force does, over this spherical surface, represent numerically the strength of the field.

At 2 centimètres distant from O these marked lines will have spread out over a sphere of four times the area; and therefore there will be Q lines piercing each 1 square centimètre of the sphere whose radius is 2 centimètres. And, in general, at any distance from O the lines will have so thinned out that the number piercing each 1 square centimètre of the sphere, whose Q

radius is 7, will be

But

Q Q

4 9

Q

also represent the strengths of the field at distances 2 centimètres, 3 centimètres, r centimètres, from O respectively. Therefore these marked lines will represent numerically the strength of the field at any point, by the number of them that at that point cut 1 square centimètre held perpendicular to them.

The reader should carefully study this method of marking out the lines. He will see that it consists in marking out such a number that they do over one equipotential surface (viz. over the sphere of unit radius) represent, by the number piercing each 1 square centimètre, the field-strength at all points over this surface. He will see also that the result found to hold is that these lines so chosen will indicate the field-strength at any distance from O, in just the same manner.

Of course where the force on + unit is less than 1 dyne, or the field-strength is less than unity, there will be less than I line piercing each 1 square centimètre. We must then take several square centimètres and divide the number of lines piercing them by the number of square centimètres taken.

Corollary. Uniform field.-Hence it follows that in a uniform field the marked lines of force are parallel and equidistant from one another.

§ 14. General Case.-We cannot in an elementary Course prove that the same result hoids in the general case; but at least the above discussion will prepare the reader for the following statement. It may be well to point out that 'equipotential surface,' and 'surface lying perpendicularly to the lines of force' are synonymous expressions.

If we mark out such a number of the lines of force that over any one equipotential surface they measure, by the number piercing each square centimètre, the field-strengths all over that equipotential surface, then will these same marked lines in any part of the field measure the field-strength by the number piercing 1 square centimètre held perpendicularly to the lines of force at the place in question.'

It is to be noted that when we use the expression 'number of lines of force,' we refer to these marked lines.

§ 15. Total Number of marked Lines of Force.-In the simple case given in § 13, since we mark Q lines of force for each 1 square centimètre of the sphere of 1 centimètre radius, and since such a sphere has an area of 4π square centimètres, it follows that we have a number 47 Q of such marked lines.

And if there be any system of quantities of electricity + or -, and if the whole system of quantities be surrounded by an envelope, then the total number of + lines piercing this envelope will be 42Q; where ΣQ is the algebraic sum of the quantities.

§ 16. Tubes of Force. Fo is Constant.'-We may, out of the infinite crowd of lines of force, construct hollow tubes bounded by the lines. Such tubes are called tubes of force.

r.

Now lines of force cannot cross; for, if they did, it would mean that at the point where they crossed. there were two resultant

forces, which would be absurd. Hence, if such a tube once include any number n of the marked lines, it will always include these same lines, neither more nor fewer.

Let us consider the simple case of § 13, and let us suppose the area of the perpendicular cross section of such a tube to be • square centimètres. Then, if the tube enclose n lines of force, there are n lines of force piercing σ, square centimètres; or the

n

average force on our + unit will be measured by according to

the results of § 13. If we name this average force F1, then the product of (average force over cross section) into (the cross section) will be

If we take another section 2 of the same tube we have for

Hence we have for the same tube of force the result that F11 = F22= = = n = constant; which is more shortly

1

Fo constant.

= σ sec 8,

If we take an oblique section σ', and take the force F' resolved perpendicular to this oblique section, then F' = F cos 0, and o' where is the angle that the oblique section makes with the perpendicular section. Hence the product F'o'

=

Εσ

= constant.

We may state this property of tubes of force as follows.

If o be the area of cross section of a tube of force, and F be the average force on + unit over this area o resolved perpendicularly to it, then the product Fo is constant; it equals n, where n is the number of marked lines of force enclosed by the tube in question.

We have proved the above only for the simple case of § 13; but we here state that it is true for any field, however complex.

A tube of force in which one marked line is enclosed, or a tube where Fo= 1, is called a unit tube of force.

§ 17. Statement of some further Theorems on Lines of Force.

(1) It can be shown that a tube of force terminates against equal and opposite quantities of electricity, + q and q respectively. This statement includes Faraday's law of electrostatic induction referred to and discussed in Chapter IV. § 16.

(2) Lines of force and tubes of force do so terminate when they meet a conducting surface. Any lines of force on the other side of the conducting surface are either lines that have curved round part the edge of the conductor, or are independent lines due to other charged systems.

(3) Hollow conductor acting as a screen.—The diagram represents in section a hellow conductor X Y separating two systems of electrical charges, a b c external, and p q r internal.

Now theory and experiment show that no lines of force due to abc exist inside X Y (see Chapter IV. § 13 (ii.)). Hence, if XY is to earth the system par is totally unaffected by abc; and if

X Y be not to earth the only influence of a be will be to raise or lower the potential of X Y and of its interior as a whole; they do not in either case alter the nature of the field inside.

Again, both theory and experiment show that the external effect of the presence of the system pqr inside X Y is to cause there

So we may use large conducting plates put to earth as fairly A wire-gauze case will completely screen an

complete screens.

electroscope (see note).

(4) Potential at an external point due to uniformly charged concentric spheres.-It can be shown by the integral calculus that if we have

Note on insulated spherical screens.-We will consider here two simple cases of uninsulated and uncharged screens, showing what effect their presence has upon the potential of an external point. The reader can easily draw for himself the simple diagrams that represent the two cases considered.

(i.) Let there be a charged sphere, or a charged particle, A; and let there be an external point P at a distance R from A.

Q units, the potential Vp of P is measured by

R

Then if the charge on A be

If now an insulated and un

charged spherical screen B be placed round A concentrically with it, between it and the point P, we know both by theory and experiment that the field at P is due only to the 'free' charge that has been ‘repelled to the surface of B ; the charge + Q on A, and the induced charge Q on the inner surface of B, together give a zero field at any point P outside. But, by (4) above, the potential at P will be the same as if the charge Q on the sphere B were collected at its centre. That is, we have still Vp. Hence, when A and B

R

are concentric, the presence of the insulated spherical screen B makes no difference in the potential of external points.

(ii.) Next, let B be no longer concentric with A; but let the distance from P to the centre of A be R, while to the centre of B the distance from P is r. Then

without the screen we have Vp

; since it is only the free charge on the screen that gives a field

at the external point P.

(iii.) If any charge Q' be given to B, we have merely to add it to the induced charge Q that has been inductively given to its surface; and the potential at P becomes

$ 18. The Potential of an 'Isolated' Body.-Potential is a property of a point in space, as explained earlier. But any equipotential surface has one potential, the same for all points on it. Now any continuous homogeneous conductor (as a brass sphere or cylinder, a tin vessel, &c.) forms an equipotential surface; for no differences of potential could exist on a surface where electrical readjustment takes place instantly. Hence we can speak of 'the potential of a conductor.'

The potential may be a hard matter to calculate, owing to the continuous nature of the distribution of electricity over the body, which makes the expression of § 9 (iii.) in general a matter for the integral calculus. But we can easily deal with the case of a sphere.

r

$ 19. Potential of an Isolated Sphere.-The figure represents in section a sphere of centre O and radius R. Let it have on