204207] Greerts Equivalent Stratum 181 



207. Equation (122) expresses the potential at any point in the space 



outside S in terms of the values of V and ^ over the boundary of this space. 



We have seen, however, that the value of the potential is uniquely determined 



3F 

 by the values either of V or of -~ over the boundary of the space. In actual 



electrostatic problems, the boundaries are generally conductors, and therefore 

 equipotentials. In this case equation (123) expresses the values of the 



dV 

 potential in terms of = only, amounting in fact simply to 



\jtvt 



What is generally required is a knowledge of the value of V P in terms of the 

 values of V over the boundaries, and this the present method is unable to 

 give. For special shapes of boundary, solutions have been obtained by 

 various special methods, and these it is proposed to discuss in the next 

 chapter. 



REFERENCES. 



On Green's Theorem and its applications : 



MAXWELL. Electricity and Magnetism, Chapters iv and v. 

 GREEN. Mathematical Papers of George Green. (Edited by N. M. Ferrers.) 

 London (Macmillan and Co., 1870). 



On Forces on dielectrics and stresses in a dielectric medium : 



HELMHOLTZ. Wiedemanri's Annalen der Physik, Vol. 13 (1881), p. 385. 



EXAMPLES. 



1. If the electricity in the field is confined to a given system of conductors at given 

 potentials, and the inductive capacity of the dielectric is slightly altered according to any 

 law such that at no point is it diminished, and such that the differential coefficients of the 

 increment are also small at all points, prove that the energy of the field is increased. 



2. A slab of dielectric of inductive capacity K and of thickness x is placed inside a 

 parallel plate condenser so as to be parallel to the plates. Shew that the surface of the 

 slab experiences a tension 



3. For a gas K=l+6p, where p is the density and 6 is small. A conductor is 

 immersed in the gas : shew that if 2 is neglected the mechanical force on the conductor 

 is 27T0- 2 per unit area. Give a physical interpretation of this result. 



