ELECT ROSTATICAL THEORY.] ELECTRICITY 29 Potential energy of a system of con ductors. or, in words, the coefficient of induction of on 2 is equal to that of 2 on 1. There is one more general theorem on electrical distribu tion which, from its great practical importance, deserves a place here. Suppose we take a hollow conductor of any form, place any electrical system inside it, and connect the conductor with the earth, then equilibrium will be estab lished, in such a way that the potential of every portion of the conductor is zero. Now, the potential being zero at all infinitely distant points, we may regard the outside space as inclosed by a surface of zero potential ; hence the potential at every point in this space must be the same, and there can be no electrical action anywhere outside. Again, removing the internal system, let us place any system outside the conductor, and, besides, charge it to any desired extent, keeping it insulated this time. Then the outer and inner surfaces of the conductor will be level surfaces ; and, since there is no electricity inside the inner surface, the potential in the interior will be constant. Hence the external. system, in a state of equilibrium, exerts no action whatever within. Now we may evidently, with out mutual disturbance, superpose such an internal and external system as we have described, and still get a system in equilibrium. It is, moreover, clear that we can in this way satisfy the most general conditions that can be assigned. Hence, since we know that there can be only one solution of the problem of electrical equilibrium, the synthetical one thus obtained represents the actual state of affairs. When, therefore, a hollow conductor with any external and internal systems is in equilibrium, the equili brium of the internal is independent of that of the external system. Moreover, if we draw any surface in the substance of the hollow conductor, no lines of force cross it in one direction or the other; therefore the whole amount of electricity within must be zero ; in other words, the charge on the in ternal surface of the conductor is equal and opposite to the algebraical sum of the charges on all the bodies ivithin. . These propositions contain the principle of what are called electrical screens, i.e. sheets of metal used to defend electrical instruments, fcc., from external influences. On the practical efficiency of gratings in this way, see Maxwell ( 203) ; on the application to the theory of lightning conductors, see a paper by him in the reports of the British Association for 1876. If we take the simple case where there is no external system, but only a charge on the hollow conductor, we get a complete explanation of Faraday s ice-pail experiment. The potential energy of a system of charged conductors is the work required to bring them from a neutral state to the charges and potentials which they have at any time. The state of zero potential energy here contemplated is of course that in which there is an equal amount of + and - electricity everywhere in the system, or, as we might put it, the state in which there is no electrical separation. Now if Q denote the potential energy of the system, we have with the notation of (21) Q-2^ (25), the summation including every pair of elements in the system. If the system be in equilibrium, then, reasoning as above, it is obvious that 2EV is just twice 2^, inas much as each pair of elements will come in twice. Hence we get Q = i2EV (26). This is an expression of the greatest importance. W r e can give it various forms ; by means of (18) and (20) we get = 422 j r .V,V.= } 2 2 i r=l =! r=l =1 (27). So that Q is a homogeneous quadratic function of the potentials or of the charges. If, therefore, we increase the potentials of all the conductors, or the charges of all the conductors in any ratio, we increase thereby the potential energy in the duplicate of that ratio. We can by a transformation, which is a particular case of a theorem of Green s, obtain a very remarkable volume integral for the potential energy of an electrical system. Let V denote the potential at any point in the field. Consider Green s the integral theorem, l /"/~/"/jTf 2 ju |S jw s - ///( +~ + tedydz,
- JJJ dx ^ dy dz)
where the integration is to be extended throughout the whole of the space unoccupied by conductors. We have by partial integration and two similar equations. Hence dx dV dz where the surface integration extends" over the surface of all the conductors, and it is to be noticed that dv is drawn from the con ductor into the insulating medium. If p and & be volume and surface densities, . -T-, andp= - v 2 V. 4?r dv iir Thus we get ^tfff <
- //+*///*
dxdydz (28). This result includes a more general case than our present one ; for it shows that the potential energy of an electrical system is given by the integral on the left hand side in all cases, whether there is equilibrium or not. It is not even restricted to the case of perfect conductors and perfect non conductors, for a slight modification of our preliminary statements would include that case as well. At present, however, we have p = everywhere, and V constant at the surface and in the substance of each conductor, so that the right hand side is simply the expression |2EV which we have already found for the potential energy ; we may there fore write iy Iffj *" dz (29), E, being the resultant force at any point of the field, and dv the element of volume. It is clear that we may if we like extend the integration over the whole field, since in the substance of any conductor R = 0. When we know the potential energy of an electrical sys- tern it is very easy to find the force which resists or tends to produce any change of configuration. Two particular cases are of common occurrence and of considerable interest. First, let the charges on all the conductors be kept con- stant. Let the variable which is altered by the supposed change of configuration be $, and let 4> be the correspond ing force 1 tending to increase <. Then, since no energy is supplied from without, if we suppose the displacement made infinitely slowly, so that no kinetic energy is gene rated, we have Force tending to pro- ratiou. 1 Or generalized force component, i.e., the amount of work per
unit of <p done in increasing <f>.