Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/356

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324
ELECTROMAGNETIC INSTRUMENTS.
[719.

he volume of this layer is

,
(9)

where is the length of wire in this layer.

This gives us in terms of . Substituting this in equations (3) and (4), we find

, (10)
, (11)

where and represent the portions of the values of and of due to this layer of the coil.

Now if be the given electromotive force,

,

where is the resistance of the external part of the circuit, independent of the galvanometer, and the force at the centre is

.

We have therefore to make a maximum, by properly adjusting the section of the wire in each layer. This also necessarily involves a variation of because depends on .

Let and be the values of and of when the given layer is excluded from the calculation. We have then

,
(12)

and to make this a maximum by the variation of the value of for he given layer we must have

.
(13)

Since is very small and ultimately vanishes, will be sensibly, and ultimately exactly, the same whichever layer is excluded, and we may therefore regard it as constant. We have therefore, by (10) and (11),

.
(14)

If the method of covering the wire and of winding it is such that the proportion between the space occupied by the metal of