of a long narrow cylinder 383 



Let a be the surface-density on the flat ends. 



Let ^r be the potential at a point on the axis for which x = . 



* dx + /*27r<ry [(/ - )* + v^dy 

 Jo 



...... (i) 



the first integral representing the part of the potential due to the curved surface, 

 and the other two the parts due to the positive and the negative flat ends 

 respectively. 



The condition of equilibrium of the electricity is that t/t must be constant 

 for all points within the cylinder, and therefore for all points on the axis 

 between the two ends. 



If, by giving proper values to A and a, we can make the value of <p constant 

 for any finite length along the axis, then, by Art. 144 of "Electricity and 

 Magnetism," 1(1 will be constant for all points within the surface of the cylinder. 



It was shown in Note 2 that the distribution of electricity in equilibrium 

 on a straight line without breadth is a uniform one. We may expect, therefore, 

 that the distribution on a cylinder will approximate to uniformity as the radius 

 of the cylinder diminishes. 



If we suppose A and a to be each of them constant, 







where /j and / 2 are the distances of the point () on the axis from the edges of 

 the curved surface at the + and ends of the cylinder respectively. 



Just within the positive flat end of the cylinder, where $ is just less than /, 





/ 



If the electricity were in equilibrium, this would be zero, and if the cylinder 

 is so long that we may neglect the reciprocal of / 2 , we find 



A = 2-rrbcr, ...... (4) 



or the surface-density on the end must be equal to the surface-density on the 

 curved surface. 



The whole charge is therefore 



E = X(2l+b). ...... (5) 



The greatest value of the potential is at the middle of the axis, where = o. 

 Calling it ^r (0| and putting/ = I, 



(6) 



The potential at the end of the axis is 



^ (l , = A (log | + i . ...... (7) 



