Fm. 119. 



480-482] The Magnetic Field producedty Electric Currents 415 



481. Instead of regarding this field as due to a particle of moment jointly 

 proportional to the area PQRS and to the current-strength, we may regard 

 it as due to a small magnetic shell, coinciding with the area PQRS, and of 

 strength simply proportional to the current flowing in PQRS. 



482. Next, let us consider the current flowing in a closed circuit of any 

 shape we please, and not necessarily in 



one plane. Let us cover in the closed 

 circuit by an area of any kind having the 

 circuit for its boundary, and let us cut 

 up this area into infinitely small meshes 

 by two systems of lines. A current of 

 strength i flowing round the boundary 

 circuit, is exactly equivalent to a current 

 of strength i flowing round each mesh in 

 the same direction as the current in the 

 boundary. For, if we imagine this latter 

 system of currents in existence, any line 

 such as AB in the interior will have two currents flowing through it, one 

 from each of the two meshes which it separates, and these currents will 

 be equal but in opposite directions. Thus all the currents in the lines 

 which have been introduced in the interior of the circuit annihilate one 

 another as regards total effect, while the currents in those parts of the 

 meshes which coincide with the original circuit just combine to repro- 

 duce the original current flowing in this circuit. 



Thus the original circuit is equivalent, as regards magnetic effect, to a 

 system of currents, one in each mesh. By taking the meshes sufficiently 

 small, we may regard each mesh as plane, so that the magnetic effect of a 

 current circulating in it is known : the magnetic effect of the current in a 

 single mesh is that of a magnetic shell of strength proportional to the current 

 and coinciding in position with the mesh. Thus, by addition, we find that 

 the whole system of currents produces the same magnetic effects as a single 

 magnetic shell coinciding with the surface of which the original current- 

 circuit is the boundary, and of strength proportional to the current. This 

 shell, then, produces the same magnetic effect as the original single current. 

 The magnetic shell is spoken of as the " equivalent magnetic shell." 



Thus we have obtained the following result : 



"A current flowing in any closed circuit produces the same magnetic field 

 as a certain magnetic shell, known as the 'equivalent magnetic shell! This 

 shell may be taken to be any shell having the circuit for its boundary, its 

 strength being uniform and proportional to that of the current" 



