Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/241

and secondary circuits respectively, and M is the coefficient of their mutual induction.

Let us suppose that no electromotive force acts on the secondary circuit except that due to the induction of the primary current. We have then

$E_{2}=R_{2}{\dot {y}}_{2}+{\frac {d}{dt}}(M{\dot {y}}_{1}+N{\dot {y}}_{2})=0.$

Integrating this equation with respect to t, we have

$R_{2}y_{2}+M{\dot {y}}_{1}+N{\dot {y}}_{2}=C,{\text{ a constant,}}$

where y₂ is the integral current in the secondary circuit.

The method of measuring an integral current of short duration will be described in Art. 748, and it is easy in most cases to ensure that the duration of the secondary current shall be very short.

Let the values of the variable quantities in the equation at the end of the time t be accented, then, if y₂ is the integral current, or the whole quantity of electricity which flows through a section of the secondary circuit during the time t,

$R_{2}y_{2}=M{\dot {y}}_{1}+N{\dot {y}}_{2}-(M'{\dot {y}}_{1}'+N'{\dot {y}}_{2}').$

If the secondary current arises entirely from induction, its initial value ${\dot {y}}_{2}$ must be zero if the primary current is constant, and the conductors at rest before the beginning of the time t.

If the time t is sufficient to allow the secondary current to die away, ${\dot {y}}_{2}'$, its final value, is also zero, so that the equation becomes

$R_{2}y_{2}=M{\dot {y}}_{1}-M'{\dot {y}}_{1}'.$

The integral current of the secondary circuit depends in this case on the initial and final values of $M{\dot {y}}_{1}$.

Induced Currents.

582.] Let us begin by supposing the primary circuit broken, or ${\dot {y}}_{1}=0$, and let a current ${\dot {y}}_{1}'$ be established in it when contact is made.

The equation which determines the secondary integral current is

$R_{2}y_{2}=-M{\dot {y}}_{1}'.\,$

When the circuits are placed side by side, and in the same direction, M is a positive quantity. Hence, when contact is made in the primary circuit, a negative current is induced in the secondary circuit.

When the contact is broken in the primary circuit, the primary current ceases, and the induced current is y₂ where

$R_{2}y_{2}=M{\dot {y}}_{1}.\,$

The secondary current is in this case positive.