ELECTROMAGNETIC ACTION OF A COIL. 77 



In this case, for a given channel the surface of the coil is pro- 

 portional to the number of windings, and therefore inversely as the 

 square of the diameter of the wire. 



For a given wire, a*nd similar volumes of the channel, the surface 

 is on the one hand proportional to the number of windings that is 

 to say, to the section of the channel and on the other to the surface 

 of the mean section ; it varies therefore as the fourth power of the 

 ratio of the homologous dimensions. 



728. ELECTROMAGNETIC ACTION. The law of the distribution 

 of force in the field of a coil only depends on the shape of the coil, 

 and the strength of the field at each point is proportional to the 

 strength of the current. We need then only calculate the action for 

 unit current. 



Let us first suppose that the channel is filled by a homogenous 

 mass of metal, and the current uniformly distributed in its meridian 

 section ; the intensity for unit surface may be called the density of 

 the current. 



Let us now suppose the meridian section divided into equal 

 squares by a series of lines, one set parallel and the other perpen- 

 dicular to the axis, there being n\ squares for each unit of surface. 

 Let us also suppose the mass of metal divided into a series of con- 

 centric rings insulated from each other, and corresponding to the 

 squares of the meridian section ; this operation will produce no 

 change either in the distribution of the current or in its external 

 action. If we suppose that each of these rings is traversed by unit 

 current, the density of the current is n\, and the electromagnetic 

 action is proportional to n\ that is to say, it varies inversely as the 

 dimensions of each elementary square. 



If we replace each ring by a cylindrical wire which occupies the 

 central part of the corresponding square, we form a coil uniformly 

 wound. If each wire is traversed by the quantity of electricity 

 which previously traversed the square, the density of the current is 

 no longer uniform ; but its mean value remains the same, whatever 

 be the diameter of the wire. 



On the other hand, if the current is uniformly distributed in the 

 section of the wire, its external action is appreciably the same, 

 whether we suppose the wire reduced to its axis, or whether its 

 diameter is that of the side of the square. 



The difference between the actions exerted by the coil, when we 

 replace the cylindrical wire by a wire with a square section, may be 

 neglected for any point which is at a great distance compared with . 

 the diameter of the wire (722). 



