131, 132] ZERO POTENTIAL. 259 



132. Zero Potential. If we have a single charged conduc- 

 tor in the form of a sphere, uninfluenced by other bodies, since the 

 surface density will be constant, the potential at the center will 



p 



be - where r is the radius of the sphere, and since the potential 



S) 



throughout the conductor is constant, its value will be - . As r 



increases the potential decreases in absolute value. Now the 

 earth may be considered as a conductor of a radius which is in- 

 finite in comparison with the dimensions of our apparatus. Its 

 potential may be therefore regarded as zero, and any conductor may 

 be kept at zero potential by being connected with the earth. 



If within a conductor there is a hollow space, not containing 

 any electrified body, the function V is harmonic in the cavity, 

 and being equal to a constant at the inner surface of the conduc- 

 tor, must by Kelvin and Dirichlet's principle be constant through- 

 out the cavity, consequently there is no force at any point in the 

 cavity. Or a closed conductor screens the interior from the effect 

 of an external field of force. This explains Faraday's Experi- 

 ment VII. 



If the system is composed of a single hollow conductor in 

 communication with the earth, containing within several rigidly 

 electrified bodies D, then the total potential V being zero at the 

 outer surface of the conductor and at infinity is by Dirichlet's 

 principle zero everywhere in the external space, and there is no 

 force there. That is, a closed conductor connected to earth shields 

 external space from the action of an electric field of force within. 

 This principle of electric screens is of great importance in practice 

 in connection with electrostatic instruments. 



The surface density at the outer surface of the conductor 



_1 ^T 



4-7T dn e 



then vanishes, and the outside is without charge. On the inside 

 surface, the charge is 



the normal being, as usual, drawn aiuay from the conductor. But 



172 



