234] . ELECTROMAGNETISM. 459 



so that its integral is 



(5) H=C-2 n^wlogrdadb, 



where r 2 = (x - a) 2 + (y - &) 2 , 



and C is a constant, which, though infinite, does not affect the 



value of the force. 



If the conductors are concentric circular cylindrical tubes and 

 the current-density is uniform, we may find the magnetic force 

 without finding the vector-potential, in the same way as in 231, 

 for it is evident that the lines of force are all circles in planes 

 perpendicular to the conductors. At points outside the outer 

 tube, at a distance p from the axis, the line integral of magnetic 

 force (which we will denote by P instead of H, to prevent con- 

 fusion with the vector-potential) around a circle is 



4-7T/, 



*-" 



where 

 (7) 



is the total current through all the conductors. Accordingly at 

 external points the field is the same as if the current were con- 

 centrated in the axis of the conductor. If different tubes are 

 made part of the same circuit, so that all the current flowing 

 in one direction is returned in the other direction by concentric 

 conductors, the total current is equal to zero, and the force is zero at 

 all external points. Such a double tubular conductor accordingly, 

 like a toroidal coil, emits no tubes of magnetic induction. For 

 this reason, when it is wished to protect delicate magnetic instru- 

 ments from the action of strong currents, the circuit should be 

 formed of concentric conductors. The mutual inductance of any 

 external circuit with such a concentric conductor is accordingly 

 zero, so that, as we shall see in the next chapter, no currents 

 would be induced in the concentric conductors by external cur- 

 rents. Such a conductor would thus be suitable for telephone 

 circuits. 



In the space outside the conductors the magnetic potential is 

 (8) n 



