68 Conductors and Condensers [OH. in 



so that if a is the radius of the cylinder, and if it has a charge e per unit 

 length, we have 



cr = 



To find the intensity at any point outside the conductor, construct a Gauss' 

 surface by first drawing a cylinder of radius r, coaxal with the original 

 cylinder, and then cutting off a unit length by two parallel planes at 

 unit distance apart, perpendicular to the axis. From sym- 

 metry the force at every point is perpendicular to the axis 

 of the cylinder, so that the normal intensity vanishes at 

 every point of the plane ends of this Gauss' surface. The 

 surface integral of normal intensity will therefore consist 

 entirely of the contributions from the curved part of the 

 surface, and this curved part consists of a circular band, of 

 unit width and radius r hence of area 2?r?\ If R is the 

 outward intensity at every point of this curved surface, 

 Gauss' Theorem supplies the relation 



2-TrrR = 4-Tre, 



so that R = -. FIG. 28. 



r 



This, we notice, is independent of a, so that the intensity is the same as 

 it would be if a were very small, i.e., as if we had a fine wire electrified with 

 a charge e per unit length. 



In the foregoing, we must suppose r to be so small, that at a distance r 

 from the cylinder the influence of the ends is still negligible in comparison 

 with that of the nearer parts of the cylinder, so that the investigation does 

 not hold for large values of r. It follows that we cannot find the potential 

 by integrating the intensity from infinity, as has been done in the cases of 

 the point charge and of the sphere. We have, however, the general 

 differential equation 



so that in the present case, so long as r remains sufficiently small 



dV = _2e 

 dr r ' 



giving upon integration 



V=C-2e\ogr. 



The constant of integration C cannot be determined without a knowledge 

 of the conditions at the ends of the cylinder. Thus for a long cylinder, the 

 intensity at points near the cylinder is independent of the conditions at the 

 ends, but the potential and capacity depend on these conditions, and are 

 therefore not investigated here. 



