489-491] Magnetic Potential of Field 421 



when r < a, and 



i P, (cos 0) - P 3 (cos 0) - ... 



when r > a. 



r s 

 At points so near to the origin that may be neglected, the potential is 



(l- - cos 0V 

 V a /' 



= 29rt l- - cos 



-*rf(l-g, 



where z = rcosO, and the magnetic force is a uniform force -^ =^- 



dz a 



parallel to the axis. 



Solenoids. 



490. A cylinder, wound uniformly with wire through which a current 

 can be sent, is called a ' solenoid.' 



Consider first a circular cylinder of radius a and 

 height h, having a wire coiled round it at the uniform 

 rate of n turns per unit length, the wire carrying a 

 current i. Let z be a coordinate measuring the 

 distance of any cross-section from the base of the 

 solenoid. Then the small layer between z and z + dz, 

 being of thickness dz, will contain ndz turns of wire. 

 The currents flowing in all these turns may be re- FlG> 127 ' 



garded as a single current mdz flowing in a circle, this circle being of radius 

 a and at distance z from the base of the solenoid. The magnetic potential 

 of this current may be written down from the formula of the last section, and 

 the potential of the whole solenoid follows by integration. 



491. Endless Solenoid. In the limiting case in which the solenoid is of 

 infinite length (or in which the ends are so far away that the solenoid may 

 be treated as though it were of infinite length), the field can be determined 

 in a simpler manner. 



Consider first the field outside the solenoid. In taking a unit pole round 

 any path outside the solenoid which completely surrounds the solenoid, the 

 work done is, by 485, 4nri. The current flowing per unit length of the 

 solenoid is ni. In general we are concerned with cases in which this is finite 

 n being very large and i being very small. The quantity 4>7ri may accordingly 

 be neglected, and we can suppose that the work done in taking unit pole 

 round the solenoid is zero. 



