Experiment 17: Coulomb’s Proof Plane

Superposition of electric systems.

77.] We have already (Art. 29) given some examples of the superposition of electric effects, but we must now state the principle of superposition more definitely.

If the same system is electrified in three different ways, then if the potential at any point in the third case is the sum of the potentials in the first and second cases, the electrification of any part of the system in the third case will be the sum of the electrifications of the same part in the first and second cases. By reversing the sign of the electrifications and potentials in one of these cases, we may enunciate the principle with respect to the case in which the potential and the electrification are at every point the excess of what they are in the first case over what they are in the second.

78.] We may now establish a theorem which is of the greatest importance in the theory of electricity.

If the electric field under consideration consist of a finite portion of a di- electric medium, and if at every point of the boundary of this region the potential is given, and if the distribution of electrification within the region be also given, then the potential at any point within the region can have one and only one value consistent with these conditions. One value at least of the potential must be possible, because the conditions of the theorem are physically possible. Again, if at any point two values of the potential were possible, then by taking the excess of the first value over the second for every point of the system, a third case might be formed in whichTHOMSON’S THEOREM.

the potential is everywhere the excess of the first case above the second. At the boundary of the region the potential in the third case is everywhere zero. Within the region the electrification is everywhere zero. Hence, by (Art. 75), at every point within the region the potential in the third case is zero. There is, therefore, no difference between the distribution of potential in the first case and in the second, or, in other words, the potential at any point within the region can have only one value. If in any case we can find a distribution of potential which satisfies the given conditions, then by this theorem we are assured that this distribution is the only possible solution of the problem. Hence the importance of this theorem in the theory of electricity. 79.] For instance, let A be an electrified body and let B be one of the equipotential surfaces surrounding the body. Let the po- tential of the surface B be equal to P. Now let a conduct- ing body be constructed and placed so that its external surface coincides with the closed surface B, and let it be so electrified that its potential is P. Then the condi- tions of the region outside B are the same as when it was acted on by the body A only. For the potential over the Fig. 18. whole bounding surface of the region is P, the same as before, and whatever electrified bodies exist outside of B remain unchanged. Hence the potential at every point outside of B may, consistently with the conditions, be the same as before. By our theorem, therefore, the potential at every point outside B must be the same, when, instead of the body A, we have a conducting surface B, raised to the potential P. 80.] The charge of every part of the surface of a conductor is of the same sign as its potential, unless there is another body in the field whose potential is of the same sign but numerically greater. Let us suppose the potential of the body to be positive; then, if on any part of its surface there is negative electricity, lines of force must terminate on this part of the surface, and these lines of force must begin at some electrified surface whose potential is higher than that of the body. Hence, if there is no other body whose potential is higher than that of the given body, no part of the surface of the given body can be charged with negative electricity.INDUCED ELECTRIFICATION. 66 If an uninsulated conductor is placed in the same field with a charged con- ductor, the charge on every part of the surface of the uninsulated conductor is of the opposite sign to the charge of the charged conductor. For since the potential of the uninsulated body is zero, there can be no line of force between it and the walls of the room, or infinite space where the potential is always zero. The line of force which has one end at any point of the surface of this body must therefore have its other end at some point of the charged body, and since the two extremities of a line of force are oppositely electrified, the electrification of the surface of the uninsulated body must be everywhere opposite to the charge of the charged body. The charged body in this experiment is called the Inductor, and the other body the induced body. When the induced body is uninsulated, the electricity spread over every part of its surface is, as we have just proved, of the opposite sign to that of the inductor. The total charge, EA , of the induced body, which we may call A, may be found by multiplying PB , the potential of the inductor B, by QBA , the mutual coefficient of induction between the bodies, which is always a negative quantity. This electrification induced on an uninsulated body is called by some writ- ers on electricity the Induced Electrification of the First Species. Since the potential of A is already zero, it is manifest that if any part of its surface is touched by a fine wire communicating with the ground there will be no discharge. Next, let us suppose that the body, A, instead of being uninsulated is insu- lated, but originally without charge. Under the action of the inductor B part of its surface, on the side next to B, will become electrified oppositely to B; but since the algebraic sum of its electrification is zero, some other part of its surface must be electrified similarly to B. This electrification, of the same name as that of B, is called by writers on electricity the Induced Electrification of the Second Species. If a wire connected with the ground be now made to touch any part of the surface of A, electricity of the same name as that of B will be discharged, its amount being equal and opposite to the negative charge (of the first species) whichCOEFFICIENTS OF CAPACITY. 67 remains on the body A, which is now reduced to potential zero. In order to obtain a clearer idea of the distribution of electricity on the surface of A under various conditions, let us begin by supposing the potential of A to be zero and that of B to be unity. Let the surface-density at a given point P on the surface of A be −σ1 , and let the whole charge of A be −qAB . The negative sign is prefixed to the symbols of these quantities because the quantities themselves are always negative. The charge of B in this case is qB . Let us next suppose the potential of A to be unity and that of B to be zero, and let the surface-density at the point P be now σ2 , and the whole charge on A, qA . These quantities are both essentially positive, and qA is called the capacity of A. The value of both is increased on account of the presence of B in the field. Let us now suppose that the potentials of A and B are PA and PB respec- tively; then the surface-density at the point P is σ = P A σ2 − P B σ1 , and the charge of A is EA = PA qA − PB qAB , and that of B is EB = PB qB − PA qAB . [See Art. 39.] If A is insulated and without charge EA = 0, which gives q PA = PB AB , qA and the surface-density at P is P σ = B (qAB σ2 − qA σ1 ). qA On a region of the surface of A next to B, σ will be of the opposite sign from PB ; and on a region on the other side from B, σ will be of the same sign with PB . The boundary between these two regions forms what is called theCOEFFICIENTS OF CAPACITY. 68 neutral line, the form and position of which depend on the form and position of A and B.