Lines of Force of a Circular Current. 21 



then 'b<i>'b'^ '64* ^^— p 



and ylr = const. 



must represent lines o£ force. 



The potential of a circular current is equivalent to that of 

 a magnetic shell, which is derivable from the Newtonian 

 potential of a uniform disk, by differentiation with respect 

 to a normal, 



3. Potential of a uniform circular disk. — Let the surface- 

 density be unity ; taking x, y axes in the plane of the disk, 

 and z axis perpendicular to it through the centre, the poten- 

 tial at point ct', 0, z is evidently given by 



pdpdQ , 



J a r»2ir 



'2a;p cos 6 + p^ + z^ 



p and 6 being polar coordinates, and a the radius of the 

 disk. 



Writing B!^ = £c^-''2j:p cosO + p"^, and making use of 

 Lipschitz^s integral concerning the Bessel's function, we 

 obtain 



^ = J^-^^^Jo(XR)^X.(2) 



The addition theorem of Bessel's function gives 



Jo(>^R) = Jo(>^^) Jo(V) + 2ij,^(X.r) j;(Xp) COS n(9. . (3) 



1 



Substituting the two expressions (2) and (3) in (1), and 

 integrating between the limits and 27r, 



U = 27r I 



Jo Jo 



'pjQ(Xx)Jo(\p)d\dp. 



But since t /^ \ d(pJi{Xp)) 



P'^'^^P^= xdp "' 



U = 27rar 



a o — \z 



■Jo(\.r)Ji(\a)c?\. ... (4) 

 '0 '^ 



This expression for the Newtonial potential of a uniform 

 circular disk seems to have been first obtained by H.Weber*. 



4. Potential and lines of force of a circular magnetic shell. 

 — The potential of a circular magnetic shell of unit strength 

 is evidently given by differentiating (4) with respect to z, 



* H. Weber, Orelle's Journal Ixxv. p. 88. 



