is called a magnetic field.

In so far as such force is due to any particular magnetic body, we speak of the field due to that body.

The direction in which our + unit-pole at any point is urged is called the line of force at that point.

If a + unit-pole is urged with 1 dyne, the field at that point is defined to have unit strength.

If with 5 dynes, then the field has a strength 5.

In general we shall designate the numerical value of the strength of a field by the letters H or I.

Thus a pole of strength in a field of strength H is urged by a force of μ x H dynes If a field is uniform it means that the strength is the same at all places in it, the lines of force being moreover, as a necessary consequence, parallel to one another; as will be seen in Chapter X. § 13.

§ 9. Magnetic Moment of a Needle.-Let us consider a magnetic needle in a uniform field. Since the poles, ʼn and s, are equal and opposite, and since the lines of force are in a uniform

field parallel, it follows that the forces acting on the poles will be equal and opposite; or that a magnetic needle in a uniform field is acted on by a pure couple.

We have seen that each force will be μ H dynes. Hence if the needle lie at right angles to the lines of force, and if its

length be / cm., the couple acting on the needle will be measured by Ημι units of couple.

The part μl depends on the needle alone; and into whatever field we introduce the needle, so

long as it remains mole

cularly unaltered, this μl remains constant.

We give to this quantity μl the name of the magnetic moment of the needle. We call it m. When μl=1, both and being measured in C.G.S. units as given in §§ 2 and 7, then the needle is said to be of unit moment.

A needle of unit moment in unit field, placed perpendicularly to the lines of force, will be acted on with unit couple.

It will be useful for the student to prove for himself that when a needle of pole-strength μ, and length 1, is placed in a field H so as to make an angle with the lines of force, then the couple acting on the needle is

Hμl sin 0, or H m sin 0.

If we make up a cube of needles by first laying similar needles side by side so as to form a layer, and then piling layer on layer, we can see that the following statement will be true; the magnetic moment of a uniformly magnetised bar is proportional to the volume of the bar. The student should puzzle this out by the 'building up' method just suggested.

§ 10. Magnetic Moment of Practical, not Ideal, Magnets.We have considered our magnet as consisting of two poles occu pying points, separated by a distance called 'the length of the needle.' But the actual magnet may be considered to consist of an indefinite number of such pairs of poles, the poles of each pair being of equal strength and of opposite sign; thus we may