This expression represents the weight of water times the average height it has been lifted. Similarly, when a condenser is charged, the potential difference between the plates is gradually raised from zero to the final potential, V, and the work done in charging it is

where Q is the final charge. Since the energy of a charged condenser must be equal to the work done in charging it, this expression gives the energy possessed by the condenser.

If the charge in equation (102) is measured in coulombs, and the potential difference in volts, the energy will be in joules. If both Q and V are measured in C.G.S. units, the energy will be in ergs.

Since any charged conductor may be regarded as a condenser, the other "coating" being the surrounding objects (walls, ceiling, table, etc.), equation (102), for the energy of a charged condenser, applies to any charged conductor.

517. Summary. It is customary to take the potential of the earth as zero and to call all potentials higher than this positive, and all lower potentials negative. When the potential of the earth is taken as zero, the expression "the potential of a conductor" really means the difference of potential between that conductor and the ground.

Rule 1. Bringing a positive charge near an insulated conductor raises the potential of that conductor.

Rule 2. Bringing a negative charge near an insulated conductor lowers the potential of that conductor.

Rule 3. Bringing a grounded conductor near an insulated charged conductor reduces the difference of potential between the charged conductor and the ground.

When two oppositely charged conductors are moved closer together, their difference of potential will be decreased if they are insulated. However, if they are connected to some source which maintains a constant difference of potential, the quantity, or charge, on the conductors will increase as they are moved nearer each other. When the conductors are close together, they will have, for the same difference of potential, relatively large charges. Such a pair of conductors is called a condenser. A condenser may also consist of two groups of plates placed near together. A Leyden jar is an example of one type of condenser. As these jars may be charged to high potentials, relatively large charges may be stored in them.

When a condenser is connected in an alternating-current circuit, there will be a current due to the charging, discharging, and charging in an opposite direction, which follow one another at a rapid rate. When a condenser is connected in a direct-current circuit, there is no continuous current in the connecting wires.

The equation Q = CV (equation (88)) gives the quantitative relation between the charge on the condenser and the difference of potential, V. C, called the capacitance or capacity, is a constant for any one condenser. The capacitance of a condenser is equal to the ratio of the charge on the condenser to the difference of potential; that is,

In other words, the capacitance of a condenser is equal to the charge when the voltage is equal to unity.

The capacitance of a condenser depends on the distance between the conductors: the less this distance, the greater the capacitance. It depends also on the size of the conductors. A very important discovery, first made by Cavendish, and independently by Faraday, showed that the capacitance of two conductors depends on the medium between them. From this it can be deduced that the attraction and repulsion of charges are different for different media.

The insulating medium surrounding any conductor is called the dielectric. The dielectric constant of any material is equal to the ratio of the capacitance of a condenser when that material is used as the dielectric, to the capacitance when there is a vacuum between the conductors.

When two charges, q1 and q2, are on two conductors the sizes of which are small compared with their distance apart, the law of force becomes equation (90),

F=9192,

kd2

where d is the distance apart, and k is the dielectric constant of the medium in which the charges are immersed.

When Q is measured in coulombs and V in volts, the capacitance, C, in equation (88) is measured in farads. The farad is such a large unit that the microfarad is commonly used.

The C. G.S. electrostatic unit of capacitance is relatively small, for 1 microfarad = 9 × 105 electrostatic units.

The capacitance in electrostatic units of a sphere which is surrounded by air, but isolated from other conductors, is equal to the radius of the sphere expressed in centimeters. The capacitance of two parallel plates d centimeters apart (provided the plates are large compared with d) is (equation (91)) Ak/4 πd electrostatic units, where A is the area of one side of one of the plates, and k is the dielectric constant of the medium between the plates.

フ

When the capacitances C1, C2, C3, C4, etc. are connected in parallel, the resultant capacitance is (equation (94))

When the capacitances C1, C2, C3, C4, etc. are connected in series, the

resultant capacitance is obtained from

1 1 1 1 1

+

CCCCC4

- + ....

The energy of a charged condenser is given by equation (102),

W = + QV.

If Q is measured in coulombs and V in volts, the energy, as computed from this equation, will be expressed in joules. If both Q and V are in C. G. S. units, the energy is given in ergs. This expression for the energy may be applied to any charged conductor if V is regarded as the difference of potential between it and the ground.

1. Find the capacitance (a) in electrostatic units and (b) in microfarads of a sphere 60 cm. in diameter.

ym

2. The radius of the earth is approximately 6400 km. What is its capacitance in microfarads?

3. A pair of parallel circular plates are 30 cm. in diameter and 0.5 cm. apart, with air between them. Compute the capacitance (a) in electrostatic units; (b) in microfarads.

4. What would be the capacitance, in microfarads, of the pair of plates of Problem 3 if they were submerged in oil the dielectric constant of which is 2.0?

5. The capacitance of a small plate condenser was found by experiment to be 0.00015 microfarad in air, and 0.00033 microfarad when submerged in engine oil. What is the dielectric constant of the oil ?

6. Each plate of the condenser of Fig. 324 was 25 × 30 cm., and they were separated by 3 mm. of glass the dielectric constant of which was 6.0. Find the total capacitance in microfarads.

7. Three condensers, of capacitance 0.5, 1.0, and 2.0 microfarads, respectively, are connected in series. (a) Find the resulting capacitance. (b) What will be the charge when a voltage of 500 volts is applied to the combination?

8. Three condensers, of capacitance 0.5, 1.5, and 2.0 microfarads, re

spectively, are connected in parallel to a 220-volt circuit. (a) Compute the resulting capacitance. (b) Compute the charge in coulombs on each condenser.

9. The capacitance of each of ten Leyden jars is 0.0016 microfarad. They are connected in parallel and charged to a potential difference of 30,000 volts. Compute the total energy in joules.

10. Two charged balls attract each other with a force of 50 dynes. If they were submerged in oil without discharging them, what would be the attractive force? (The dielectric constant of the oil is 3.1.)

11. A condenser of 2.0 microfarads was charged to a difference of potential of 200 volts and then insulated. An uncharged condenser of 1.0 microfarad was then connected in parallel with the first. What is the voltage at their terminals?

CHAPTER XXXVI

ELECTRICAL UNITS

Introduction, 518. The C. G. S. electrostatic system, 519. The C.G.S. electromagnetic system, 520. The numerical relation between electromagnetic and electrostatic units, 521. The practical system, 522. International units, 523.

518. Introduction. Even a very elementary knowledge of electricity is incomplete without a knowledge of the principal electrical units and the logical processes on which they are built.

The units used in electricity beautifully illustrate the advantages of a logical system. In the C.G.S. system there is an entire absence of arbitrary units: they are all based on the units of mechanics-the centimeter, the second, the dyne, and the erg. It is a derived system of units. To appreciate such a system one has only to compare it with the British system of weights and measures, with its large number of arbitrary units. The practical advantage in a derived system lies in the ease with which computations can be made. For example, watts are obtained by multiplying amperes by volts, no numerical constant appearing in the equation. The same is true of many of the other electrical equations. Again, in changing from one unit to another in measuring the same electrical quantity, all that is usually necessary is to shift the decimal point. Contrast this with the calculations necessary in changing from one to the other of two units of volume-from cubic inches to bushels, for example, or from cubic inches to cubic yards.

There are two fundamental systems of units in electrical work: the C.G.S. electrostatic and the C.G.S. electromagnetic. There is also the practical system, which is based on the electromagnetic. There is a very good reason for the use of all these systems. Certain types of problems can often be solved more easily by using one system than by using another. For example, it is often simpler in dealing with certain problems involving stationary charges, such