Chapter II: Front Matter (2)

In the absolute determination of capacity we have to measure the ratio
of the charge of a condenser to its plate potential difference. One of
the best methods for doing this is to charge the condenser by the
known voltage of a battery, and then discharge it through a
galvanometer and repeat this process rapidly and successively. If a
condenser of capacity C is charged to potential V, and discharged n
times per second through a galvanometer, this series of intermittent
discharges is equivalent to a current nCV. Hence if the galvanometer
is calibrated by a potentiometer (q.v.) we can determine the value of
this current in amperes, and knowing the value of n and V thus
determine C. Various forms of commutator have been devised for
effecting this charge and discharge rapidly by J.J. Thomson, R.T.
Glazebrook, J.A. Fleming and W.C. Clinton and others.[11] One form
consists of a tuning-fork electrically maintained in vibration of
known period, which closes an electric contact at every vibration and
sets another electromagnet in operation, which reverses a switch and
moves over one terminal of the condenser from a battery to a
galvanometer contact. In another form, a revolving contact is used
driven by an electric motor, which consists of an insulating disk
having on its surface slips of metal and three wire brushes a, b, c
(see fig. 2) pressing against them. The metal slips are so placed
that, as the disk revolves, the middle brush, connected to one
terminal of the condenser C, is alternately put in conductive
connexion with first one and then the other outside brush, which are
joined respectively to the battery B and galvanometer G terminals.
From the speed of this motor the number of commutations per second can
be determined. The above method is especially useful for the
determinations of very small capacities of the order of 100
electrostatic units or so and upwards.

_Dielectric constant._--Since all electric charge consists in a state of strain or polarization of the dielectric, it is evident that the physical state and chemical composition of the insulator must be of great importance in determining electrical phenomena. Cavendish and subsequently Faraday discovered this fact, and the latter gave the name "specific inductive capacity," or "dielectric constant," to that quality of an insulator which determines the charge taken by a conductor embedded in it when charged to a given potential. The simplest method of determining it numerically is, therefore, that adopted by Faraday.[12] He constructed two equal condensers, each consisting of a metal ball enclosed in a hollow metal sphere, and he provided also certain hemispherical shells of shellac, sulphur, glass, resin, &c., which he could so place in one condenser between the ball and enclosing sphere that it formed a condenser with solid dielectric. He then determined the ratio of the capacities of the two condensers, one with air and the other with the solid dielectric. This gave the dielectric constant K of the material. Taking the dielectric constant of air as unity he obtained the following values, for shellac K = 2.0, glass K = 1.76, and sulphur K = 2.24.

TABLE I.--_Dielectric Constants (K) of Solids (K for Air = 1)._

+----------------------------------+-------+--------------------+
| Substance. | K. | Authority. |
+----------------------------------+-------+--------------------+
| Glass, double extra dense flint, | | |
| density 4.5 | 9.896 | J. Hopkinson |
| Glass, light flint, density 3.2 | 6.72 | " |
| Glass, hard crown, density 2.485 | 6.61 | " |
| / | 2.24 | M. Faraday |
| | | 2.88 | Coullner |
| Sulphur . . . . . . . . . . .< | 3.84 | L. Boltzmann |
| | | 4.0 | P.J. Curie |
| \ | 2.94 | P.R. Blondlot |
| / | 2.05 | Rosetti |
| | | 3.15 | Boltzmann |
| Ebonite . . . . . . . . . . .< | 2.21 | Schiller |
| \ | 2.86 | Elsas |
| India-rubber, pure brown | 2.12 | Schiller |
| India-rubber, vulcanized, grey | 2.69 | " |
| Gutta-percha | 2.462 | J.E. H. Gordon |
| / | 1.977 | Gibson and Barclay |
| | | 2.32 | Boltzmann |
| Paraffin . . . . . . . . . . .< | 2.29 | J. Hopkinson |
| \ | 1.99 | Gordon |
| / | 2.95 | Wallner |
| Shellac . . . . . . . . . . .< | 2.74 | Gordon |
| \ | 3.04 | A.A. Winkelmann |
| / | 6.64 | I. Klemencic |
| | | 8.00 | P.J. Curie |
| Mica . . . . . . . . . . . . .< | 7.98 | E.M.L. Bouty |
| \ | 5.97 | Elsas |
| Quartz-- | | |
| along optic axis | 4.55 | P.J. Curie |
| perp. to optic axis | 4.49 | P.J. Curie |
| Ice at -23 deg. |78.0 | Bouty |
+----------------------------------+-------+--------------------+

Since Faraday's time, by improved methods, but depending essentially upon the same principles, an enormous number of determinations of the dielectric constants of various insulators, solid, liquid and gaseous, have been made (see tables I., II., III. and IV.). There are very considerable differences between the values assigned by different observers, sometimes no doubt due to differences in method, but in most cases unquestionably depending on variations in the quality of the specimens examined. The value of the dielectric constant is greatly affected by the temperature and the frequency of the applied electric force.

TABLE II.--_Dielectric Constant (K) of Liquids._

+----------------------------+--------+--------------+
| Liquid. | K. | Authority. |
+----------------------------+--------+--------------+
| Water at 17 deg. C. | 80.88 | F. Heerwagen |
| " " 25 deg. C. | 75.7 | E.B. Rosa |
| " " 25.3 deg. C. | 78.87 | Franke |
| Olive oil | 3.16 | Hopkinson |
| Castor oil | 4.78 | " |
| Turpentine | 2.15 | P.A. Silow |
| " | 2.23 | Hopkinson |
| Petroleum | 2.072 | Silow |
| " | 2.07 | Hopkinson |
| Ethyl alcohol at 25 deg. C.| 25.7 | Rosa |
| Ethyl ether | 4.57 | Doule |
| " " | 4.8 | Bouty |
| Acetic acid | 9.7 | Franke |
+----------------------------+--------+--------------+

TABLE III.--_Dielectric Constant of some Bodies at a very low
Temperature (-185 deg. C.) (Fleming and Dewar)._

+----------------+-----------+------------+
| Substance. | K | K |
| | at 15 | at -185 |
| | deg. C. | deg. C. |
+----------------+-----------+------------+
| Water | 80 | 2.4 to 2.9 |
| Formic acid | 62 | 2.41 |
| Glycerine | 56 | 3.2 |
| Methyl alcohol | 34 | 3.13 |
| Nitrobenzene | 32 | 2.6 |
| Ethyl alcohol | 25 | 3.1 |
| Acetone | 21.85 | 2.62 |
| Ethyl nitrate | 17.7 | 2.73 |
| Amyl alcohol | 16 | 2.14 |
| Aniline | 7.5 | 2.92 |
| Castor oil | 4.78 | 2.19 |
| Ethyl ether | 4.25 | 2.31 |
+----------------+-----------+------------+

The above determinations at low temperature were made with either a steady or a slowly alternating electric force applied a hundred times a second. They show that the dielectric constant of a liquid generally undergoes great reduction in value when the liquid is frozen and reduced to a low temperature.[13]

The dielectric constants of gases have been determined by L. Boltzmann and I. Klemencic as follows:--

TABLE IV.--_Dielectric Constants (K) of Gases at 15 deg. C. and 760 mm.
Vacuum = 1._

+---------------------+------------+------------+------------+
| | Dielectric | | Optical |
| Gas. | Constant | [root]K. | Refractive |
| | K. | | Index. |
| | | | [mu]. |
+---------------------+------------+------------+------------+
| Air | 1.000590 | 1.000295 | 1.000293 |
| Hydrogen | 1.000264 | 1.000132 | 1.000139 |
| Carbon dioxide | 1.000946 | 1.000475 | 1.000454 |
| Carbon monoxide | 1.000690 | 1.000345 | 1.000335 |
| Nitrous oxide | 1.000994 | 1.000497 | 1.000516 |
| Ethylene | 1.001312 | 1.000656 | 1.000720 |
| Marsh gas (methane) | 1.000944 | 1.000478 | 1.000442 |
| Carbon bisulphide | 1.002900 | 1.001450 | 1.001478 |
| Sulphur dioxide | 1.00954 | 1.004770 | 1.000703 |
| Ether | 1.00744 | 1.003720 | 1.00154 |
| Ethyl chloride | 1.01552 | 1.007760 | 1.001174 |
| Ethyl bromide | 1.01546 | 1.007730 | 1.00122 |
+---------------------+------------+------------+------------+

In general the dielectric constant is reduced with decrease of temperature towards a certain limiting value it would attain at the absolute zero. This variation, however, is not always linear. In some cases there is a very sudden drop at or below a certain temperature to a much lower value, and above and below the point the temperature variation is small. There is also a large difference in most cases between the value for a steadily applied electric force and a rapidly reversed or intermittent force--in the last case a decrease with increase of frequency. Maxwell (_Elec. and Magn._ vol. ii. S 788) showed that the square root of the dielectric constant should be the same number as the refractive index for waves of the same frequency (see ELECTRIC WAVES). There are very few substances, however, for which the optical refractive index has the same value as K for steady or slowly varying electric force, on account of the great variation of the value of K with frequency.

There is a close analogy between the variation of dielectric constant of an insulator with electric force frequency and that of the rigidity or stiffness of an elastic body with the frequency of applied mechanical stress. Thus pitch is a soft and yielding body under steady stress, but a bar of pitch if struck gives a musical note, which shows that it vibrates and is therefore stiff or elastic for high frequency stress.

_Residual Charges in Dielectrics._--In close connexion with this lies the phenomenon of residual charge in dielectrics.[14] If a glass Leyden jar is charged and then discharged and allowed to stand awhile, a second discharge can be obtained from it, and in like manner a third, and so on. The reappearance of the residual charge is promoted by tapping the glass. It has been shown that this behaviour of dielectrics can be imitated by a mechanical model consisting of a series of perforated pistons placed in a tube of oil with spiral springs between each piston.[15] If the pistons are depressed and then released, and then the upper piston fixed awhile, a second discharge can be obtained from it, and the mechanical stress-strain diagram of the model is closely similar to the discharge curve of a dielectric. R.H.A. Kohlrausch called attention to the close analogy between residual charge and the elastic recovery of strained bodies such as twisted wire or glass threads. If a charged condenser is suddenly discharged and then insulated, the reappearance of a potential difference between its coatings is analogous to the reappearance of a torque in the case of a glass fibre which has been twisted, released suddenly, and then gripped again at the ends.

For further information on the qualities of dielectrics the reader is
referred to the following sources:--J. Hopkinson, "On the Residual
Charge of the Leyden Jar," _Phil. Trans._, 1876, 166 [ii.], p. 489,
where it is shown that tapping the glass of a Leyden jar permits the
reappearance of the residual charge; "On the Residual Charge of the
Leyden Jar," ib. 167 [ii.], p. 599, containing many valuable
observations on the residual charge of Leyden jars; W.E. Ayrton and J.
Perry, "A Preliminary Account of the Reduction of Observations on
Strained Material, Leyden Jars and Voltameters," _Proc. Roy. Soc._,
1880, 30, p. 411, showing experiments on residual charge of condensers
and a comparison between the behaviour of dielectrics and glass fibres
under torsion. In connexion with this paper the reader may also be
referred to one by L. Boltzmann, "Zur Theorie der elastischen
Nachwirkung," _Wien. Acad. Sitz.-Ber._, 1874, 70.

_Distribution of Electricity on Conductors._--We now proceed to
consider in more detail the laws which govern the distribution of
electricity at rest upon conductors. It has been shown above that the
potential due to a charge of q units placed on a very small sphere,
commonly called a point-charge, at any distance x is q/x. The
mathematical importance of this function called the potential is that
it is a scalar quantity, and the potential at any point due to any
number of point charges q1, q2, q3, &c., distributed in any manner, is
the sum of them separately, or

q1/x1 + q2/x2 + q3/x3 + &c. = [Sigma](q/x) = V (17),

where x1, x2, x3, &c., are the distances of the respective point
charges from the point in question at which the total potential is
required. The resultant electric force E at that point is then
obtained by differentiating V, since E = -dV/dx, and E is in the
direction in which V diminishes fastest. In any case, therefore, in
which we can sum up the elementary potentials at any point we can
calculate the resultant electric force at the same point.

We may describe, through all the points in an electric field which
have the same potential, surfaces called equipotential surfaces, and
these will be everywhere perpendicular or orthogonal to the lines of
electric force. Let us assume the field divided up into tubes of
electric force as already explained, and these cut normally by
equipotential surfaces. We can then establish some important
properties of these tubes and surfaces. At each point in the field the
electric force can have but one resultant value. Hence the
equipotential surfaces cannot cut each other. Let us suppose any other
surface described in the electric field so as to cut the closely
compacted tubes. At each point on this surface the resultant force has
a certain value, and a certain direction inclined at an angle [theta]
to the normal to the selected surface at that point. Let dS be an
element of the surface. Then the quantity E cos [theta]dS is the
product of the normal component of the force and an element of the
surface, and if this is summed up all over the surface we have the
total electric flux or induction through the surface, or the surface
integral of the normal force mathematically expressed by [int]E cos
[theta]dS, provided that the dielectric constant of the medium is
unity.

We have then a very important theorem as follows:--If any closed
surface be described in an electric field which wholly encloses or
wholly excludes electrified bodies, then the total flux through this
surface is equal to 4[pi]- times the total quantity of electricity
within it.[16] This is commonly called Stokes's theorem. The proof is
as follows:--Consider any point-charge E of electricity included in
any surface S, S, S (see fig. 3), and describe through it as centre a
cone of small solid angle d[omega] cutting out of the enclosing
surface in two small areas dS and dS' at distances x and x'. Then the
electric force due to the point charge q at distance x is q/x, and the
resolved part normal to the element of surface dS is q cos[theta]/x^2.
The normal section of the cone at that point is equal to dS
cos[theta], and the solid angle d[omega] is equal to dS cos[theta]/x^2.
Hence the flux through dS is qd[omega]. Accordingly, since the total
solid angle round a point is 4[pi], it follows that the total flux
through the closed surface due to the single point charge q is 4[pi]q,
and what is true for one point charge is true for any collection
forming a total charge Q of any form. Hence the total electric flux
due to a charge Q through an enclosing surface is 4[pi]Q, and
therefore is zero through one enclosing no electricity.

Stokes's theorem becomes an obvious truism if applied to an
incompressible fluid. Let a _source_ of fluid be a point from which an
incompressible fluid is emitted in all directions. Close to the source
the stream lines will be radial lines. Let a very small sphere be
described round the source, and let the strength of the source be
defined as the total flow per second through the surface of this small
sphere. Then if we have any number of sources enclosed by any surface,
the total flow per second through this surface is equal to the total
strengths of all the sources. If, however, we defined the strength of
the source by the statement that the strength divided by the square
of the distance gives the velocity of the liquid at that point, then
the total flux through any enclosing surface would be 4[pi] times the
strengths of all the sources enclosed. To every proposition in
electrostatics there is thus a corresponding one in the hydrokinetic
theory of incompressible liquids.

Let us apply the above theorem to the case of a small
parallel-epipedon or rectangular prism having sides dx, dy, dz
respectively, its centre having co-ordinates (x, y, z). Its angular
points have then co-ordinates (x [+-] 1/2 dx, y [+-] 1/2 dy, z [+-]
1/2 dz). Let this rectangular prism be supposed to be wholly filled up
with electricity of density [rho]; then the total quantity in it is
[rho] dx dy dz. Consider the two faces perpendicular to the x-axis.
Let V be the potential at the centre of the prism, then the normal
forces on the two faces of area dy.dx are respectively

/dV 1 d^2V \ /dV 1 d^2V \
- ( -- + -- ---- dx ) and ( -- - -- ---- dx ),
\dx 2 dx^2 / \dx 2 dx^2 /

and similar expressions for the normal forces to the other pairs of
faces dx.dy, dz.dx. Hence, multiplying these normal forces by the
areas of the corresponding faces, we have the total flux parallel to
the x-axis given by -(d^2V/dx^2)dx dy dz, and similar expressions for
the other sides. Hence the total flux is

/d^2V d^2V d^2V\
- ( ---- + ---- + ---- ) dx dy dz
\dx^2 dy^2 dz^2/

and by the previous theorem this must be equal to 4[pi][rho]dx dy dz.

d^2V d^2V d^2V
Hence ---- + ---- + ---- + 4[pi][rho] = 0 (18).
dx^2 dy^2 dz^2

This celebrated equation was first given by S.D. Poisson, although
previously demonstrated by Laplace for the case when [rho] = 0. It
defines the condition which must be fulfilled by the potential at any
and every point in an electric field, through which [rho] is finite
and the electric force continuous. It may be looked upon as an
equation to determine [rho] when V is given or vice versa. An exactly
similar expression holds good in hydrokinetics, provided that for the
electric potential we substitute velocity potential, and for the
electric force the velocity of the liquid.

The Poisson equation cannot, however, be applied in the above form to
a region which is partly within and partly without an electrified
conductor, because then the electric force undergoes a sudden change
in value from zero to a finite value, in passing outwards through the
bounding surface of the conductor. We can, however, obtain another
equation called the "surface characteristic equation" as
follows:--Suppose a very small area dS described on a conductor having
a surface density of electrification [sigma]. Then let a small, very
short cylinder be described of which dS is a section, and the
generating lines are normal to the surface. Let V1 and V2 be the
potentials at points just outside and inside the surface dS, and let
n1 and n2 be the normals to the surface dS drawn outwards and inwards;
then -dV1/dn1 and -dV2/dn2 are the normal components of the force over
the ends of the imaginary small cylinder. But the force perpendicular
to the curved surface of this cylinder is everywhere zero. Hence the
total flux through the surface considered is -{(dV1/dn1) +
(dV2/dn2)}dS, and this by a previous theorem must be equal to
4[pi][sigma]dS, or the total included electric quantity. Hence we have
the surface characteristic equation,[17]

(dV1/dn1) + (dV2/dn2) + 4[pi][sigma] = 0 (19).

Let us apply these theorems to a portion of a tube of electric force.
Let the part selected not include any charged surface. Then since the
generating lines of the tube are lines of force, the component of the
electric force perpendicular to the curved surface of the tube is
everywhere zero. But the electric force is normal to the ends of the
tube. Hence if dS and dS' are the areas of the ends, and +E and -E'
the oppositely directed electric forces at the ends of the tube, the
surface integral of normal force on the flux over the tube is

EdS - E'dS' (20),

and this by the theorem already given is equal to zero, since the tube
includes no electricity. Hence the characteristic quality of a tube of
electric force is that its section is everywhere inversely as the
electric force at that point. A tube so chosen that EdS for one
section has a value unity, is called a unit tube, since the product of
force and section is then everywhere unity for the same tube.

In the next place apply the surface characteristic equation to any
point on a charged conductor at which the surface density is [sigma].
The electric force outward from that point is -dV/dn, where dn is a
distance measured along the outwardly drawn normal, and the force
within the surface is zero. Hence we have

-dV/dn = 4.0[pi][sigma] or [sigma] = -(1/4 [pi])dV/dn = E/4[pi].

The above is a statement of Coulomb's law, that _the electric force at
the surface of a conductor is proportional to the surface density of
the charge at that point and equal to 4[pi] times the density_.[18]

If we define the positive direction along a tube of electric force as
the direction in which a small body charged with positive electricity
would tend to move, we can summarize the above facts in a simple form
by saying that, _if we have any closed surface described in any manner
in an electric field, the excess of the number of unit tubes which
leave the surface over those which enter it is equal to 4[pi]-times
the algebraic sum of all the electricity included within the surface_.

Every tube of electric force must therefore begin and end on
electrified surfaces of opposite sign, and the quantities of positive
and negative electricity on its two ends are equal, since the force E
just outside an electrified surface is normal to it and equal to
[sigma]/4[pi], where [sigma] is the surface density; and since we have
just proved that for the ends of a tube of force EdS = E^1dS', it
follows that [sigma]dS = [sigma]'dS', or Q = Q', where Q and Q' are
the quantities of electricity on the ends of the tube of force.
Accordingly, since every tube sent out from a charged conductor must
end somewhere on another charge of opposite sign, it follows that the
two electricities always exist in equal quantity, and that it is
impossible to create any quantity of one kind without creating an
equal quantity of the opposite sign.

We have next to consider the energy storage which takes place when
electric charge is created, i.e. when the dielectric is strained or
polarized. Since the potential of a conductor is defined to be the
work required to move a unit of positive electricity from the surface
of the earth or from an infinite distance from all electricity to the
surface of the conductor, it follows that the work done in putting a
small charge dq into a conductor at a potential v is v dq. Let us then
suppose that a conductor originally at zero potential has its
potential raised by administering to it small successive doses of
electricity dq. The first raises its potential to v, the second to v'
and so on, and the nth to V. Take any horizontal line and divide it
into small elements of length each representing dq, and draw vertical
lines representing the potentials v, v', &c., and after each dose.
Since the potential rises proportionately to the quantity in the
conductor, the ends of these ordinates will lie on a straight line and
define a triangle whose base line is a length equal to the total
quantity Q and height a length equal to the final potential V. The
element of work done in introducing the quantity of electricity dq at
a potential v is represented by the element of area of this triangle
(see fig. 4), and hence the work done in charging the conductor with
quantity Q to final potential V is 1/2 QV, or since Q = CV, where C is
its capacity, the work done is represented by 1/2 CV^2 or by 1/2 Q^2/C.

If [sigma] is the surface density and dS an element of surface, then
[int][sigma]dS is the whole charge, and hence 1/2 [int] V[sigma]dS is
the expression for the energy of charge of a conductor.

We can deduce a remarkable expression for the energy stored up in an
electric field containing electrified bodies as follows:[19] Let V
denote the potential at any point in the field. Consider the integral
_ _ _ _ _
1 / / / | /dV\^2 /dV\^2 /dV\^2 |
W = ----- | | | |(----) + (----) + (----) | dx dy dz. (21)
8[pi]_/_/_/ |_\dx/ \dy/ \dz/ _|

where the integration extends throughout the whole space unoccupied by
conductors. We have by partial integration
_ _ _ _ _ _ _ _
/ / / /dV\^2 / / dV / / / d^2V
| | |(----) dx dy dz = | | V -- dy dz - | | | V ---- dx dy dz,
_/_/_/ \dx/ _/_/ dx _/_/_/ dx^2

and two similar equations in y and z. Hence
_ _ _ _ _
1 / / / | /dV\^2 /dV\^2 /dV\^2 |
----- | | | | (----) + (----) + (----) | dx dy dz =
8[pi]_/_/_/ |_ \dx/ \dy/ \dz/ _|
_ _ _ _ _
1 / / dV 1 / / /
----- | | V -- dS - ----- | | | V[nabla]V dx dy dz (22)
8[pi]_/_/ dn 8[pi]_/_/_/

where dV/dn means differentiation along the normal, and [nabla] stands

d^2 d^2 d^2
for the operator ---- + ---- + ----. Let E be the resultant electric
dx^2 dy^2 dz^2

force at any point in the field. Then bearing in mind that [sigma] =
(1/4 [pi])dV/dn, and [rho] = -(1/4 [pi])[nabla]V, we have finally
_ _ _ _ _ _ _ _
1 / / / 1 / / 1 / / /
----- | | | E^2dv = -- | | V[sigma]dS + -- | | | V[rho]dv.
8[pi] _/_/_/ 2 _/_/ 2 _/_/_/

The first term on the right hand side expresses the energy of the
surface electrification of the conductors in the field, and the second
the energy of volume density (if any). Accordingly the term on the
left hand side gives us the whole energy in the field.

Suppose that the dielectric has a constant K, then we must multiply
both sides by K and the expression for the energy per unit of volume
of the field is equivalent to 1/2 DE where D is the displacement or
polarization in the dielectric.

Furthermore it can be shown by the application of the calculus of
variations that the condition for a minimum value of the function W,
is that [nabla]V = 0. Hence that distribution of potential which is
necessary to satisfy Laplace's equation is also one which makes the
potential energy a minimum and therefore the energy stable. Thus the
actual distribution of electricity on the conductor in the field is
not merely a stable distribution, it is _the only_ possible stable
distribution.

_Method of Electrical Images._--A very powerful method of attacking
problems in electrical distribution was first made known by Lord
Kelvin in 1845 and is described as the method of electrical
images.[20] By older mathematical methods it had only been possible to
predict in a few simple cases the distribution of electricity at rest
on conductors of various forms. The notion of an electrical image may
be easily grasped by the following illustration: Let there be at A
(see fig. 5) a point-charge of positive electricity +q and an infinite
conducting plate PO, shown in section, connected to earth and
therefore at zero potential. Then the charge at A together with the
induced surface charge on the plate makes a certain field of electric
force on the left of the plate PO, which is a zero equipotential
surface. If we remove the plate, and yet by any means can keep the
identical surface occupied by it a plane of zero potential, the
boundary conditions will remain the same, and therefore the field of
force to the left of PO will remain unaltered. This can be done by
placing at B an equal negative point-charge -q in the place which
would be occupied by the optical image of A if PO were a mirror, that
is, let -q be placed at B, so that the distance BO is equal to the
distance AO, whilst AOB is at right angles to PO. Then the potential
at any point P in this ideal plane PO is equal to q/AP - q/BP = O,
whilst the resultant force at P due to the two point charges is
2qAO/AP^3, and is parallel to AB or normal to PO. Hence if we remove
the charge -q at B and distribute electricity over the surface PO with
a surface density [sigma], according to the Coulomb-Poisson law,
[sigma] = qAO/2[pi]AP^3, the field of force to the left of PD will
fulfil the required boundary conditions, and hence will be the law of
distribution of the induced electricity in the case of the actual
plate. The point-charge -q at B is called the "electrical image" of
the point-charge +q at A.

We find a precisely analogous effect in optics which justifies the
term "electrical image." Suppose a room lit by a single candle. There
is everywhere a certain illumination due to it. Place across the room
a plane mirror. All the space behind the mirror will become dark, and
all the space in front of the mirror will acquire an exalted
illumination. Whatever this increased illumination may be, it can be
precisely imitated by removing the mirror and placing a second lighted
candle at the place occupied by the optical image of the first candle
in the mirror, that is, as far behind the plane as the first candle
was in front. So the potential distribution in the space due to the
electric point-charge +q as A together with -q at B is the same as
that due to +q at A and the negative induced charge erected on the
infinite plane (earthed) metal sheet placed half-way between A and B.

The same reasoning can be applied to determine the electrical image of
a point-charge of positive electricity in a spherical surface, and
therefore the distribution of induced electricity over a metal sphere
connected to earth produced by a point-charge near it. Let +q be any
positive point-charge placed at a point A outside a sphere (fig. 6) of
radius r, and centre at C, and let P be any point on it. Let CA = d.
Take a point B in CA such that CB.CA = r^2, or CB = r^2/d. It is easy
then to show that PA : PB = d : r. If then we put a negative
point-charge -qr/d at B, it follows that the spherical surface will be
a zero potential surface, for

q/PA - rq/d . 1/PB = 0 (24).

Another equipotential surface is evidently a very small sphere
described round A. The resultant force due to these two point-charges
must then be in the direction CP, and its value E is the vector sum of
the two forces along AP and BP due to the two point-charges. It is not
difficult to show that

E = -(d^2 - r^2)q/rAP^3 (25),

in other words, the force at P is inversely as the cube of the
distance from A. Suppose then we remove the negative point-charge, and
let the sphere be supposed to become conductive and be connected to
earth. If we make a distribution of negative electricity over it,
which has a density [sigma] varying according to the law

[sigma] = -(d^2 - r^2)q/4[pi]rAP^3 (26),

that distribution, together with the point-charge +q at A, will make a
distribution of electric force at all points outside the sphere
exactly similar to that which would exist if the sphere were removed
and a negative point charge -qr/d were placed at B. Hence this charge
is the electrical image of the charge +q at A in the spherical
surface.

We may generalize these statements in the following theorem, which is
an important deduction from a wider theorem due to G. Green. Suppose
that we have any distribution of electricity at rest over conductors,
and that we know the potential at all points and consequently the
level or equipotential surfaces. Take any equipotential surface
enclosing the whole of the electricity, and suppose this to become an
actual sheet of metal connected to the earth. It is then a zero
potential surface, and every point outside is at zero potential as far
as concerns the electric charge on the conductors inside. Then if U is
the potential outside the surface due to this electric charge inside
alone, and V that due to the opposite charge it induces on the inside
of the metal surface, we must have U + V = 0 or U = -V at all points
outside the earthed metal surface. Therefore, whatever may be the
distribution of electric force produced by the charges inside taken
alone, it can be exactly imitated for all space outside the metal
surface if we suppose the inside charge removed and a distribution of
electricity of the same sign made over the metal surface such that its
density follows the law

[sigma]= -(1/4 [pi])dU/dn (27),

where dU/dn is the electric force at that point on the closed
equipotential surface considered, due to the original charge alone.

BIBLIOGRAPHY.--For further developments of the subject we must refer
the reader to the numerous excellent treatises on electrostatics now
available. The student will find it to be a great advantage to read
through Faraday's three volumes entitled _Experimental Researches on
Electricity_, as soon as he has mastered some modern elementary book
giving in compact form a general account of electrical phenomena. For
this purpose he may select from the following books: J. Clerk Maxwell,
_Elementary Treatise on Electricity_ (Oxford, 1881); J.J. Thomson,
_Elements of the Mathematical Theory of Electricity and Magnetism_
(Cambridge, 1895); J.D. Everett, _Electricity_, founded on part iii.
of Deschanel's _Natural Philosophy_ (London, 1901); G.C. Foster and
A.W. Porter, _Elementary Treatise on Electricity and Magnetism_
(London, 1903); S.P. Thompson, _Elementary Lessons on Electricity and
Magnetism_ (London, 1903).

When these elementary books have been digested, the advanced student
may proceed to study the following: J. Clerk Maxwell, _A Treatise on
Electricity and Magnetism_ (1st ed., Oxford, 1873; 2nd ed. by W.D.
Niven, 1881; 3rd ed. by J.J. Thomson, 1892); Joubert and Mascart,
_Electricity and Magnetism_, English translation by E. Atkinson
(London, 1883); Watson and Burbury, _The Mathematical Theory of
Electricity and Magnetism_ (Oxford, 1885); A. Gray, _A Treatise on
Magnetism and Electricity_ (London, 1898). In the collected
_Scientific Papers_ of Lord Kelvin (3 vols., Cambridge, 1882), of
James Clerk Maxwell (2 vols., Cambridge, 1890), and of Lord Rayleigh
(4 vols., Cambridge, 1903), the advanced student will find the means
for studying the historical development of electrical knowledge as it
has been evolved from the minds of some of the master workers of the
19th century. (J. A. F.)

FOOTNOTES:

[1] See Maxwell, _Elementary Treatise on Electricity_ (Oxford, 1881),
p. 47.

[2] See Maxwell, _Treatise on Electricity and Magnetism_ (3rd ed.,
Oxford, 1892), vol. i. p. 80.

[3] Maxwell, Ibid. vol. i. S 74a; also _Electrical Researches of the
Hon. Henry Cavendish_, edited by J. Clerk Maxwell (Cambridge, 1879),
p. 104.

[4] Laplace (_Mec. Cel._ vol. i. ch. ii.) gave the first direct
demonstration that no function of the distance except the inverse
square can satisfy the condition that a uniform spherical shell
exerts no force on a particle within it.

[5] The solution of the problem of determining the distribution on an
ellipsoid of a fluid the particles of which repel each other with a
force inversely as the nth power of the distance was first given by
George Green (see Ferrer's edition of Green's _Collected Papers_, p.
119, 1871).

[6] See Thomson and Tait, _Treatise on Natural Philosophy_, S 519.

[7] See article "Electricity," _Encyclopaedia Britannica_ (9th
edition), vol. viii. p. 30. The reader is also referred to an article
by Lord Kelvin (_Reprint of Papers on Electrostatics and Magnetism_,
p. 178), entitled "Determination of the Distribution of Electricity
on a Circular Segment of a Plane, or Spherical Conducting Surface
under any given Influence," where another equivalent expression is
given for the capacity of an ellipsoid.

[8] See Maxwell, _Electricity and Magnetism_, vol. i. pp. 284-305
(3rd ed., 1892).

[9] It is an interesting fact that Cavendish measured capacity in
"globular inches," using as his unit the capacity of a metal ball, 1
in. in diameter. Hence multiplication of his values for capacities by
2.54 reduces them to E.S. units in the C.G.S. system. See _Elec.
Res._ p. 347.

[10] For fuller details of these methods of comparison of capacities
see J.A. Fleming, _A Handbook for the Electrical Laboratory and
Testing Room_, vol. ii. ch. ii. (London, 1903).

[11] See Fleming, _Handbook for the Electrical Laboratory_, vol. ii.
p. 130.

[12] Faraday, _Experimental Researches on Electricity_, vol. i. S
1252. For a very complete set of tables of dielectric constants of
solids, liquids and gases see A. Winkelmann, _Handbuch der Physik_,
vol. iv. pp. 98-148 (Breslau, 1905); also see Landolt and Bornstein's
_Tables of Physical Constants_ (Berlin, 1894).

[13] See the following papers by J.A. Fleming and James Dewar on
dielectric constants at low temperatures: "On the Dielectric Constant
of Liquid Oxygen and Liquid Air," _Proc. Roy. Soc._, 1897, 60, p.
360; "Note on the Dielectric Constant of Ice and Alcohol at very low
Temperatures," ib., 1897, 61, p. 2; "On the Dielectric Constants of
Pure Ice, Glycerine, Nitrobenzol and Ethylene Dibromide at and above
the Temperature of Liquid Air," id. ib. p. 316; "On the Dielectric
Constant of Certain Frozen Electrolytes at and above the Temperature
of Liquid Air," id. ib. p. 299--this paper describes the cone
condenser and methods used; "Further Observations on the Dielectric
Constants of Frozen Electrolytes at and above the Temperature of
Liquid Air," id. ib. p. 381; "The Dielectric Constants of Certain
Organic Bodies at and below the Temperature of Liquid Air," id. ib.
p. 358; "On the Dielectric Constants of Metallic Oxides dissolved or
suspended in Ice cooled to the Temperature of Liquid Air," id. ib. p.
368.

[14] See Faraday, _Experimental Researches_, vol. i. S 1245; R.H.A.
Kohlrausch, _Pogg. Ann._, 1854, 91; see also Maxwell, _Electricity
and Magnetism_, vol. i. S 327, who shows that a composite or
stratified dielectric composed of layers of materials of different
dielectric constants and resistivities would exhibit the property of
residual charge.

[15] Fleming and Ashton, "On a Model which imitates the behaviour of
Dielectrics." _Phil. Mag._, 1901 [6], 2, p. 228.

[16] The beginner is often puzzled by the constant appearance of the
factor 4[pi] in electrical theorems. It arises from the manner in
which the unit quantity of electricity is defined. The electric force
due to a point-charge q at a distance r is defined to be q/r^2, and
the total flux or induction through the sphere of radius r is
therefore 4[pi]q. If, however, the unit point charge were defined to
be that which produces a unit of electric flux through a
circumscribing spherical surface or the electric force at distance r
defined to be 1/4 [pi]r^2, many theorems would be enunciated in
simpler forms.

[17] See Maxwell, _Electricity and Magnetism_, vol. i. S 78b (2nd
ed.).

[18] Id. ib. vol. i. S 80. Coulomb proved the proportionality of
electric surface force to density, but the above numerical relation E
= 4[pi][sigma] was first established by Poisson.

[19] See Maxwell, _Electricity and Magnetism_, vol. i. S 99a (3rd
ed., 1892), where the expression in question is deduced as a
corollary of Green's theorem.

[20] See Lord Kelvin's _Papers on Electrostatics and Magnetism_, p.
144.

ELECTROTHERAPEUTICS, a general term for the use of electricity in therapeutics, i.e. in the alleviation and cure of disease. Before the different forms of medical treatment are dealt with, a few points in connexion with the machines and currents, of special interest to the medical reader, must first be given.

_Faradism._--For the battery required either for faradism or galvanism, cells of the Leclanche type are the most satisfactory. Being dry they can be carried in any position, are lighter, and there is no trouble from the erosion of wires and binding screws, such as so often results from wet cells. The best method of producing a smooth current in the secondary coil is for the interruptor hammer to vibrate directly against the iron core of the primary coil. For this it is best that the interruptor be made of a piece of steel spring, as a high rate of interruption can then be maintained, with a fairly smooth current in the secondary coil. This form of interruptor necessitates that the iron core be fixed, and variation in the primary induced current is arranged for by slipping a brass tube more or less over the iron core, thus cutting off the magnetic field from the primary coil. The secondary current (that obtained from the secondary coil) can be varied by keeping the secondary coil permanently fixed over the primary and varying the strength of the primary current. Where, as suggested above, the iron core is fixed, the primary and secondary induced currents will be at their strongest when the brass tube is completely withdrawn. As there is no simple means of measuring the strength of the faradic current, it is best to start with a very weak current, testing it on the muscles of one's own hand until these begin to contract and a definite sensory effect is produced; the current can then be applied to the part, being strengthened only very gradually.

_Galvanism._--For treatment by galvanism a large battery is needed, the simplest form being known as a "patient's battery," consisting of a variable number of dry cells arranged in series. The cells used are those of Leclanche, with E.M.F. (or voltage) of 1.5 and an internal resistance of .3 ohm. Thus the exact strength of the current is known; the number of cells usually employed is 24, and when new give an E.M.F. of about 36 volts. By using the formula C = E/R, where E is the voltage of the battery, R the total resistance of battery, electrodes and the patient's skin and tissues, and C the current in amperes, the number of cells required for any particular current can be worked out. The resistance of the patient's skin must be made as low as possible by thoroughly wetting both skin and electrodes with sodium bicarbonate solution, and keeping the electrodes in very close apposition to the skin. A galvanometer is always fitted to the battery, usually of the d'Arsonval type, with a shunt by means of which, on turning a screw, nine-tenths of the inducing current can be short-circuited away, and the solenoid only influenced by one-tenth of the current which is being used on the patient. In districts where electric power is available the continuous current can be used by means of a switchboard. A current of much value for electrotherapeutic purposes is the sinusoidal current, by which is meant an alternating current whose curve of electromotive force, in both positive and negative phase, varies constantly and smoothly in what is known as the sine curve. In those districts supplied by an alternating current, the sinusoidal current can be obtained from the mains by passing it through various transformers, but where the main supply is the direct or constant current, a motor transformer is needed.

_Static Electricity._--For treatment by static electricity the Wimshurst type of machine is the one most generally used. A number of electrodes are required; thus for the application of sparks a brass ball and brass roller electrode, for the "breeze" a single point and a multiple point electrode, and another multiple point electrode in the form of a metal cap that can be placed over the patient's head. The polarity of the machine must always be tested, as either knob may become positive or negative, though the polarity rarely changes when once the machine is in action. The oldest method of subjecting a patient to electric influence is that in which static electricity is employed. The patient is insulated on a suitable platform and treated by means of charges and discharges from an electrical machine. The effect is to increase the regularity and frequency of the pulse, raise the blood pressure and increase the action of the skin. The nervous system is quieted, sleep being promoted, the patient often becoming drowsy during the application. If while the patient is being treated a point electrode is brought towards him he feels the sensation of a wind blowing from that point; this is an electric breeze or brush discharge. The breeze is negative if the patient is positively charged and vice versa. The "breeze discharge" treatment is especially valuable in subduing pain of the superficial cutaneous nerves, and also in the treatment of chronic indolent ulcers. Quite recently this form of treatment has been applied with much success to various skin lesions--psoriasis, eczema and pruritus. Static electricity is also utilized for medical purposes by means of "sparks," which are administered with a ball electrode, the result being a sudden muscular contraction at the point of application. The electrode must be rapidly withdrawn before a second spark has time to leap across, as this is a severe form of treatment and must be administered slowly. It is mainly employed for muscular stimulation, and the contractions resulting from spark stimulation can be produced in cases of nerve injury and degeneration, even when the muscles have lost their reaction to faradism. The sensory stimulation of this form of treatment is also strong, and is useful in hysterical anaesthesia and functional paralysis. Where a milder sensory stimulation is required friction can be used, the electrode being in the form of a metal roller which is moved rapidly outside the patient's clothing over the spine or other part to be treated. The clothing must be dry and of wool, and each additional woollen layer intensifies the effect.

Another method of employing electricity at high potential is by the employment of high frequency currents. There are two methods of application: that in which brush discharges are made use of, with undoubtedly good effects in many of the diseases affecting the surface of the body, and that in which the currents of the solenoid are made to traverse the patient directly. The physiological value of the latter method is not certain, though one point of interest in connexion with it is that whereas statical applications raise the blood pressure, high frequency applications lower it. It has been used in the case of old people with arterio-sclerosis, and the reduction of blood pressure produced is said to have shown considerable permanence.

_The Faradic Current._--G.B. Duchenne was the first physician to make use of the induced current for treatment, and the term "faradization" is supposed to be due to him. But in his day the differences between the two currents available, the primary and the secondary, were not worked out, and they were used somewhat indiscriminately. Nowadays it is generally accepted that the primary current should be used for the stimulation of deep-lying organs, as stomach and intestines, &c., while the secondary current is employed for stimulation of the limb muscles and the cutaneous sensory nerves. The faradic current is also used as a means of diagnosis for neuro-muscular conditions. When the interrupted current is used to stimulate the skin over a motor nerve, all the muscles supplied by that nerve are thrown into rapid tetanic contraction, the contraction both beginning and ceasing sharply and suddenly with the current. This is the normal reaction of the nerve to faradism. If the muscle be wasted from disuse or some local cause unconnected with its nerve-supply, the contraction is smaller, and both arises and relaxes more slowly. But if the lesion lies in the nerve itself, as in Bell's palsy, the muscles no longer show any response when the nerve is stimulated, and this is known as the reaction of degeneration in the nerve. It is usually preceded by a condition of hyperexcitability. These results are applied to distinguish between functional paralysis and that due to some organic lesion, as in the former case the reaction of faradism will be as brisk as usual. Also at the beginning of most cases of infantile paralysis many more groups of muscles appear to be affected than ultimately prove to be, and faradism enables the physician to distinguish between those groups of muscles that are permanently paralysed owing to the destruction of their trophic centre, and those muscles which are only temporarily inhibited from shock, and which with proper treatment will later regain their full power. In the testing of muscles electrically that point on the skin which on stimulation gives the maximum contraction for that muscle is known as the "motor point" for that muscle. It usually corresponds to the entry of the motor nerve. Faradic treatment may be employed in the weakness and emaciation depending on any long illness, rickets, anaemia, &c. For these cases it is best to use the electric bath, the patient being placed in warm water, and the two electrodes, one at the patient's back and the other at his feet, being connected with the secondary coil. The patient's general metabolism is stimulated, he eats and sleeps better and soon begins to put on weight. This is especially beneficial in severe cases of rickets. In the weakness and emaciation due to neurasthenia, especially in those cases being treated by the Weir Mitchell method (isolation, absolute confinement to bed, massage and overfeeding), a similar faradic bath is a very helpful adjunct. In tabes dorsalis faradic treatment will often diminish the anaesthesia and numbness in the legs, with resulting benefit to the ataxy. Perhaps the most beneficial use of the faradic current is in the treatment of chronic constipation--especially that so frequently met with in young women and due to deficient muscular power of the intestinal walls. In long-standing cases the large intestine becomes permanently dilated, and its muscular fibres so attenuated as to have no power over the intestinal contents. But faradism causes contraction at the point of stimulation, and the peristaltic wave thus started slowly progresses along the bowel. All that is needed is a special electrode for introduction into the bowel and an ordinary roller electrode. The rectal electrode consists of a 6-inch wire bearing at one end a small metal knob and fitted at the other into a metal cup which screws into the handle of the electrode. The only part exposed is the metallic knob; the rest is coated with some insulating material. The patient reclines on a couch on his back, the rectal electrode is connected, and having been vaselined is passed some three inches into the rectum. A current is started with the secondary coil in such a position as to give only an extremely weak current. The roller electrode is then wetted with hot water and applied to the front of the abdomen. At first the patient should feel nothing, but the current should slowly be increased until a faint response is perceptible from the abdominal muscles. This gives the required strength, and the roller electrode, pressed well into the abdominal wall, should very slowly be moved along the course of the large intestine beginning at the right iliac fossa. Thus a combination of massage and faradic current is obtained, and the results are particularly satisfactory. Treatment should be given on alternate days immediately after breakfast, and should be persevered with for six or eight weeks. The patient can be taught to administer it to himself.