CURRENTS IN CONDUCTING SHELLS OP SMALL THICKNW8. 309 



currents as produced from instant to instant in a thin shell, whatever its thickness 

 may be. 



T/ie Potential Induced on a Conductor. 



18. Let now dF^dt, dG^dt, dH^dt denote the time variation of the components 

 of vector potential for the external system. Then dF /dt, dGg/dt, dllo/dt, 

 have an associated function, defined as in (11), which we will call i/<,. 



Similarly, if dF/dt, dG/dt, dB./dt relate to the induced currents, they have 

 an associated function \j/. 



The functions dF^dt dG^dt dH^dt are the components of an electromotive 

 force, and, therefore, by (12), produce on the conductor a distribution having 

 potential / . 



Similarly the functions dF/dt dG/dt dH/dt produce on the conductor a 

 distribution having potential 1(1. Initially on the formation of the induced currents 

 they satisfy the conditions 



. it -f --<*+*)=*** 



19. Hence we arrive at the conclusion that any variation of the magnetic field 

 outside of a conductor causes on the surface of the conductor 



(1.) A system of closed currents whose magnetic potential at the instant of their 

 creation is equal and opposite to the time variation of the magnetic potential of the 

 given system at all points on or within the conductor ; that is, a complete magnetic 

 screen. 



(2.) It creates and maintains a difference of potential at different points on the 

 conductor, and this may be used to produce an electric current in a system connected 

 with the conductor. 



20. The electrostatic distribution has energy, but such energy exists side by side 

 with the electrokinetic or magnetic energy of the closed currents, without (so to 

 speak) mixing. That is, there is no term involving products of U, V, W with 

 difi/dx, &c. 



For, if we have any system whatever of closed currents within any closed surface S, 

 and in the field of a potential i|r, 



fff { M S+ V ^y + w *dz] tedydz = f | $(lu + mv + nw)dS 



because we may take S so distant that lu + my -j- nw shall be zero everywhere 

 upon it. 



