The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 5 of 39

In the particular case in which the cone is one of revolution (e.g., if the whole field is symmetrical about an axis, as in figures 16 and 20), the equation of the cone must become

p + v'2 _ 2£'» = 0,

where the axis of £' is the axis of symmetry. The section of the equipotential made by any plane through the axis, say that of £'§", must now become

£/s_2£'2 = o

in the neighbourhood of the point of equilibrium, and this shews that the tangents to the equipotentials each make a constant angle tan-1 /2 (= 54° 44') with the axis of symmetry.

In the more general cases in which there is not symmetry about an axis, the two branches of the surface will in general intersect in a line, and the cone reduces to two planes, the equation being

dp + br}'2 = 0,

where the axis of £' is the line of intersection. We now have a + b = 0, so that the tangent planes to the equipotential intersect at right angles.

An analogous theorem can be proved when n sheets of an equipotential intersect at a point. The theorem states that the n sheets make equal angles 7r/n with one another. (Rankin's Theorem, see Maxwell's Electricity and Magnetism, § 115, or Thomson and Tait's Natural Philosophy, § 780.)

70. A conductor is always an equipotential, and can be constructed so as to cut itself at any angle we please. It will be seen that the foregoing theorems can fail either through the a, b and c of equation (24) all vanishing, or through their all becoming infinite. In the former case the potential near a point at which the conductor cuts itself, is of the form (cf. equation (25)),

/ d3V cPV \

**<-K+t(pg+«P,&5+...).

so that the components of intensity are of the forms

-»(■ aF + 2&5S* + "

The intensity near the point of equilibrium is therefore a small quantity of the second order, and since by Coulomb's Law R = 4nrcr, it follows that the

69-71]

Equipotentials and Lines of Force

61

surface density is zero along the line of intersection, and is proportional to the square of the distance from the line of intersection at adjacent points.

If, however, a, b and c are all infinite, we have the electric intensity also infinite, and therefore the surface density is infinite along the line of inter- section.

It is clear that the surface density will vanish when the conducting surface cuts itself in such a way that the angle less than two right angles is external to the conductor; and that the surface density will become infinite when the angle greater than two right angles is external to the conductor. This becomes obvious on examining the arrangement of the lines of force in the neighbourhood of the angle.

Fio. 24. Angle greater than two right angles external to conductor.

Fig. 25. Angle less than two right angles external to conductor.

71. The arrangement shewn in fig. 25 is such as will be found at the point of a lightning conductor. The object of the lightning conductor is to ensure that the intensity shall be greater at its point than on any part of the buildings it is designed to protect. The discharge will therefore take

62

Electrostatics— Field of Force

[ch. n

place from the point of the lightning conductor sooner than from any part of the building, and by putting the conductor in good electrical communication with tie earth, it is possible to ensure that no harm shall be done to the main braidings by the electrical discharge.

An application of the same principle will explain the danger to a human being or animal of standing in the open air in the presence of a thunder cloud, or of standing under an isolated tree. The upward point, whether the head of mar or animal, or the summit of the tree, tends to collect the lines of force which pass from the cloud to the ground, so that a discharge of electricity will take place from the head or tree rather than from the ground.

72. The  property  of  lines  of  force  of  clustering  together  in  this  way  is 
    

utilised also in the manufacture of electrical instruments. A cage of wire is

Fio. 27.

placed round the instrument and almost all the lines of force from any charges which there may be outside the instrument will cluster together on the convex surfaces of the wire. Very few lines of force escape through this cage, so that the instrument inside the cage is hardly affected at all by any electric phenomena which may take place outside it. Fig. 27 shews the way in which lines of force are absorbed by a wire grating. It is drawn to represent the lines of force of a uniform field meeting a plane grating placed at right angles to the field of force.

71, 72] Examples 63

The protection of a wire cage is not adequate for the most sensitive in- struments, and it is usual to enclose them entirely in a metal case, except only for one small window through which readings can be taken. When this arrangement is adopted, no lines of force at all can pass from external charges to the instrument inside the metal case except for an infinitesimal number passing through the window. Lines of force which encounter the case termi- nate on it without in any way affecting the electric field inside, and the in- strument is almost perfectly screened from any external electric field. (Cf. § 114 below.)

EXAMPLES.

1. Two particles each of mass m and charged with e units of electricity of the same sign are suspended by strings each of length a from the same point; prove that the inclination 6 of each string to the vertical is given by the equation

imga? sin3 6 = e2 cos 6.

2. Charges +4e, — e are placed at the points A, B, and Cis the point of equilibrium. Prove that the line of force which passes through C meets AB at an angle of 60° at A and at right angles at C.

3. Find the angle at A (question 2) between AB and the line of force which leaves B at right angles to AB.

4. Two positive charges ex and e2 are placed at the points A and B respectively. Shew that the tangent at infinity to the line of force which starts from ex making an angle a with BA produced, makes an angle

2«n-i(/-S-«ii^

\ e! + e2 2J

with BA, and passes through the point G in AB such that

AC : CB=e2 : ev

5. Point charges +e, — e are placed at the points A, B. The line of force which leaves A making an angle a with AB meets the plane which bisects A B at right angles, in P. Shew that

.a /5 . PAB sin- = N/2sin^— .

6. If any closed surface be drawn not enclosing a charged body or any part of one, shew that at every point of a certain closed line on the surface it intersects the equi- potential surface through the point at right angles.

7. The potential is given at four points near each other and not all in one plane. Obtain an approximate construction for the direction of the field in their neighbourhood.

64 Electrostatics — Field of Force [ch. u

8. The potentials at the four corners of a small tetrahedron A, B, C, D are Vly F2, V3, Vi respectively. G is the centre of gravity of masses Mx at A, i/2 at B, M3 at C, J/4 at 0. Shew that the potential at O is

MlV1 + M2V% + M3V3 + MiVtl Mt + Ms + Mi+Mi

9. Charges Ze, —e, —e are placed at A, B, C respectively, where B is the middle point of AC. Draw a rough diagram of the Hues of force; shew that a line of force which starts from A making an angle a with AB>cos~1( — £) will not reach B or 0, and shew that the asymptote of the line of force for which a=cos-1 ( — §) is at right angles to AC.

10. If there are three electrified points A, B, C in a straight line, such that AC=f,

f — 6CL

BC — -7, and the charges are e, — ^— and Va respectively, shew that there is always a spherical equipotential surface, and discuss the position of the points of equilibrium on the line ABC when V=e — r<. and when V=e -

11. A and Care spherical conductors with charges e+e' and — e respectively. Shew that there is either a point or a line of equilibrium, depending on the relative size and positions of the spheres, and on e'/e. Draw a diagram for each case giving the lines of force and the sections of the equipotentials by a plane through the centres.

12. An electrified body is placed in the vicinity of a conductor in the form of a surface of anticlastic curvature. Shew that at that point of any line of force passing from the body to the conductor, at which the force is a minimum, the principal curvatures of the equipotential surface are equal and opposite.

13. Shew that it is not possible for every family of non-intersecting surfaces in free space to be a family of equipotentials, and that the condition that the family of surfaces

/(X, x, y, z)=0 shall be capable of being equipotentials is that

a^x a^x a^x

dx2 dy2 dz2

\ox/ \pyj \czj shall be a function of X only.

14. In the last question, if the condition is satisfied find the potential.

15. Shew that the confocal ellipsoids

2 +J^+=1

aHA^ + A c2 + A' can form a system of equipotentials, and express the potential as a function of A.

16. If  two  charged  concentric  shells  be  connected  by  a  wire,  the  inner  one  is  wholly 
    

discharged. If the law of force were -3^, prove that there would be a charge B on the

inner shell such that if A were the charge on the outer shell, and /, g the sum and differ- ence of the radii,

2gB= - Ap {{f-g) log (f+g) -flogf+glogg} approximately.

Examples

65

17. Three infinite parallel wires cut a plane perpendicular to them in the angular points A, B, C of an equilateral triangle, and have charges e, e, —e' per unit length respectively. Prove that the extreme lines of force which pass from A to C make at

2g 5g' 2e + a'

starting angles — ^— - ir and — — — n with AC, provided that e'^>2e.

6e

6e

18. A negative point charge — e2 lies between two positive point charges ex and e3 on the line joining them and at distances a, /3 from them respectively. Shew that, if the magnitudes of the charges are given by

«i _ «3_ e2\3

a

a + /3

, and if 1< X2<

m>

there is a circle at every point of which the force vanishes. Determine the general form of the equipotential surface on which this circle lies.

19. Charges  of  electricity  elt   —  e2,  e3,   (e3>ei)  are  placed   in   a  straight  line,  the 
    

negative charge being midway between the other two. Shew that, if 4e2 He between

(e33 - e^)3 and (e33 + e^)3, the number of unit tubes of force that pass from ex to e2 is

• («i + e2 - e3) + -^= (e3i - ef) («j* - 2**,* + e£fi.

4V 2

CHAPTER III

CONDUCTORS AND CONDENSERS

73. By a conductor, as previously explained, is meant any body or system of bodies, such that electricity can flow freely over the whole. When electricity is at rest on such a conductor, we have seen (§ 44) that the charge will reside entirely on the outer surface, and (§ 37) that the potential will be constant over this surface.

A conductor may be used for the storage of electricity, but it is found that a much more efficient arrangement is obtained by taking two or more conductors — generally thin plates of metal — and arranging them in a certain way. This arrangement for storing electricity is spoken of as a " con- denser." In the present Chapter we shall discuss the theory of single conductors and of condensers, working out in full the theory of some of the simpler cases.

Conductors.

A Spherical Conductor.

74. The simplest example of a conductor is supplied by a sphere, it being supposed that the sphere is so far removed from all other bodies that their influence may be neglected. In this case it is obvious from symmetry that the charge will spread itself uniformly over the surface. Thus if e is the charge, and a the radius, the surface density <r is given by

total charge e

total area of surface 4ura2 *

The electric intensity at the surface being, as we have seen, equal to 47ro-, is e/a\

From symmetry the direction of the intensity at any point outside the sphere must be in a direction passing through the centre. To find the amount of this intensity at a distance r from the centre, let us draw a sphere of radius r, concentric with the conductor. At every point of this sphere the amount of the outward electric intensity is by symmetry the same, say R,

73-75] Spheres and Cylinders 67

and its direction as we have seen is normal to the surface. Applying Gauss' Theorem to this sphere, we find that the surface integral of normal intensity

\NdS becomes simply R multiplied by the area of the surface 47rra, so that

4sirr-R = 4<7re,

or R —

~.2

This becomes e/a2 at the surface, agreeing with the value previously obtained.

Thus the electric force at any point is the same as if the charged sphere were replaced by a point charge e, at the centre of the sphere. And, just as in the case of a single point charge e, the potential at a point outside the sphere, distant r from its centre, is

J*r2 r

a

so that at the surface of the sphere the potential is -

Inside the sphere, as has been proved in § 37, the potential is constant, and therefore equal to e/a, its value at the surface, while the electric intensity vanishes.

As we gradually charge up the conductor, it appears that the potential at the surface is always proportional to the charge of the conductor.

It is customary to speak of the potential at the surface of a conductor as

" the potential of the conductor," and the ratio of the charge to this potential

is defined to be the " capacity " of the conductor. From a general theorem,

which we shall soon arrive at, it will be seen that the ratio of charge to

potential remains the same throughout the process of charging any conductor

or condenser, so that in every case the capacity depends only on the shape

and size of the conductor or condenser in question. For a sphere, as we

have seen,

charge e capacity = — - — ^— r = - = a, 1 J potential e

a so that the capacity of a sphere is equal to its radius.

A Cylindrical Conductor,

75. Let us next consider the distribution of electricity on a circular cylinder, the cylinder either extending to infinity, or else having its ends so far away from the parts under consideration that their influence may be neglected.

As in the case of the sphere, the charge distributes itself symmetrically,

5—2

68

Conductors and Condensers

[ch. Ill

so that if a is the radius of the cylinder, and if it has a charge e per unit length, we have

Vrra

To find the intensity at any point outside the conductor, construct a Gauss' surface by first drawing a cylinder of radius r, coaxal with the original cylinder, and then cutting off a unit length by two parallel planes at unit distance apart, perpendicular to the axis. From sym- metry the force at every point is perpendicular to the axis of the cylinder, so that the normal intensity vanishes at every point of the plane ends of this Gauss' surface. The surface integral of normal intensity will therefore consist entirely of the contributions from the curved part of the surface, and this curved part consists of a circular band, of unit width and radius r — hence of area 2irr. If R is the outward intensity at every point of this curved surface, Gauss' Theorem supplies the relation

2irrR = 4nre,

so that

r

Fig. 28.

This, we notice, is independent of a, so that the intensity is the same as it would be if a were very small, i.e., as if we had a fine wire electrified with a charge e per unit length.

In the foregoing, we must suppose r to be so small, that at a distance r from the cylinder the influence of the ends is still negligible in comparison with that of the nearer parts of the cylinder, so that the investigation does not hold for large values of r. It follows that we cannot find the potential by integrating the intensity from infinity, as has been done in the cases of the point charge and of the sphere. We have, however, the general differential equation

dV

dr

= -R,

so that in the present case, so long as r remains sufficiently small

dV 2e

or r

giving upon integration

V=C-2e\ogr.

The constant of integration C cannot be determined without a knowledge of the conditions at the ends of the cylinder. Thus for a long cylinder, the intensity at points near the cylinder is independent of the conditions at the ends, but the potential and capacity depend on these conditions, and are therefore not investigated here.

75-77] Infinite Plane, 69

An Infinite Plane.

76. Suppose we have a plane extending to infinity in all directions, and electrified with a charge a per unit area. From symmetry it is obvious that the lines of force will be perpendicular to the plane at every point, so that the tubes of force will be of uniform cross-section. Let us take as Gauss' surface the tube of force which has as cross-section any element w of area of the charged plane, this tube being closed by two cross-sections each of area <u at distance r from the plane. If R is the intensity over either of these cross-sections the contribution of each cross-section to Gauss' integral is Rco, so that Gauss' Theorem gives at once

2Ro> = 4sTT<JGi,

whence R = 2tto-.

The intensity is therefore the same at all distances from the plane.

The result that at the surface of the plane the intensity is lira-, may at first seem to be in opposition to Coulomb's Theorem (§ 57) which states that the intensity at the surface of a conductor is 47r<x. It will, however, be seen from the proof of this theorem, that it deals only with conductors in which the conducting matter is of finite thickness; if we wish to regard the electrified plane as a conductor of this kind we must regard the total electrification as being divided between the two faces, the surface density being \a on each, and Coulomb's Theorem then gives the correct result.

If the plane is not actually infinite, the result obtained for an infinite plane will hold within a region which is sufficiently near to the plane for the edges to have no influence. As in the former case of the cylinder, we can obtain the potential within this region by integration. If r measures the perpendicular distance from the plane

-!^ = i2 = 27nr, or

so that V= G — 27rcrr,

and, as before, the constant of integration cannot be determined without a knowledge of the conditions at the edges.

77. It is instructive to compare the thr^ee expressions which have been obtained for the electric intensity at points outside a charged sphere, cylinder and plane respectively. Taking r to be the distance from the centre of the

70

Conductors and Condensers

[ch. Ill

sphere, from the axis of the cylinder, and from the plane, respectively, we have found that

outside the sphere, R is proportional to — ,

outside the cylinder, R is proportional to - ,

outside the plane, R is constant.

From the point of view of tubes of force, these results are obvious enough deductions from the theorem that the intensity varies inversely as the cross- section of a tube of force. The lines of force from a sphere meet in a point, the centre of the sphere, so that the tubes of force are cones, with cross- section proportional to the square of the distance from the vertex. The lines of force from a cylinder all meet a line, the axis of the cylinder, at right angles, so that the tubes of force are wedges, with cross-section proportional to the distance from the edge. And the lines of force from a plane all meet the plane at right angles, so that the tubes of force are prisms, of which the cross-section is constant.

78. We may also examine the results from the point of view which regards the electric intensity as the resultant of the attractions or repulsions from different elements of the charged surface.

Let us first consider the charged plane. Let P, P' be two points at distances r, r from the plane, and let Q be the foot of the perpendicular from either on to the plane. If P is near to Q, it will be seen that almost the whole of the intensity at P is due to the charges in the immediate neighbourhood of Q. The more distant parts contribute forces which make angles with QP nearly equal to a right angle, and after being resolved along QP these forces hardly contribute anything to the resultant intensity at P.

Owing to the greater distance of the point P', the forces from given elements of the plane are smaller at P' than at P, but have to be resolved through a smaller angle. The forces from the regions near Q are greatly diminished from the former cause and are hardly affected by the latter. The forces from remote regions are hardly affected by the former circumstance, bufc their effect is greatly increased by the latter. Thus on moving

Fig. 29.

77-79] Spherical Condenser 71

from P to P' the forces exerted by regions near Q decrease in efficiency, while those exerted by more remote regions gain. The result that the total resultant intensity is the same at P' as at P, shews that the decrease of the one just balances the gain of the other.

If we replace the infinite plane by a sphere, we find that the force at a near point P is as before contributed almost entirely by the charges in the neighbourhood of Q. On moving from P to P', these forces are diminished just as before, but the number of distant elements [ Qj:

of area which now add contributions to the intensity at P' is much less than before. Thus the gain in the contributions Fig. 30.

from these elements does not suffice to

balance the diminution in the contributions from the regions near Q, so that the resultant intensity falls off on withdrawing from P to P'

The case of a cylinder is of course intermediate between that of a plane and that of a sphere.

Condensers.

Spherical Condenser.

79. Suppose that we enclose the spherical conductor of radius a dis- cussed in § 74, inside a second spherical conductor of internal radius b, the two conductors being placed so as to be concentric and insulated from one another.

It again appears from symmetry that the intensity at every point must be in a direction passing through the common centre of the two spheres, and must be the same in amount at every point of any sphere concentric with the two conducting spheres. Let us imagine a concentric sphere of radius r drawn between the two conductors, and when the charge on the inner sphere is e, let the intensity at every point of the imaginary sphere of radius r be R. Then, as before, Gauss' Theorem, applied to the sphere of radius r, gives the relation

4nrr2R = 4nre,

Q

so that R — - ,

r2

This only holds for values of r intermediate between a and b, so that to obtain the potential we cannot integrate from infinity, but must use the differential equation. This is

or r-

.« »

72 Conductors and Condensers [ch. m

which upon integration gives

V=C + - (27).

r

We can determine the constant of integration as soon as we know the potential of either of the spheres. Suppose for instance that the outer sphere is put to earth so that V=0 over the sphere r = b, then we obtain at once from equation (27)

so that G = — e/b, and equation (27) becomes

r b

On taking r = a, we find that the potential of the inner sphere is e( — rj ,

and its charge is e, so that the capacity of the condenser is

1 ah

or

11 b-a'

a b

80. In the more general case in which the outer sphere is not put to earth, let us suppose that Va, Vb are the potentials of the two spheres of radii a and b, so that, from equation (27)

a

F6 = C + |.

Then we have on subtraction

«-*>-(H)'

so that the capacity is ~ — y. .

The lines of force which start from the inner sphere must all end on the inner surface of the outer sphere, and each line of force has equal and opposite charges at its two ends. Thus if the charge on the inner sphere is e, that on the inner surface of the outer sphere must be — e. We can there- fore regard the capacity of the condenser as being the charge on either of the two spheres divided by the difference of potential, the fraction being taken always positive. On this view, however, we leave out of account any charge which there may be on the outer surface of the outer sphere : this is not regarded as part of the charge of the condenser.

79-82] Cylindrical Condenser 73

An examination of the expression for the capacity,

ab

will shew that it can be made as large as we please by making b — a sufficiently small. This explains why a condenser is so much more efficient for the storage of electricity than a single conductor.

81. By taking more than two spheres we can form more complicated condensers. Suppose, for instance, we take concentric spheres of radii a, b, c in ascending order of magnitude, and connect both the spheres of radii a and c to earth, that of radius b remaining insulated. Let V be the potential of the middle sphere, and let ex and e2 be the total charges on its inner and outer surfaces. Regarding the inner surface of the middle sphere and the surface of the innermost sphere as forming a single spherical condenser, we have

Vab 6l~b-a'

and again regarding the outer surface of the middle sphere and the outermost sphere as forming a second spherical condenser, we have

Vbc c — b

Hence the total charge E of the middle sheet is given by

E = ex + e2

ab be

= V -7 +

b — a c — b

so that regarded as a single condenser, the system of three spheres has a capacity

ab be

b — a c — b'

which is equal to the sum of the capacities of the two constituent condensers into which we have resolved the system. This is a special case of a general theorem to be given later (§ 85).

Coaxal Cylinders.

82. A conducting circular cylinder of radius a surrounded by a second coaxal cylinder of internal radius b will form a condenser. If e is the charge on the inner cylinder per unit length, and if V is the potential at any point between the two cylinders at a distance r from their common axis, we have, as in § 75,

V=C-2e\ogr,

74 Conductors and Condensers [ch. in

and it is now possible to determine the constant C as soon as the potential of either cylinder is known.

Let Va, Vt, be the potentials of the inner and outer cylinders, so that

Va = C - 2e log a,

Vb=C-2e\ogb.

By subtraction Va - Vb = 2e log f - J ,

so that the capacity is . ■ ,

per unit length.

Parallel Plate Condenser.

83. This condenser consists of two parallel plates facing one another,

say at distance d apart. Lines of force will pass from the inner face of one

to the inner face of the other, and in regions sufficiently far removed from

the edges of the plate these lines of force will be perpendicular to the plate

throughout their length. If cr is the surface density of electrification of one

plate, that of the other will be — a. Since the cross-section of a tube

remains the same throughout its length, and since the electric intensity

varies as the cross-section, it follows that the intensity must be the same

throughout the whole length of a tube, and this, by Coulomb's Theorem,

will be 47ro-, its value at the surface of either plate. Hence the difference of

potential between the two plates, obtained by integrating the intensity 4tto-

along a line of force, will be

Arrrcrd.

The capacity per unit area is equal to the charge per unit area a divided by this difference of potential, and is therefore

1

4>nd'

The capacity of a condenser formed of two parallel plates, each of area A,

is therefore

A

4>ird'

except for a correction required by the irregularities in the lines of force near the edges of the plates.

Inductive Capacity.

84. It was found by Cavendish, and afterwards independently by Faraday, that the capacity of a condenser depends not only on the shape and size of the conducting plates but also on the nature of the insulating material, or dielectric to use Faraday's word, by which they are separated.

82-85]

Series of Condensers

75

It is further found that on replacing air by some other dielectric, the capacity of a condenser is altered in a ratio which is independent of the shape and size of the condenser, and which depends only on the dielectric itself. This constant ratio is called the specific inductive capacity of the dielectric, the inductive capacity of air being taken to be unity.

We shall discuss the theory of dielectrics in a later Chapter. At present it will be enough to know that if C is the capacity of a condenser when its plates are separated by air, then its capacity, when the plates are separated by any dielectric, will be KG, where K is the inductive capacity of the particular dielectric used. The capacities calculated in this Chapter have all been calculated on the supposition that there is air between the plates, so that when the dielectric is different from air each capacity must be multi- plied by K.

The following table will give some idea of the values of E actually observed for different dielectrics. For a great many substances the value of K is found to vary widely for different specimens of the material and for different physical conditions.

Sulphur

2-8 to 4-0.

Ethyl Alcohol

26-5

Mica

6-0 to 8-0.

Water at 17° C.

80-0

Glass

6 6 to 9-9.

Ice at -7-5° C.

70-8

Paraffin

2-0 to 2-3.

Ice at- 200° C.

2-43

The values of K for some gases are given

on p. 132.

Compound Condensers.

Condensers in Parallel.

85. Let  us  suppose  that  we  take  any  number  of  condensers  of  capacities 
    

d,C2, ... and connect all their high potential plates together by a conducting

s

Fig. 31.

wire, and all their low potential plates together in the same way. This is known as connecting the condensers in 'parallel.

The high potential plates have now all the same potential, say Vlt while the low potential plates have all the same potential, say V0. If elt e2, the charges on the separate high potential plates, we have

el = C1(V1-V0),

e2 = C2(V1- F0), etc.,

are

76

Conducto7,s and Condensers

[ch. m

and the total charge E is given by

E = e1 + e2 + ...

= (C1 + C2+...)(F1-F0).

Thus the system of condensers behaves like a single condenser of capacity

Cx + C7a + C3 + . . . .

It will be noticed that the compound condenser discussed in § 81 con- sisted virtually of two simple spherical condensers connected in parallel.

Condensers in Cascade.

86. We might, however, connect the low potential plate of the first to the high potential plate of the second, the low potential plate of the second to the high potential plate of the third, and so on. This is known as arranging the condensers in cascade.

Fig. 32.

Suppose that the high potential plate of the first has a charge e. This induces a charge — e on the low potential plate, and since this plate together with the high potential plate of the second condenser now form a single insulated conductor, there must be a charge + e on the high potential plate of the second condenser. This induces a charge - e on the low potential plate of this condenser, and so on indefinitely ; each high potential plate will have a charge + e, each low potential plate a charge — e.

Thus the difference of potential of the two plates of the first condenser will be e/Cly that of the second condenser will be e/C2, and so on, so that the total fall of potential from the high potential plate of the first to the low potential plate of the last will be

1 1

...).

We see that the arrangement acts like a single condenser of capacity

1 JL_ _1

85-89]

The Leyden Jar

77

Pkactical Condensers. Practical Units.

87. As will be explained more fully later, the practical units of electricians are entirely different from the theoretical units in which we have so far supposed measurements to be made. The practical unit of capacity is called the farad, and is equal, very approximately, to 9 x 10" times the theoretical C.G.S. electrostatic unit, i.e., is equal to the actual capacity of a sphere of radius 9 x 10u cms. This unit is too large for most purposes, so that it is convenient to introduce a subsidiary unit — the microfarad — equal to a millionth of the farad, and therefore to 9 x 105 C.G.S. electrostatic units. Standard condensers can be obtained of which the capacity is equal to a given fraction, frequently one-third or one-fifth, of the microfarad.

The Leyden Jar.

88. For experimental purposes the commonest form of condenser is the Leyden Jar. This consists essentially of a glass vessel, bottle-shaped, of which the greater part of the surface is coated inside and outside with tinfoil. The two coatings form the two plates of the condenser, contact with the inner coating being established by a brass rod which comes through the neck of the bottle, the lower end having attached to it a chain which rests on the inner coating of tinfoil.

O

csssai^p

Fig. 33.

To form a rough numerical estimate of the capacity of a Leyden Jar, let us suppose that the thickness of the glass is \ cm., that its specific inductive capacity is 7, and that the area covered

with tinfoil is 400 sq. cms. Neglecting corrections required by the irregu- larities in the lines of force at the edges and at the sharp angles at the bottom of the jar, and regarding the whole system as a single parallel plate condenser, we obtain as an approximate value for the capacity

KA

4nrd

electrostatic units,

in which we must put K = 7, A = 400 and d = |. On substituting these values the capacity is found to be approximately 450 electrostatic units,

or about ^q$ microfarad.

Parallel Plates. A more convenient condenser for some purposes is a modification of

the parallel plate condenser. Let us suppose that we arrange n plates, each

78

Conductors and Condensers

[ch. m

of area A, parallel to one another, the distance between any two adjacent plates being d. If alternate plates are joined together so as to be in electrical contact the space between each adjacent pair of plates may be regarded as

Fig. 34.

KA