ELECTRIC FIELD. 155 



The small sphere has a charge of + 2 coulombs, and the inside 

 surface of the outer sphere has a charge of g coulombs; and 

 it is required to find the intensity of the electric field at any point 

 p distant r from the common center of the two spheres. 

 It is evident that the lines of force of the electric field are radial 

 straight lines because of the entire symmetry of the arrangement. 

 Therefore the electric field at p is in the direction of the arrow e. 

 Imagine a spherical surface of radius r with its center at 0. 

 The area of this spherical surface is ^irr* and the electric field 

 is everywhere at right angles to its surface and equal to e. 

 Therefore the flux across the spherical surface is 4xr 2 e, and 

 according to Gauss's theorem this flux must be equal to Bq 

 (see Art. 86). Therefore we have: 



-5-5 <" 



The expression for e in equation (i) does not depend upon the 

 radius of either sphere in Fig. 100. Therefore equation (i) 

 gives the electric field intensity e in volts per centimeter at any 

 point at a distance of r centimeters from the center of a sphere 

 of any size over which q coulombs of charge is uniformly dis- 

 tributed, provided the given sphere is surrounded by a hollow 

 concentric metal sphere of any radius greater than r. Of course 

 r must be greater than the radius of the inner sphere. 



93. Electric field due to a concentrated charge. Imagine a 

 charge of q coulombs on a sphere of indefinitely small radius. 

 Such an imagined charge is called a concentrated charge. It is 

 evident from what is said above that equation (i) of Art. 92 

 gives the intensity e of the electric field at a distance of r centi- 

 meters from a concentrated charge of q coulombs. 



Attraction or repulsion of concentrated charges. Consider 

 two concentrated charges g f and q" at a distance of d centi- 

 meters apart. The electric field intensity at q" due to tf is 



e' = , and the force F exerted on q" by this field is equal 



Air a 



