CHAPTER V

CAPACITY-ELECTRICAL ENERGY

450. Capacity of a Conductor.-There is a constant relation between the charge of a conductor and its potential, for if the density of the charge at every point of a conductor is doubled the total charge will also be doubled, and the force exerted on a unit charge, placed anywhere in the neighbourhood of the charged conductor, will also be doubled, so that the work done in removing the unit charge from the neighbourhood of the conductor to a place of zero potential will be doubled, that is, the potential of the conductor will be doubled. This constant ratio of the charge of a conductor to its potential is called the capacity of the conductor. Thus if a charge Q raises the potential of a conductor to V, the capacity, C, is given by the relation C=QV. If the conductor is charged to unit potential, then VI and the capacity is numerically equal to the charge necessary to charge the conductor to unit potential. Hence we may also define the capacity of a conductor as the charge which must be communicated to it to raise its potential by one unit.

451. Condensers.-We have seen in § 447 that if an uninsulated conductor is brought near a charged body, the potential of this latter is diminished on account of the induced charge on the uninsulated conductor. Hence the potential of the insulated conductor produced by a given charge is less when the uninsulated conductor is near than it is when this conductor is absent; in other words, the effect of bringing the uninsulated conductor near the charged one is to increase the capacity of this latter. We may consider the same problem in a somewhat more direct way, if we suppose that a given conductor, say a plane AB (Fig. 440), is insulated and then charged to a potential when at a distance from all other conductors.

Let a second plane, which is connected with earth, be placed at such a distance from AB that its presence does not appreciably affect the electrical condition of AB. Then the work that is done in carrying a unit of positive

CA

P.

FIG. 440.

electricity from a point P near AB to a point P', which is at zero potential, is equal to V. Next suppose that the uninsulated plane is moved near to AB, into the position CD), so that an appreciable charge is induced on it.

The work that must now be done to move the unit charge from P to P' by the same path as before will be less than before, for, on account of the attraction exerted on the unit charge by the negative charge induced on CD), the force exerted on the unit is everywhere less than it was before. Hence the potential of AB is less than it was before. As the plane CD is moved nearer to AB the amount of the induced negative charge increases, and the influence of this negative induced charge in diminishing the repulsive force exerted on the unit charge becomes greater and greater, and hence the potential of AB becomes less and less. The charge on AB remains however the same, and therefore, since the potential to which this charge is capable of raising AB diminishes as the uncharged and uninsulated conductor CD is brought near, it follows that the capacity of AB must increase as the conductor CB is brought near. If, instead of keeping the charge on AB constant, we had kept the potential constant, then we should have had to increase the charge on AB as the conductor CD was brought up.

An arrangement of two conductors, one of which is insulated and the other is uninsulated, placed near one another with an insulator between, is called a condenser. The name condenser was given to such an arrangement on account of the fact that the presence of the second uninsulated conductor appears to exert a condensing action on the electrical charge on the insulated conductor, so that for a given potential it can receive a much greater charge than it could without the presence of the uninsulated conductor.

The capacity of a condenser is the charge which must be communicated to the insulated conductor to raise its potential through one unit of potential. The two conductors of a condenser are sometimes called the armatures of the condenser.

The commonest form of condenser is that shown in Fig. 441, and is called a Leyden jar. It consists of a glass jar, the interior of which is coated with tinfoil up to within an inch or so of the top, and a metal knob which is in conducting communication with this inside coating. This tinfoil forms the insulated armature of the condenser, the uninsulated armature being formed by a coating of tinfoil on the outside of the jar.

-B

Another form of condenser which is commonly used consists of a plate of glass or some other insulating material, which is coated on each side with a sheet of tinfoil or some other conductor, a margin of an inch or so being allowed all round the edge of the glass. One coating is connected with earth, and the other forms the insulated armature of the condenser. This arrangement is sometimes called a fulminating pane.

FIG. 441.

If the insulated armature of a condenser is charged to a potential of

V, the other armature being at a potential zero, and this armature is then insulated, while the armature which was at first insulated is put to earth, this armature will not lose much of its charge, as now the rolls of the two armatures are reversed, for what was originally the induced charge is now the inducing charge, while the former inducing charge is now the induced charge. When a condenser is charged, most of the lines of force stretch across from one armature to the other, few stretching from the insulated armature to surrounding objects. Now, in order to discharge a charged conductor, the bodies on which the other ends of the tubes of force which leave the conductor terminate must be put in conducting communication with the conductor. For we may imagine that when a charged body is put to earth by means of a conducting communication, such as a wire, that the two ends of each tube of force travel along the conducting wire towards one another, the tube of force shortening up in virtue of the tension which exists along every such tube, until the two ends come together and the tube of force shrinks to nothing. In the case of the condenser, if the two armatures are put in conducting communication all the tubes of force are able to shrink to nothing, that is, the condenser becomes completely discharged. If, however, after charging the uninsulated armature is insulated, and the other armature is put in conducting communication with earth, only those tubes of force which stretch from this armature to the surrounding uninsulated conductors, such as the walls of the room, will be able to shrink and vanish. The great majority of the tubes which stretch from one armature to the other will not be able to shrink, for the armatures are not in conducting communication.

A C

452. Specific Inductive Capacity.-If a condenser is formed by two conducting plates AB and CD (Fig. 442), placed parallel to one another, the intervening insulator being air, the capacity will have a definite value, say C. If now, while the two armatures are kept at the same distance apart, the air between the plates is replaced by some other insulator, say paraffin, the capacity of the condenser will be altered, in the case taken the capacity will be increased. We thus see that the capacity of a condenser depends not only on the geometrical conditions of the armatures, such as their size, shape, and distance apart, but also on the nature of the medium which fills the space between the plates. This fact is expressed by saying that dielectrics, as the media between the armatures are called, have different specific inductive capacities.

B D

FIG. 442.

The specific inductive capacity of the air is taken as unity, and that of any other dielectric is measured by the ratio of the capacity of a condenser, of which the given substance is the dielectric, to the capacity of the same condenser when the given medium is replaced by air.

Thus if the capacity of a given condenser with air as the dielectric is C,

its capacity when the air is replaced by a dielectric of which the specific inductive capacity is K will be CK.

In order to compare the specific inductive capacities of different dielectrics, Faraday used a condenser of the form shown in Fig. 443- It

FIG. 443.

consisted of an outer brass sphere, PQ, made up of two hemispheres, which fitted accurately together. This formed the uninsulated armature of the condenser, the other armature being formed by a brass sphere, C, which was held in a position concentric with the outer sphere by means of an insulating rod, A. A metal wire passing down through a allowed the inside sphere to be charged.

Two exactly similar condensers of this form were made, and one of them was charged by means of the rod B, the outside hollow sphere being connected to earth. The magnitude of the charge imparted to the condenser was then determined by touching B with a proof-plane, the charge taken away by the plane being measured with the torsion balance. The knob B was then connected with the similar knob of the other condenser, so that the two shared the charge. The charge of each was then tested by means of the proof-plane as before, and was found to be the same, thus showing that the capacity of the two condensers was the same, as from their equal size and shape ought to be the case.

Next the space, mn, between the inside and outside spheres in one of the condensers was filled with the medium of which the specific inductive capacity was to be determined. The other condenser was then again charged, the amount of the charge being measured as before. The knobs of the two condensers were then connected together, and the potential again measured. In the case of such a dielectric as paraffin, the potential of the two is considerably less than half the potential of the air-condenser before the two are put into communication, hence the paraffin-condenser has taken more than half the charge of the air-condenser; and since when they are connected the potential to which they are charged must be the same, it follows that the capacity of the condenser in which the dielectric is paraffin must be greater than that of the one in which the dielectric is air. In order to calculate the specific inductive capacity, K, of the paraffin, suppose that the potential to which the air-condenser was originally charged was V1, while the potential of the two condensers when joined together is V. If C, and C are the capacities of the condensers

of which the dielectrics are air and paraffin respectively, then the original charge of the air-condenser is C1V1, while its charge after it has been put into communication with the paraffin-condenser is VC. The charge of the paraffin-condenser is equal to VC, but this must also be equal to the charge lost by the air-condenser, that is, to V11- V1C1. Thus the specific inductive capacity of the paraffin, which by definition is equal to the ratio of the capacity of a condenser of which the dielectric is paraffin to the capacity of the same condenser when the dielectric is air, can be found. For V1C2=C1(V1- Va),

As we shall see later, the determination of the specific inductive capacity of different dielectrics is of great interest from its bearing on the electromagnetic theory of light. In the following table the values of the specific inductive capacity of some dielectrics are given. The values obtained depend, in the case of solids, on the physical condition of the solid as well as on the duration of the electrical charge employed in the measure

453. Energy of a Charged Condenser.—Suppose that a condenser of capacity C is charged to a potential V, the uninsulated armature being at the potential zero. Since the potential of the one armature is V, and that of the other is O, the work done in moving a unit charge from one armature to the other will be V. Hence if we suppose that the condenser is discharged by the process of carrying the charge, one unit at a time, from one armature to the other, the work done during the transference of the first unit will be V. On account of the loss of this anfount of the charge the potential will be reduced to V-1/C, for the original charge was VC, and the charge after the abstraction of one unit is VC-1, and this charge will raise the potential of a condenser of

capacity C to a potential VCI. Hence the work done in carrying the

C

second unit from one armature to the other will be V-1/C, and so on.

1 It will really be a little less than V, since the removal of the first unit will reduce the charge on the armature. If, however, is great, so that the removal of a single unit makes little effect, or if we remove less than a unit each time, the error on this account can be made negligable.