The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 7 of 39

From equations (34), it is clear that qu is the value of Ex when Tf=l, Vi = Vi = ... = 0. This leads to an extended definition of the capacity of a conductor, in which account is taken of the influence of the other conductors in the field. We define the capacity of the conductor 1, when in the presence of conductors 2, 3, 4, ..., to be qn, namely, the charge required to raise conductor 1 to unit potential, all the other conductors being put to earth.

Energy of a System of charged Conductors.

106. Suppose we require to find the energy of a system of conductors, their charges being Eu E2) ••• En, so that their potentials are Vlf %, ... Vn given by equations (32).

Let W denote the energy when the charges are kElt kE2, ... JcEn. Corresponding to these charges, the potentials will be kV1} kV2, ... kVn. It we bring up an additional small charge dk . Ex from infinity to conductor 1, the work to be done will be dkEx . kV[; if we bring up dkE2 to conductor 2 the work will be dkE2kV2 and so on. Let us now bring charges dkEx to 1, dkE2 to 2, dkE3 to 3, ... dkEn to n. The total work done is

kdk(E1Vl + E.zV2+...+EnVn) (36),

and the final charges are

(k + dk) Ely (k + dk) E2,...(k + dk) En.

The energy in this state is the same function of k + dk as W is of k, and may therefore be expressed as

W+d-J^dk.

dW Expression (36), the increase in energy, is therefore equal to -^-r dk, whence

d^ = k(ElVl + E2V2 + ... + EnVu),

so that on integration

W = W (E& + E.2V2+... + EnVn).

No constant of integration is added, since W must vanish when k = 0. Taking k = 1, we obtain the energy corresponding to the final charges Ei, Ei} ... En, in the form

W=&EV , (37).

105-109] Energy 95

If we substitute for the V's their values in terms of the charges as given by equations (31), we obtain

W = $(puEii + 2puE1E2 + p22E2* + ...) (38),

and similarly from equations (34),

W = $(qnV?+2q12VX+q*J?+...) ....(39).

107. If W is expressed as a function of the E's, we obtain by differ- entiation of (38),

^r = pnE1 + p12E2 + ...+ pmEn

= Vi, by equation (32).

This result is clear from other considerations. If we increase the charge

dW on conductor 1 by dE1} the increase of energy is ^rr dElt and is also V^dEx

since this is the work done on bringing up a new charge dEx to potential ~VX. Thus on dividing by dEx, we get

wrv~ (40)-

Soalso W=El (41)

as is at once obvious on differentiation of (39).

108. In changing the charges from EX,E2, ... to E-f, E2', ... let us suppose that the potentials change from Vlt V2, ... to Vi,V2,.... The work done, W — W, is given by

W- W=%(E'Y'-EV).

Since, however, by § 103, 1EV = HE'V, this expression for the work done can either be written in the form

\ 2 [E'V - EV-(EV - E'V)}, which leads at once to

W -W = ^(E'-E)(V'+ V) (42);

or in the form |2 {E'V - EV+(EV - E'V)},

which leads to W - W = %%(V- V)(E' + E) (43).

109. If the changes in the charges are only small, we may replace E' by E + dE, and find that equation (42) reduces to

dW=ZVdE,

from which equation (40) is obvious, while equation (43) reduces to

dW = $EdV,

leading at once to (41).

96 Systems of Conductors [oh. iv

110. It is worth noticing that the coefficients of potential, capacity and induction can be expressed as differential coefficients of the energy ; thus

_32TT

Pr*~dErdEa>

and so on.

The last two equations give independent proofs of the relations

Prs == Psr> Qrs ~ Qsr*

Properties of the Coefficients.

111. A certain number of properties can be deduced at once from the fact that the energy must always be positive. For instance since the value of W given by equation (38) is positive for all values of Eu E2, ... En, it follows at once that

Pn, P-2-2, p33, ••• are positive,

that pnp^ - Pu is positive, that

PuPkPiz

PviP^p-a is positive

.P13.P23.P33 and so on. Similarly from equation (39), it follows that

tfn. #22, £33, ••• are positive, and there are other relations similar to those above.

112. More valuable properties can, however, be obtained from a con- sideration of the distribution of the lines of force in the field.

Let us first consider the field when

E1 = l, E2 = Ez = . . . = 0.

The potentials are V1=p11, V2 = pn, etc.

Since conductors 2, 3, ... are uncharged, their potentials must be inter- mediate between the highest and lowest potentials in the field. Thus the potential of 1 must be either the highest or the lowest in the field, the other extreme potential being at infinity. It is impossible for the potential of 1 to be the lowest in the field ; for if it were, lines of force would enter in at every point, and its charge would be negative. Thus the highest potential in the field must be that of conductor 1, and the other potentials must all

110-114] Properties of the Coefficients 97

be intermediate between this potential and the potential at infinity, and must therefore all be positive. Thus pn> p12, pa, ... pm are all positive and the first is the greatest.

Next let us put Vx = l, V2=V,= ...=0,

so that the charges are ■ qu, ql2, q^, ••• <?m-

The highest potential in the field is that of conductor 1. Thus lines of force leave but do not enter conductor 1. The lines may either go to the other conductors or to infinity. No lines can leave the other conductors. Thus the charge on 1 must be positive, and the charges on 2, 3, . .. all negative, i.e., qn is positive and ql2, q13, ... are all negative. Moreover the total strength of the tubes arriving at infinity is qn + qu + qi3+ ••• + 5i»i so that this must be positive.

113. To sum up, we have seen that

(i) All the coefficients of potential (pn,pi2, •••) are positive, (ii) All the coefficients of capacity (qn, q^, ...) are positive,

(iii) All the coefficients of induction (q12, q13, ...) are negative, and we have obtained the relations

(pu — p12) is positive, (qn + <?i2 + • ■ • +, qm) is positive.

In limiting cases it is of course possible for any of the quantities which have been described as always positive or always negative, to vanish.

Values of the Coefficients in Special Cases.

Electric Screening.

114. The first case in which we shall consider the values of the coefficients is that in which one conductor, say 1, is completely surrounded by a second conductor 2.

0

Fig. 38.

If Ez = 0, the conductor 2 becomes a closed conductor with no charge inside, so that the potential in its interior is constant, and therefore K—K- Putting E1 = 0> the relation ~%= V2 gives the equation

(P12 -P22) E2 + (p13 -p,3) Es + . . . = 0. J. 7

98 Systems of Conductors [en. iv

This being true for all values of E2, E3, ... we must have

Pm = P22, ?Jis=J02s> etc.

Next let us put unit charge on 1, leaving the other conductors uncharged. The energy is %pn. If we join 1 and 2 by a wire, the conductors 1 and 2 form a single conductor, so that the electricity will all flow to the outer surface. This wire may now be removed, and the energy in the system is \p^. Energy must, however, have been lost in the flow of electricity, so that p22 must be less than pn.

Since we have already seen that pVi=p22 and pn— p12 cannot be negative, it is clear that pw cannot be greater than pn The foregoing argument, however, goes further and enables us to prove that pn — p.^ is actually positive.

Let us next suppose that conductor 2 is put to earth, so that V2 = 0. Then if E1 = 0, it follows that T[=0. Hence from the equations

E1 = q11V1 + qliV%+... + qmVn (44)

we obtain in this special case that

qi3Vi + quV±+ ••• + qmVa = o.

This is true, whatever the values of Vs, Vi} .... so that

Suppose that conductor 1 is raised to unit potential while all the other conductors are put to earth. The aggregate strength of the tubes of force which go to infinity, namely qn + q12 + ... + qm (§ 112), is in this case zero, so that ql2 = -qn-

The system of equations (44) now reduces, when V2 = 0, to

E1 = qllV1 (45),

E2 = ql2Vi + q^Vi + q.2iVi+ (46),

Es = qS3Vi + q~iVi+... Ei = qZiVi + qiiVi+ ..

.(47)

Equations (47) shew that the relations between charges and potential outside 2 are quite independent of the electrical conditions which obtain inside 2. So also the conditions inside 2 are not affected by those outside 2, as is obvious from equation (45). These results become obvious when we consider that no lines of force can cross conductor 2, and that there is no way except by crossing conductor 2 for a line of force to pass from the conductors outside 2 to those inside 2.

An electric system which is completely surrounded by a conductor at potential zero is said to be " electrically screened " from all electric systems

114, 115] Coefficients for Spherical Condenser 99

outside this conductor ; for charges outside this " screen " cannot affect the screened system. The principle of electric screening is utilised in electro- static instruments, in order that the instrument may not be affected by external electric actions other than those which it is required to observe. As a complete conductor would prevent observation of the working of the instrument, a cage of wire is frequently used as a screen, this being very nearly as efficient as a completely closed conductor (see § 72). In more delicate instruments the screening may be complete except for a small window to admit of observation of the interior.

Spherical Condenser.

115. Let us apply the methods of this Chapter to the spherical con- denser described in § 79. Let the inner sphere of radius a be taken to be conductor 1, and the outer sphere of radius b be taken to be conductor 2.

The equations connecting potentials and charges are

V1=pllE1+p21E2, V,=p12E1+p22E2.

A unit charge placed on 2 raises both 1 and 2 to potential 1/6, so that on putting E1 = 0, E2 = 1, we must have V1 = V2 = 1/6. Hence it follows that

1

Pi\ P& 7 •

If we leave 2 uncharged and place unit charge on 1, the field of force is that investigated in § 79, so that V[ = 1/a, V2 = l/b. Hence

1 1

Pn = af Pl*=b'

These results exemplify

(i) the general relation p12 = p.21,

(ii) the relation peculiar to electric screening, pi2=p,2.

The equations now become

Vl~ a b '

v* b + b ■

Solving for Ex and E2 in terms of Vx and V2, we obtain

a6_ ab_

b — a b — a

b — a b — a

L. ab ab h-

sothat ?ii = ^. <Z"=^ = -F3-a> *-TrS'

7—2

100 Systems of Conductors [ch. iv

We notice that q12 = q21 , that the value of each is negative, and that

qu = — qn, in accordance with §113. The value of qn is the capacity of

sphere 1 when 2 is to earth, and is in agreement with the result of § 79.

b2 The capacity of 2 when 1 is to earth, q22, is seen to be ^ . This can

also be seen by regarding the system as composed of two condensers, the inner sphere and the inner surface of the outer sphere form a single spherical

condenser of capacity j , while the outer surface of the outer sphere has

capacity b. The total capacity accordingly

ab . b2

• b

b — a b — a'

Two spheres at a great distance apart.

116. Suppose  we  have  two  spheres,  radii  a,  b,  placed  with  their  centres 
     

at a great distance c apart. Let us first place unit charge on the former, the

Fig. 39.

charge being placed so that the surface density is constant. This will not produce uniform potential over 2 ; at a point distant r from the centre of 1 it will produce potential 1/r. We can, however, adjust this potential to the uniform value 1/c by placing on the surface of 2 a distribution of electricity

such that it produces a potential over this surface.

Take B, the centre of the second sphere, as origin, and AB as axis of x.

Then we may write

1 1 r — ex c 1

= = — . as iar as — .

c r cr c2 c2

Let cr be the surface density required to produce this potential, then clearly a is an odd function of x, and therefore the total charge, the value of a integrated over the sphere, vanishes. Thus the potential of 2 can be adjusted to the uniform value 1/c without altering the total charge on 2 from zero, neglecting 1/c3. The new surface density being of the order of 1/c2, the additional potential produced on 1 by it will be at most of order 1/c3, so that if we neglect 1/c3 we have found an equilibrium arrangement which makes

^ = 1, L\ = 0, V^1-, K = l.

& c

115-117] Coefficients for two distant Spheres 101

Substituting these values in the equations

V2=p12E1+p22E2,

we find at once

that

Pn

1

a

neglecting

1

c3'

and similarly we can see

Pa

that

1 r

i

1 c3'

P-n

1

~b

neglecting

1

c3'

Solving the

equations

vx =

Ex E%

■■ — + — ,

a r

v2 =

' r + 6 '

we find that, neglecting - ,

c

a

ab>

ab ab „ 1

fr"g«— / a6='7asfaraSc"

b

We notice that the capacity of either sphere is greater than it would be if the other were removed. This, as we shall see later, is a particular case of a general theorem.

Two conductors in contact.

117. If two conductors are placed in contact, their potentials must be equal. Let the two conductors be conductors 1 and 2, then the equation Vi = V2 becomes

(Pn - Pn) #i + (Pu ~ Pn) E2+ ...=0,

or, say, *EX + $E2 + yE3 + . . . = 0.

If we know the total charge E on 1 and 2, we have

Ey + E2 = E,

and on solving these two equations we can obtain Ex and E„. We find that

E,_ jSE + yE3 + SEi+ ... En_ aE + yEs + hEA + ...'

102 Systems of Conductors [ch. iv

giving the ratio in which the charge E will distribute itself between the two conductors 1 and 2. If the conductors 3, 4, ... are either absent or uncharged,

E2 a pu - pn ' which is independent of E and always positive. It is to be noticed that El vanishes only if p&^piz, i.e., if 2 entirely surrounds 1.

Mechanical Forces on Conductors.

118. We have already seen that the mechanical force on a conductor is the resultant of a system of tensions over its surface of amount 27rcr2 per unit area. The results of the present Chapter enable us to find the resultant force on any conductor in terms of the electrical coefficients of the system.

Suppose that the positions of the conductors are specified by any co- ordinates £i, f2> •••> so that pn,pw, • ••■, <7u> qvt, •••> and consequently also W, are functions of the f's. If ^ is increased to £x + d%u without the charges on

dW the conductors being altered, the increase in electrical energy is -^r d%x, and

this increase must represent mechanical work done in moving the conductors. The force tending to increase £ is accordingly

%'

Since the charges on the conductors are to be kept constant, it will of course be most convenient to use the form of W given by equation (38), and the force is obtained in the form

-t(^Ef + 2&*BlE,+ ...) (48).

It is however possible, by joining the conductors to the terminals of electric batteries, to keep their potentials constant. In this case, however, we must not use the expression (39) for W, and so obtain for the force

-i{w/' + *WJ>V'+~) (49)'

for the batteries are now capable of supplying energy, and an increase of electrical energy does not necessarily mean an equal expenditure of mechanical energy, for we must not neglect the work done by the batteries. Since the resultant mechanical force on any conductor may be regarded as the resultant of tensions 27ro-2 per unit area acting over its surface, it is clear that this resultant force in any position depends solely on the charges in this position. It is therefore the same whether the charges or potentials are kept constant, and expression (48) will give this force whether the conductors are connected to batteries or not.

1 1 7-1 20] Mechanical forces 1 03

119. As  an  illustration,  we   may  consider   the  force  between  the  two 
     

charged spheres discussed in § 116.

dW The force tending to increase c, namely — =— , is

_ i (dpn F2,93P™FP , dpa ™ A

2le7Al + 2a7^2 + *^2J'

and substituting the values

1 . 1

pn=~ + terms in - , a c3

1

Pl2 = - +

c

» •}

1

p^=l +

» >J

it is found that this force is

E.E.. . 1 — - — h terms in -- c2 c

Thus, except for terms in c4, the force is the same as though the charges were collected at the centres of the spheres. Indeed, it is easy to go a stage further and prove that the result is true as far as c*. We shall, however, reserve a full discussion of the question for a later Chapter.

120. Let  us  write 
     

i(p11E1* + 2p12E1Es + ...)=We,

h(qnW +2g12KK +...) = Wy. Then We and Wr are each equal to the electrical energy ^EV, so that

We + Wv-XEV = 0 (50).

In whatever way we change the values of

-"1> -"2) •••> '1> Ki> •••» SI' £2> *••>

equation (50) remains true. We may accordingly differentiate it, treating the expression on the left as a function of all the .£"s, V's and |'s. Denoting the function on the left-hand of equation (50) by (p, the result of differentiation will be

Now ~^- = ^ - V, = 0, by equation (40),

wrwr Ei 0> - " (41)»

so that we are left with X ^ §£ = 0,

104 Systems of Conductors [en. iv

and since this equation is true for all displacements and therefore for all values of §£, Sf2> ...» it follows that each coefficient must vanish separately.

Thus ||=0, or

dWe dWr_ (

dW As we have seen, — ^ is the mechanical force tending to increase f1(

dW and this has now been shewn to be equal to -^ , which is expression (49)

with the sign reversed. Thus the mechanical force, whether the charges or the potentials are kept constant, is

»(§g* + »f|MSK + ...) (52),

a form which is convenient when we know the potentials, but not the charges, of the system.

In making a small displacement of the system such that fx is changed

dW into f j + d^i, the mechanical work done is -^ d%x. If the potentials are

kept constant the increase in electrical energy is -~ d%x. The difference of these expressions, namely

fiWy d_K\

represents energy supplied by the batteries. From equation (51), it appears

dWr that this expression is equal to 2 -^if d%n so that the batteries supply energy

equal to twice the increase in the electrical energy of the system, and of this energy half goes to an increase of the final electrical energy, while half is expended as mechanical work in the motion of the conductors.

Introduction of a new conductor into the field.

121. When a new conductor is introduced into the field, the coefficients Pn>Pn> •••» tfn, <?i2> ••• are naturally altered.

Let us suppose the new conductor introduced in infinitesimal pieces, which are brought into the field uncharged and placed in position so that they are in every way in their final places except that electric communication is not established between the different pieces. So far no work has been done and the electrical energy of the field remains unaltered.

Now let electric communication be established between the different pieces, so that the whole structure becomes a single conductor. The separate

120-122] The Attracted Disc Electrometer 105

pieces, originally at different potentials, are now brought to the same potential by the flow of electricity over the surface of the conductor. Electricity can only flow from places of higher to places of lower potential, so that electrical energy is lost in this flow. Thus the introduction of the new conductor has diminished the electric energy of the field.

If we now put the new conductor to earth there is in general a further flow of electricity, so that the energy is still further diminished.

Thus the electric energy of any field is diminished by the introduction of a new conductor, whether insulated or not.

Consider the case in which the new conductor remains insulated. Let the energy of the field before the introduction of the new conductor be

$(pnE1i+2p12E1E2+...+pnnEn>).... (53).

After introduction, the energy may be taken to be

h (PiiW + 2pi*EiE3 +...+ Pnn'Er?) (54),

where pn', etc., are the new coefficients of potential. Further coefficients of the type Pi,n+i, p-2,n+i> "->Pn+i,n+i are of course brought into existence, but do not enter into the expression for the energy, since by hypothesis En+1 = 0.

Since expression (54) is less than expression (53), it follows that

(Pn - Pn) Ef + 2 (p12 - pa') E1E2 + ...

is positive for all values of Elt E2, Hence pa — pn is positive, and other

relations may be obtained, as in § 111.

Electrometers. I. The Attracted Disc Electrometer.

Fig. 40.

122. This instrument is, as regards its essential principle, a balance in which the beam has a weight fixed at one end and a disc suspended from the other. Under normal conditions the fixed weight is sufficiently heavy

106 Systems of Conductors [ch. iv

to outweigh the disc. In using the instrument the disc is made to become one plate of a parallel plate condenser, of which the second plate is adjusted until the electric attraction between the two plates of the condenser is just sufficient to restore the balance.

The inequalities in the distribution of the lines of force which would otherwise occur at the edges of the disc are avoided by the use of a guard- ring (§ 90), so arranged that when the beam of the balance is horizontal the guard-ring and disc are exactly in one plane, and fit as closely as is practicable.

Let us suppose that the disc is of area A and that the disc and guard- ring are raised to potential V. Let the second plate of the condenser be placed parallel to the disc at a distance h from it, and put to earth. Then the intensity between the disc and lower plate is uniform and equal to Vfk, so that the surface density on the lower face of the disc is a = Vj^irh. The mechanical force acting on the disc is therefore a force lira^A or V2A/87rh* acting vertically downwards through the centre of the disc. If this just suffices to keep the beam horizontal, it must be exactly equal to the weight, say W, which would have to be placed on this disc to maintain equilibrium if it were uncharged. This weight is a constant of the instrument, so that

the equation

V-A

8tt/i2

enables us to determine V in terms of known quantities by observing h.

The instrument is arranged so that the lower plate can be moved parallel

to itself by a micrometer screw, the reading of which gives h with great

accuracy. We can accordingly determine V in absolute units, from the

equation

A '

If we wish to determine a difference of potential we can raise the upper plate to one potential Vl; and the lower plate to the second potential V2, and we then have

f8TrW A

A more accurate method of determining a difference of potential is to keep the disc at a constant potential v, and raise the lower plate successively to potentials Vx and V2. If h^ and h2 are the values of h which bring the disc to its standard position when the potentials of the lower plate are Vx and J£, we have

V1-Vi = hJi

v-Vx = hlsJ

8ttW A '

/Hit W

122, 123]

The Quadrant Electrometer

107

so that

V^V-<Jh-K*J

8ttW

It is now only necessary to measure h^ — h2, the distance through which the lower plate is moved forward, and this can be determined with great accuracy, as it depends solely on the motion of the micrometer screw.

II. The Quadrant Electrometer.

123. Measurement  of  Potential  Difference.     This   instrument   is   more 
     

delicate than the disc electrometer just described, but enables us only to compare two potentials, or potential differ- ences; we cannot measure a single potential in terms of known units.

The principal part of the instrument consists of a metal cylinder of height small compared with its radius, divided into four quadrants A, B, C, D by two diameters at right angles. These quadrants are insulated separately, and then opposite quadrants are connected in pairs, two by wires joined to a point E and two by wires joined to some other point F.

The inside of the cylinder is hollow and inside this a metal disc or " needle " is free to move, being suspended by a delicate fibre, so that it can rotate without touching the quadrants. Before using the instrument the needle is charged to a high potential, say v, either by means of the fibre, if this is a conductor, or by a small conducting

wire hanging from the needle which passes through the bottom of the cylinder. The fibre is adjusted so that when the quadrants are at the same potential the needle rests, as shewn in the figure, in a symmetrical position with respect to the quadrants. In this state either surface of the needle and the opposite faces of the quadrants may be regarded as forming a parallel plate condenser.

If, however, the potential of the two quadrants joined to E is different from that of the two quadrants joined to F, there is an electrical force tending to drag the needle under that pair of quadrants of which the potential is more nearly equal to v. The needle accordingly moves in this direction until the electric forces are in equilibrium with the torsion of the fibre, and an observation of the angle through which the needle turns will give an

Fig. 41.

108 Systems of Conductors [ch. iv

indication of the difference of potential between the two pairs of quadrants. This angle is most easily observed by attaching a small mirror to the fibre just above the point at which it emerges from the quadrants.

Let us suppose that when the needle has turned through an angle 6, the total area A of the needle is placed so that an area S is inside the pair of quadrants at potential K, and an area A — S inside the pair at potential V>. Let h be the perpendicular distance from either face of the needle to the faces of the quadrants. Then the system may be regarded as two parallel plate condensers of area S, distance h, and difference of potential v — V[, and two parallel plate condensers for which these quantities have the values A — S, h, v — Vz. There are two condensers of each kind because there are two faces, upper and lower, to the needle. The electrical energy of this system is accordingly

(v-vys (v - vy (A - S)

4>Trh 4nrh

The energy here appears as a quadratic function of the three potentials concerned: it is expressed in the same form as the Wr of § 120. The mechanical force tending to increase 6, i.e., the moment of the couple tending

to turn the needle in the direction of 6 increasing, is therefore -^- . Now

in Wv the only term in the coefficients of the potentials which varies with 6 is S, so that on differentiation we obtain

Wy _ (v - V,f - (v - TQ2 d_S dd ~ 4ttA d0°

If r is the radius of the needle — measured from its centre, which is under the line of division of the quadrants — we clearly have r^ = r2, so that we can write the equation just obtained in the form

9TTr (2t>-K-K)G?-K)

d9 " 4ttA

r\

In equilibrium this couple is balanced by the torsion couple of the fibre, which tends to decrease 6. This couple may be taken to be k6, where k is a constant, so that the equation of equilibrium is

W 4^h {°b)-

For small displacements of the needle, r2 may be replaced by a2, the radius of the needle at its centre line. Also v is generally large compared with K and V2. The last equation accordingly assumes the simpler form

123, 124] The Quadrant Electrometer 109

shewing that 6 is, for small displacements of the needle, approximately proportional to the difference of potential of the two pairs of quadrants. The instrument can be made extraordinarily sensitive owing to the possibility of obtaining quartz-fibres for which the value of k is very small.

If the difference of potential to be measured is large, we may charge the needle simply by joining it to one of the pairs of quadrants, say the pair at potential J£. We then have v= V,, and equation (55) becomes

kff ~ 4tt/* '

so that 0 is now proportional to the square of the potential difference to be measured.

a2

Writing „ — j-% = C, so that G is a constant of the instrument, we have,

when v is large

e = Cv{Vl-Vi) (56),

when v = V2,

e^iciv.-vy (57).

124. Measurement of charge. Let us speak of the pairs of quadrants at potentials Yx, V2 as conductors 1, 2 respectively, and let the needle be conductor 3. When the quadrants are to earth and the needle is at potential T^, the charge E induced on the first pair of quadrants by the charge on the needle will be given by

where q13 is the coefficient of induction. This coefficient is a function of the angle 6 which defined the position of the needle. If the instrument is adjusted so that 0 = 0 when both pairs of quadrants are to earth, we must use the value of q13 corresponding to 6 = 0, say {ql3, so that

E = (qiz\V3 (58).

Now suppose that the first pair of quadrants is insulated and receives an additional charge Q, the second pair being still to earth. Let the needle be deflected through an angle 6 in consequence. Since the charge on the first pair of quadrants is now E + Q, we have

E + Q = (qn)eV1 + (q13)eV3. On subtracting equation (58) from this we obtain

If 6 is small this may be written

110 Systems of Conductors [ch. iv

where qn , -^ are supposed calculated for 6 = 0. Since V2 = 0, we have from

equation (56),

0 = OV%Ylt

so that <3 = (^+^3^)^'

shewing that for small values of 0, Q is directly proportional to 0.

Let us suppose that we join the first pair of quadrants (conductor 1) to a condenser of known capacity T which is entirely outside the electro- meter. Since the needle (3) is entirely screened by the quadrants the value of q13 remains unaltered, while qn will become qu + V. If 0' is now the deflection of the needle, we have

'qn + T dqv

-{&+%*)'■

so that, by combination with the last equation, we have

If 0" is the deflection obtained by joining the pairs of quadrants to the terminals of a battery of known potential difference D, we have from equation (56),

CVy.

D'

and on substituting this value for GV3, our equation becomes

Q =

0" 0" ' l'~~0

giving Q in terms of the known quantities V, D and the three readings 0, 0' and 0".

An ordinary quadrant electrometer will measure differences of potential down to about j^W electrostatic units. Thus in spite of its somewhat high capacity of about 50 electrostatic units, it forms an extremely efficient instru- ment for the measurement or detection of small electric charges.

An improved form of the instrument has recently been introduced by Dolazalek, in which the electrostatic capacity is very small. This is capable of measuring potential differences down to too'ooo electrostatic units, and is correspondingly more sensitive for the measurement of charges.

124] Examples 111

EXAMPLES.

1. If the algebraic sum of the charges on a system of conductors be positive, then on one at least the surface density is everywhere positive.

2. There are a number of insulated conductors in given fixed positions. The capacities of any two of them in their given positions are C\ and C2, and their mutual coefficient of induction is B. Prove that if these conductors be joined by a thin wire, the capacity of the combined conductor is

Ci+Ca+25.

3. A system of insulated conductors having been charged in any manner, charges are

transferred from one conductor to another till they are all brought to the same potential V.

Shew that

V= Eft* +2s2),

where «i , s2 are the algebraic sums of the coefficients of capacity and induction respectively, and E is the sum of the charges.

4. Prove that the effect of the operation described in the last question is a decrease of the electrostatic energy equal to what would be the energy of the system if each of the original potentials were diminished by V.

5. Two equal similar condensers, each consisting of two spherical shells, radii a, b, are insulated and placed at a great distance r apart. Charges e, e' are given to the inner shells. If the outer surfaces are now joined by a wire, shew that the loss of energy is approximately

6. A condenser is formed of two thin concentric spherical shells, radii a, b. A small hole exists in the outer sheet through which an insulated wire passes connecting the inner sheet with a third conductor of capacity c, at a great distance r from the condenser. The outer sheet of the condenser is put to earth, and the charge on the two connected conductors is E. Prove that approximately the force on the third conductor is

"Ai-"

Yr».

7. Two closed equipotentials Vlf VQ are such that Vi contains V0, and VP is the potential at any point P between them. If now a charge E be put at P, and both equipotentials be replaced by conducting shells and earth-connected, then the charges Ei, E0 induced on the two surfaces are given by

Ey Eq E

8. A conductor is charged from an electrophorus by repeated contacts with a plate, which after each contact is recharged with a quantity E of electricity from the electro- phorus. Prove that if e is the charge of the conductor after the first operation, the

ultimate charge is

Ee E-e

112 Systems of Conductors [ch. iv

9. Four equal uncharged insulated conductors are placed symmetrically at the corners of a regular tetrahedron, and are touched in turn by a moving spherical conductor at the points nearest to the centre of the tetrahedron, receiving charges eu e2, e3, et. Shew that the charges are in geometrical progression.

10. In question 9 replace " tetrahedron " by "square," and prove that

Oi - *>) (e^3 - e22) = ex (e2e3 - e^).

11. Shew that if the distance x between two conductors is so great as compared with the linear dimensions of either, that the square of the ratio of these linear dimensions to x may be neglected, then the coefficient of induction between them is - CO'lx, where C, C are the capacities of the conductors when isolated.