CHAPTER X.

ELECTROSTATIC POTENTIAL.

§ 1. Introductory.-In the present chapter we purpose dealing more fully with the conceptions of electrostatic potential, fields and lines of electric force, and work of electric charge and discharge. We recommend the student to read again carefully the following portions of the earlier chapters of this course.

(i.) Chapter II. § 8, &c. Here he should find no difficulty in making for himself all such necessary changes in the wording as the substitution of 'a + unit of electricity' for 'a + magnetic pole of unit strength,' &c.

(ii.) The whole of Chapter V.

(iii.) In Chapter VI., all that relates to the formulæ connecting K, V, and Q.

(iv.) Chapter VII. § 1 ; and Chapter VIII. §§ 5 and 6.

He will then be in a better position to understand on what points he needs more exact knowledge and fuller discussion.

§ 2. Definition and Measurement of Work.-—In addition to the mechanical principles and terms briefly mentioned in Chapter II., we must now assume a knowledge of the principle of work. The student should clearly understand

(i.) What 'work' means; and its essential difference from 'force.'

(ii.) What 'positive' and 'negative' work mean respectively. (iii.) In what sense, and under what general conditions, we can say that 'when a body is displaced in a field of force from a position A to a position B the work done is independent of the route followed between these two positions.'

In the C.G.S. system the unit of work is called the 'erg,' and is the work done when a force of one dyne is overcome through a distance of one centimètre along the lines of force.

§ 3. Dimensions of Work.-We see that the unit of work [W], or the erg, involves the fundamental units in the following way.

[V]

[W] = [F] × [L] = [M] × [A] × [L] = [M] × × [L];

[T]

§ 4. Energy and Conservation of Energy.-(a) If we have a supply of heat we can by a properly constructed engine do work and use up (or cause to disappear) some of the heat.

(b) If a cannon-ball be moving, we can cause it to do work such as the raising of weights, until it comes to rest.

(c) If we have a body of water at a higher level, we can cause it to do work by means of a water-mill until it is all at the lower level.

(d) If an elastic be stretched, we can make it do work until it has returned to its unstretched condition.

(e) If two substances have chemical affinity for one another, we can let them combine and give out heat, and can then cause this heat to be used up in doing work, this going on until the chemical affinity is 'satisfied.'

In each of the above cases we had a condition of things such that we could get work done (measured in ergs, foot-pounds-weight, or in any other unit involving the essential product of force overcome into distance through which it is overcome) at the expense of losing this advantageous condition of things.

Now when a body, or a system of bodies, is in such a condition that it can do work, then it or they are said to possess energy. In fact, energy may be sufficiently accurately defined as capacity for doing work, and it is measured by the work that can be done, or it is measured in ergs.

Where the energy is possessed (as in cases (a) and (b) above) in virtue of mass' and velocity combined, it is called kinetic energy. Where it is possessed (as in the other cases) in virtue of position, it is called potential energy. In case (e) we may call it chemical potential energy.

One of the greatest generalisations of modern times, the most powerful weapon we possess in discussing the problems of physical science, is the great law of conservation of energy. This law,

abundantly proved by direct experiment and corroborated by masses of more indirect evidence, means that energy is indestructible. This means that in a self-contained system-(or a system not acting upon nor acted upon by any other system)-the total amount of energy is constant, though it may undergo transformation, as, e.g., from the form of mechanical-energy into the form of heat-energy.

The student should study carefully the principle of 'conservation of energy' as given in books on mechanics, and should make himself familiar with cases of transformations of energy.

§ 5. Work against a Constant Force. Where the field of force acting on a particle is uniform, the calculation of work done is simple enough.

It is merely necessary to know the force in dynes and the distance between the initial and final positions of the particle, as measured along the lines of force, in centimètres. Then the product gives us the work in ergs.

All work done against gravitation on the surface of the earth comes practically under this head; since, for distances small as compared with the radius of the earth, we may consider the field to be constant.

$6. Work where the Force varies as the Distance.-Another case of work may here be considered, though it is of no direct interest in our present subject.

When a piece of india-rubber is stretched from the length it has when under no tension into some new length, we overcome a gradu ally increasing force. This force is proportional to the extension. If then we start with zero force and end with a force of F dynes, the extension being b centimètres, it is not difficult to see that

In this case we may call F the average force overcome.

§ 7. Work, where the Force varies as the Inverse Square of the Distance, or as -In our present subject we are con

I

cerned mainly with forces that act towards or from centres of force, the magnitude of the force, due to any such centre, varying inversely as the square of the distance from that centre. A single magnetic pole, or a single sphere charged with electricity, are simple cases of such centres of force.

K

We will first discuss this important case without making any reference to the nature of the force, whether it be electrical, magnetic, or of any other description; and then we will apply the general results obtained to electroo statics.

B

If at 1 centimètre the force be measured by F dynes, then at dis

it will be

F tance ra (r)2 r1 and r being measured in centimètres.

and at distance 7 it will be

A

What work then will be done in moving the particle from A to B? This is evidently no simple matter of arithmetic, for we cannot by any arithmetical means find the average force as we did in § 6.

We might work out the problem here without going into any advanced mathematics. But, as the calculation is one which belongs essentially to the integral calculus, and is a very easy matter to settle with the aid of this powerful mathematical weapon, we prefer in an elementary Course to give merely the result.

If A be so remote from O that the force at A is quite inappreciable as compared with that at B, or so remote that and

I

(ra)

I

I

I

(7)2

then

We

(7)'

are practically zero as compared with and we may consider A to be at an infinite distance from O. thus obtain from our formula the result that

Work done in bringing the particle

from an infinite distance up to B

The reader is here warned to remember

=

F

ergs.

B

(i.) That F is the force acting on the particle when at unit.

distance from O.

(ii.) That refers to the distance from O up to which the particle is brought, and is not the distance through which it is moved.

§ 8. Potential, and Difference of Potential.—If the reader will turn back to Chapter V. he will there see explained the general meaning of potential, and the measurement of the same with respect to some chosen zero. The explanation was, however, incomplete, because we had not then considered the question of Work.

Referring to the diagram given in § 7, we now understand that there is between A and B a certain difference of potential; and that it is measured by the work done on some sort of unit particle in bringing this from A to B.

If the field of force be one due to gravitation, and be considered to be practically uniform and acting with a force of 981 dynes on gramme mass, or with 1 dyne on gramme, then we do one erg work in raising our mass of gramme through 1 centimètre vertically. Or we could theoretically measure in centimètres the difference of gravitation potential or level between two points by finding the work done in ergs in raising a particle of gramme mass from one point to the other.

9

If the field of force be electrical, then we take as our unit particle a unit of electricity; and we measure the difference of electric-potential between the points A and B by the work that is done on this test-particle in moving it from A to B.

If the work is found to be negative, we say that the final position has a lower potential than the initial position.

-

Note. The reader must observe that all fields of force can be considered from the potential point of view, and that we may speak of gravitation, potential as well as of magnetic- or electric-potential.

§ 9. Application of § 7 to the Measurement of Electric-Potertial.-Referring to the diagram given in § 7, let us now consider the case of electrostatic forces.

Let us take as our test-particle (see above) a + unit of electricity; and let there be situated at O a quantity of electricity measured by Q units, this giving us an electrical field of force.

By the laws of electrostatics, summed up at the end of § 13, Chapter IV., the force exerted upon our + unit by a quantity of