454 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XL 



In order to find the influence of the ends of a uniform straight 

 coil of any cross-section, we may consider that each current turn 

 is. replaced by a plane shell, so that the whole current sheet is 

 replaced by a uniformly magnetized cylindrical magnet with 

 intensity of magnetization d<f)/dz = ml. The free surface charges 

 of all the shells accordingly cancel each other except for the two 

 plane ends of the magnet. These ends are single distributions 

 identical with each other except for the difference of sign. If V l 

 is the potential at any point due to a uniform single distribution 

 of unit density on the positive end 1, and F 2 that due to an 

 identical distribution on the negative end 2, then at any point 

 outside the region bounded by the cylindrical current sheet and 

 its plane ends, the potential due to the sheet is 



(12) Q, = mI(V l -V*). 



We may find the potential at a point inside the space in question 

 by the result that for an infinite cylindrical sheet the force is 

 4f7rml, so that if z is measured parallel to the generators of the 

 cylinder in the direction of the force, 



(13) . ft = 4<7rmlz (for the infinite cylinder). 



If ft' is the potential due to all of the infinite coil except the 

 portion which we are considering, we have accordingly 



(14) fl + H' = - 4nrmlz. 



But the space in question is outside the two magnets replacing 

 the two infinite parts of the sheet, so that for a point between 

 the ends, 



gvng 



(15) n = 



Now as we pass one of the ends of the coil the potential F is 

 continuous, being the potential of a single distribution, but its 

 derivative has a discontinuity of amount 4-Tr by 82, accordingly 

 the potential fl is discontinuous, but the force is continuous, the 

 discontinuity in changing from the formula (12) to (15) just can- 

 celling the discontinuity in dV/dz. In the case of a circular 

 cylindrical coil, the potentials Fi and F 2 may be found by the 

 development in spherical harmonics given in 102, and the devia- 



