A Treatise on Electricity and Magnetism/Part I/Chapter VIII

[ page ] [ page ]123.] AND LINES OF INDUCTION. 149

a corresponding series of values of 9, and if E be an integer, the number of corresponding lines of force, including the axis, will be equal to E.

We have therefore a method of drawing lines of force so that the charge of any centre is indicated by the number of lines which converge to it, and the induction through any surface cut off in the way described is measured by the number of lines of force which pass through it. The dotted straight lines on the left hand side of Fig. 6 represent the lines of force due to each of two electrified points whose charges are 10 and 10 respectively.

If there are two centres of force on the axis of the figure we may draw the lines of force for each axis corresponding to values of ^ and ^ 2 , and then, by drawing lines through the consecutive intersections of these lines, for which the value of ^ + ^2 is the same, we may find the lines of force due to both centres, and in the same way we may combine any two systems of lines of force which are symmetrically situated about the same axis. The con tinuous curves on the left hand side of Fig. 6 represent the lines of force due to the tsvo electrified points acting at once.

After the equipotential surfaces and lines of force have been constructed by this method the accuracy of the drawing may be tested by observing whether the two systems of lines are every where orthogonal, and whether the distance between consecutive eqiipotential surfaces is to the distance between consecutive lines of force as half the distance from the axis is to the assumed unit of length.

In the case of any such system of finite dimensions the line of force whose index number is ^ has an asymptote which passes

through the centre of gravity of the system, and is inclined to the

^/ axis at an angle whose cosine is 1 2 -^ , where E is the total

electrification of the system, provided ^ is less than E. Lines of force whose index is greater than E are finite lines.

The lines of force corresponding to a field of uniform force parallel to the axis are lines parallel to the axis, the distances from the axis being the square roots of an arithmetical series.

The theory of equipotential surfaces and lines of force in two dimensions will be given when we come to the theory of conjugate functions *.

• See a paper On the Flow of Electricity in Conducting Surfaces, by Prof. W. R.

Smith, Proc. R. S. Edin., 1869-70, p. 79.

�� � [ page ]CHAPTER VIII.

SIMPLE CASES OP ELECTEIFICATIOtf.

Two Parallel Planes.

124.] We shall consider, in the first place, two parallel plane conducting surfaces of infinite extent, at a distance c from each other, maintained respectively at potentials A and B.

It is manifest that in this case the potential V will be a function of the distance z from the plane A, and will be the same for all points of any parallel plane between A and J3, except near the boundaries of the electrified surfaces, which by the supposition are at an infinitely great distance from the point considered.

Hence, Laplace s equation becomes reduced to

��__

==

the integral of which is

7= C, + C 2 z;

and since when z = 0, V = A, and when z = <?, V = B,

��For all points between the planes, the resultant electrical force is normal to the planes, and its magnitude is

��c

In the substance of the conductors themselves, R = 0. Hence the distribution of electricity on the first plane has a surface- density <r, where AB

47TO- = R = --

c

On the other surface, where the potential is jB, the surface- density a- will be equal and opposite to <r, and

�� � [ page ]1 24.] SIMPLE CASES. PARALLEL PLANES. 151

Let us next consider a portion of the first surface whose area is S, taken so that no part of S is near the boundary of the surface.

The quantity of electricity on this surface is E = S<r, and, by Art. 79, the force acting on every unit of electricity is \R, so that the whole force acting on the area S, and attracting it towards the other plane, is

��Here the attraction is expressed in terms of the area S, the difference of potentials of the two surfaces (A B), and the distance between them c. The attraction, expressed in terms of the charge E } on the area S, is 2 TT

The electrical energy due to the distribution of electricity on the area S, and that on an area S on the surface B denned by projecting S on the surface B by a system of lines of force, which in this case are normals to the planes, is Q=

- 2

��- 27r E*c

- -3- A c,

= Fc.

The first of these expressions is the general expression of elec trical energy.

The second gives the energy in terms of the area, the distance, and the difference of potentials.

The third gives it in terms of the resultant force R, and the volume Sc included between the areas S and S , and shews that the energy in unit of volume isp where 8 nfl = R 2 .

The attraction between the planes is jo/S> or in other words, there is an electrical tension (or negative pressure) equal to p on every unit of area.

The fourth expression gives the energy in terms of the charge.

The fifth shews that the electrical energy is equal to the work which would be done by the electric force if the two surfaces were to be brought together, moving parallel to themselves, with their electric charges constant.

�� � [ page ]152 SIMPLE CASES.

To express the charge in terms of the difference of potentials, we have i

��1 o

The coefficient = q represents the charge due to a differ ence of potentials equal to unity. This coefficient is called the Capacity of the surface S, due to its position relatively to the opposite surface.

Let us now suppose that the medium between the two surfaces is no longer air but some other dielectric substance whose specific inductive capacity is K, then the charge due to a given difference of potentials will be K times as great as when the dielectric is air, or

��The total energy will be

��_ 2^

- gjgJSl C.

The force between the surfaces will be

_ KS (B-A)*

��--E* ~ KS l

Hence the force between two surfaces kept at given potentials varies directly as K, the specific capacity of the dielectric, but the force between two surfaces charged with given quantities of elec tricity varies inversely as K.

Two Concentric Spherical Surfaces.

125.] Let two concentric spherical surfaces of radii a and , of which I is the greater, be maintained at potentials A and B respectively, then it is manifest that the potential V is a function of r the distance from the centre. In this case, Laplace s equation becomes d*V 2 dV

~W + r ~dr =

The integral of this is

F=Q+Qr-i;

and the condition that V A when r = a, and V = B when r = 6, gives for the space between the spherical surfaces,

�� � [ page ]12 5-] CONCENTRIC SPHERICAL SURFACES. 153

Aa-Bb A-B

��r=

��d 1) & i I) dV A-B

��_ 2

��If (7 15 <r 2 are the surface-densities on the opposed surfaces of a solid sphere of radius a, and a spherical hollow of radius b, then

1 A-B 1 B-A

��If EI and ^2 be the whole charges of electricity on these surfaces,

��Tlie capacity of the enclosed sphere is therefore 7

If the outer surface of the shell be also spherical and of radius c, then, if there are no other conductors in the neighbourhood, the charge on the outer surface is

E 3 = Be.

Hence the whole charge on the inner sphere is

��and that of the outer

��If we put = oo, we have the case of a sphere in an infinite space. The electric capacity of such a sphere is a, or it is nu merically equal to its radius.

The electric tension on the inner sphere per unit of area is

(A -By 2

��STT a 2 (b-a) 2

The resultant of this tension over a hemisphere is ira 2 j) = F normal to the base of the hemisphere, and if this is balanced by a surface tension exerted across the circular boundary of the hemi sphere, the tension on unit of length being T, we have

F= 2iraT.

b* (A-B) 2 Ef Hence

��l - 8 (b of 8 a

��16iro (b-a

�� � [ page ]154 SIMPLE CASES. [126.

If a spherical soap bubble is electrified to a potential A, then, if its radius is a, the charge will be Aa, and the surface-density will be I A

47T a

The resultant electrical force just outside the surface will be 4770-, and inside the bubble it is zero, so that by Art. 79 the electrical force on unit of area of the surface will be 27ro- 2 , acting outwards. Hence the electrification will diminish the pressure of the air within the bubble by 27ro- 2 , or

��But it may be shewn that if T is the tension which the liquid film exerts across a line of unit length, then the pressure from

T within required to keep the bubble from collapsing is 2 - . If the

electrical force is just sufficient to keep the bubble in equilibrium when the air within and without is at the same pressure

A 2 = IGvaT.

��Two Infinite Coaxal Cylindric Surfaces.

126.] Let the radius of the outer surface of a conducting cylinder be , and let the radius of the inner surface of a hollow cylinder, having the same axis with the first, be I. Let their potentials be A and B respectively. Then, since the potential V is in this case a function of r, the distance from the axis, Laplace s equation becomes

d 2 F \_dV_

dr 2 + r~fo ==

whence V = Q + C 2 log r.

Since V = A when r = a, and V = B when r = b,

A log |- .Slog -

V = r a -

��If o-j, o- 2 are the surface-densities on the inner and outer surfaces,

A-B B-A

47701 = - -, 4770*2 =

�� �

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