CHAPTER XXXV POTENTIALS OF CHARGED CONDUCTORS; CAPACITANCES OF CONDENSERS Summary of facts previously explained, 504. Potentials of charged isolated conductors, 505. The effect of another charged body on the potential of a conductor, 506. A grounded conductor brought near a charged conductor, 507. The condenser, 508. A condenser connected to an alternatingcurrent circuit, 509. Quantitative relation between the charge and the voltage of a condenser, 510. Capacity, or capacitance, 511. Capacitance dependent on physical dimensions, 512. Capacitance dependent on the medium between the plates, 513. Units and formulas for capacitance, 514. Condensers in parallel and in series, 515. Energy of a charged condenser, 516. Summary, 517. 504. Summary of facts previously explained. It has been shown that in order to have a current flow from one conductor to another, the two must be at different potentials. But there is another important effect of a difference of potential. When two conductors are at different potentials, they will have static charges of different kinds of electricity on their surfaces. Even if the two conductors are connected by a wire, these static charges will exist if there is a difference of potential. There are many cases where these static charges cause important phenomena. Before proceeding to discuss these cases, however, it is important for the student to be certain that he knows the following facts, which have been previously explained: 1. A positive charge tends to move from a region of higher potential to one of lower potential. 2. A negative charge tends to move from a region of lower potential to one of higher potential... 3. If there is a difference of potential between any two points which are connected by a metallic conductor there will be a current flowing from one to the other; if there is no current, there is no difference of potential. 1 4. We are usually concerned only with the difference of potential between two different conductors and not with their absolute potentials. This difference is commonly measured in volts. 5. The difference in potential between two points is equal to the work done by some external agency in carrying a unit positive charge from the point of lower potential to that of higher potential. 505. Potentials of charged isolated conductors. It is customary to consider the potential of the earth as zero, and to call all higher potentials positive, and all lower potentials negative. For example, if an isolated positively charged conductor should be connected to the earth, positive electricity would tend to flow to the ground, or a negative charge would tend to flow from the earth to the conductor; hence, according to paragraphs 1 and 2 of the last section, such a conductor has a potential higher than that of the earth. Being at a higher potential than the earth, it is said to. have a positive potential. On the other hand, in the case of an isolated negatively charged conductor, negative electricity would tend to flow from the conductor to the earth, if it were connected to the earth. Hence, according to paragraph 2 of the last section, it has a potential lower than that of the earth. Its potential is negative. The terms positive and negative are used here in a manner similar to their use in connection with thermometer readings. All potentials above zero are called positive, while all potentials lower than zero are called negative. The student should see that the term potential, as used here, really refers to a difference of potential-the difference between the potential of the conductor and that of the earth. 506. The effect of another charged body on the potential of a conductor. Let B be a conductor with no charge on it; that is, at the same potential as the earth, or zero potential. Now bring near B a positively charged conductor, A. A will act by induction on B, producing the distribution of charges shown in Fig. 320 and explained before (sect. 378). What change has taken place in the potential of B? The student can determine this if he can answer the following question: If B is connected to the earth, in what direction will the current flow? If he knows the answer, then, by applying paragraph 1 or 2 of section 504, he can decide whether A + + B + + + + B has a higher or lower potential than the earth. If the student does this correctly, he will find that the potential of B is higher than the earth; in other words, that it has a positive potential. Moreover, he should be able to decide that all points on B are at a positive potential; for, according to paragraph 3, sect. 504, they all have the same potential. The student will thus arrive at a decision which is really a general rule: + FIG. 320 Rule 1. Bringing a positive charge near an insulated conductor raises the potential of that conductor. Take another example, -one where a negative charge is brought near a conductor. In this case, as a result of the force exerted by the negative charge of A (Fig. 321), the negative charge on B is repelled. If we should connect B to the + ground, a negative charge would flow from B to the ground. Hence we may deduce from paragraph 2, sect. 504, that the potential of B, in the condition shown by Fig. 321, is less than that of the ground: its potential has been lowered. This is an example of the following general rule : A B FIG. 321 Rule 2. Bringing a negative charge near an insulated conductor lowers the potential of that conductor. It is not difficult to prove Rules 1 and 2 experimentally. If an electrostatic voltmeter or a simple leaf electroscope is connected to the conductor B, the changes in the potential of B can readily be tested when A is moved nearer or farther away. An electrostatic voltmeter of the common type does not distinguish between positive and negative potentials; for when it is positively charged, the same kind of deflection is produced as when it is charged by an equal negative charge. Hence in performing this experiment it is best to start B with a small excess of positive electricity; that is, at a positive potential when A is removed. If A is positively charged, the deflection will be increased when A is brought near B (Rule 1); however, if A is negatively charged, the deflection will be decreased (Rule 2). 507. A grounded conductor brought near a charged conductor. Rule 3. Bringing a grounded conductor near an insulated charged conductor reduces the difference of potential between the charged conductor and the ground. This rule can either be proved by experiment or be deduced from the principles and rules already given. An experimental method will first be considered. Let A (Fig. 322) be a charged conductor connected to an electroscope or an electrostatic voltmeter farther away, the divergence of the leaves increases and the potential of A is raised. The experiment may be repeated with A charged negatively. In this way Rule 3 can be verified. A simple theoretical proof of this rule is easily obtained. When B is charged as shown in Fig. 322, moving it nearer to A is seen to be an example of Rule 2. Or, if A is negatively charged, B wiil have a positive charge by induction, and Rule 1 applies. Another theoretical method is based on the definition of difference of potential (paragraph 5 of section 504). The potential of A is equal to the work done in carrying a unit positive charge from the ground to A. Work is done in this case because the charge on A repels the unit charge. When B is near, less work will be done; for the negative charge on B will attract the unit positive charge and partly neutralize the repellent force of the charge on A. If the work done in carrying the unit charge from the ground to A is less when B is near A, the difference of potential between A and the ground must be less. 508. The condenser. Let the two plates of Fig. 323 be connected to some source-a battery of dry cells, or an electrostatic machine, or a dynamo-giving a definite difference of potential between its terminals. As soon as A and B are connected, charges will flow into them until the difference of potential between them becomes the same as that of the source. Now if A and B are moved closer together, this motion (apply Rules 1 and 2) will decrease the difference of potential between the two, and more electricity will flow into the two conductors until they are again at the same difference of potential as the source. Larger charges on A and B are required, to establish the same difference of potential, when the conductors are closer together. + + + + B-HA As an example, assume that the plates are several centimeters apart, and that they have been charged by connecting them to a battery having a large number of cells in series and giving 1500 volts. If the plates are discon nected from the battery and are well insulated, the difference of potential between them will remain at 1500 volts. If the plates are now க FIG. 323 moved closer together, so that the distance be tween them is, say, half a centimeter, the difference of potential will. fall, perhaps to 300 volts or lower. If they are again connected to the charging battery, more charge will flow in until the difference of potential is again 1500 volts. If the battery had not been disconnected, the charge would have flowed in while the plates were in motion. The relatively large charges on two conductors which have been brought close together are well shown by experiments with a Leyden jar (sect. 395). Since it was believed that electricity was some sort of fluid which could be "condensed" into two plates, such a pair of plates was long ago called a condenser. While the original idea was erroneous, the name is still used. A condenser may consist of two metallic plates insulated from each other, as in a Leyden jar, or of a large number of plates. In Fig. 324, A represents a group of five plates, and B another |