Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/315

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680.]
INDUCTION COIL.
283

That this may be a maximum, and being given, and variable,

.
(26)

This equation gives the best relation between the depths of the primary and secondary coil for an induction-machine without an iron core.

If there is an iron core of radius , then remains as before, but

, (27)
. (28)

If is given, the value of which gives the maximum value of is

.
(29)

When, as in the case of iron, is a large number, nearly. If we now make constant, and and variable, we obtain the maximum value of when

.
(30)

The coefficient of self-induction of a long solenoid whose outer and inner radii are and , and having a long iron core whose radius is , is

.
(31)

680.] We have hitherto supposed the wire to be of uniform thickness. We shall now determine the law according to which the thickness must vary in the different layers in order that, for a given value of the resistance of the primary or the secondary coil, the value of the coefficient of mutual induction may be a maximum.

Let the resistance of unit of length of a wire, such that windings occupy unit of length of the solenoid, be .

The resistance of the whole solenoid is

.
(32)

The condition that, with a given value of , may be a maximum is , where is some constant.

This gives proportional to , or the diameter of the wire of the exterior coil must be proportional to the square root of the radius.

In order that, for a given value of R, g may be a maximum

.
(33)