(relatively insignificant) condensing system with which we are not concerned in our present discussion.

The above calculation is based upon the supposition that the dielectric is air. For our quantity Q is measured in terms of a unit which repels another unit at 1 cm. distance with 1 dyne force in air; and our expressions for measurement of potential depend on this condition.

Now, if we have any other dielectric of specific inductive capacity ☛, then by the definition of a given in Chapter IX. we must write ☛. K instead of K ; and, if K' is the new capacity, we have

The reader will notice that this is the expression that we should arrive at if we assumed, all through our calculations of potential as measured by work done, that in the new medium the quantity Q acted on our unit with a force of dynes at a distance r, instead of

σ

the dynes with which it acts on the + unit when air is the medium. Hence we conclude that, in a medium of specific inductive capacity o, the force between two quantities Qand Q ́ (measured in the absolute unit I Q. Q' of Chapter V. § 1), separated by a distance r, will be

We

may

write the above relation in the form

σ

dynes.

A

§ 26. The Plate Condenser.-In the diagram we give a simple section of a plate condenser A B. Let V1 and V, represent the potentials of A and B respectively; let Q be the 'bound' charges on the two plates respectively; let be the perpendicular distance. between them, measured of course in centimètres ; let

S be the area of the inner surface of either A or B ; let

p

be the density of charge on these inside surfaces; let

be the specific inductive capacity of the dielectric.

Now we shall assume that is so small as compared with the size of the plates that the field of force between A and B is

uniform; i.e. that it consists of lines of force that are parallel and equidistant, which implies that p is uniform.

Then by § 12 we have for the field strength F, or force acting on a unit, between A and B,

VA - VB
F=

t

But by § 24 we have that at the surface of the plate

But if

In the above we have assumed air to be the dielectric. we have another dielectric of specific inductive capacity = o, then the

§ 27. Formulæ for Capacities, &c.-For convenience we here put together the chief formula that have occurred up to the present point in this chapter, adding also one or two more.

The following remarks will apply to some of them, if not

to all.

(a) All the measurements are in the C.G.S. system of units. (b) Electric quantity is measured by reference to force exerted when air is the dielectric. When there is another dielectric of specific

inductive capacity o, the force exerted is th that exerted through air

σ

see $ 25, end). The capacity of a condenser with this dielectric will by definition be times that of a similar air condenser.

(c) An ordinary condenser is really a double system, as pointed out immediately after equation (3) in § 25. The 'isolated' body is the limiting case of such a condenser, when this has become a single system. The formula for isolated bodies can be derived from these for condensers, though it is often more simple to find them independently.

I. Formulæ connecting V, Q, and K.

=

(i.) For 'isolated' bodies, Q. K.V.
(ii.) For condensers,

Q=K. (V1-V2).

In the former case Q is the sole charge; in the latter Q is the 'bound' charge.

II. Capacities of some 'isolated' bodies.

(i.) For a sphere of radius R, in air, K = R.

(ii.) For a disc of radius

and of negligible thickness, in air,

(iii.) For a cylinder of length / and of relatively small radius 7,

A wire comes under the head of (iii.). The two last formulæ have not been proved in the foregoing.

III. Capacities of some condensers.

(i.) For a spherical condenser where the radius of the inside sphere is R1, and that of the inner surface of the outer sphere is R2,

(ii.) For two co-axial cylinders of common length 7, the radius of the inner being r1, and that of the inner surface of the outer cylinder being r1⁄2,

(iii.) For two discs, that surface of either on which is the bound charge being S, and the distance between them being t,

This formula applies to any case of parallel surfaces where the dimensions are very large as compared with t. In fact, the formula (i.) above can be transformed into this when R, and R, are very great as compared with (R,-R1).

$28. Energy of Charging and Discharging.-When electricity falls from a higher to a lower level, work can be done; we lose a certain quantity of electrical potential energy, and we must have an equivalent in heat or mechanical work, &c. If conversely we raise electricity from a lower level to a higher, we expend this work and gain the above electrical potential-energy which is an equivalent (see § 4).

There is no more important matter in our present science of electricity than this question of the application of the law of 'Conservation of energy' to electrical charge and discharge. We will consider the matter under three heads.

Case I. The case where an electrical quantity Q passes between two conductors of fixed potentials V, and V2.-Now by definition and measurement of potential, we do on each + unit an amount

of + or work that is measured by (V1—V2) ergs. Hence, on the quantity of Q units we do + or

Q. (V1-V2) ergs.

1

work measured by

If one potential be V and the other be zero, the work is Q. V ergs.

Hence, in this case the energy E of charge or discharge is measured by

or

E = Q. (V1-V1) ergs

(a)

Case II. The case, more usual in electrostatics, where the potential alters during charge or discharge.--We will consider the simplest case of charging a conductor, that is originally at zero potential, with a quantity Q, up to a potential V.

Here the potential of the conductor rises in arithmetical progression with the charge given, since

and capacity is constant. The result, therefore, is that the same work is done as if the whole charge were raised to half the final potential V. Hence the work of charging, or energy expended,

This same expression gives the energy of discharge.

Analogies. (a) So, it we reckon the number of foot-pounds-weight done in building with P lbs. of bricks a tower of H feet high, it being assumed that the tower is cylindrical or rises in arithmetical progression with the amount of bricks used, we get as a result the same work as if all the bricks were raised to half the vertical height H; ¿.e. we get

=

work P. H. foot-pounds-weight.

(6) A similar expression is obtained for work done by water flowing out of a cylindrical reservoir of an average height of H above the lower level.

Case III. Case of discharge of a condenser.—In the ordinary condenser we have two plates of potentials V, and V, respectively.

In this case the discharge will cause the equal and opposite charges +Q and Q to disappear, leaving the whole condenser at some potential v due to the 'free' charge. Taking the two charges +Q and Q separately, we have for the former

which is the same as result (3), if the one coating of the condenser be at zero potential.

The reader should notice that in the case of the Leyden jar charged to potential V of inner coating, the outer coating being to earth, we have for the discharge of the jar the expression & Q V.

$29. Examples in Energy of Discharge.

(i.) 'A conductor is charged with 8 units to potential 7. What is the energy of discharge?'