Coefficients of Induction for Two Circuits

Case of two Circuits.

(37) Let R {\displaystyle R} be the primary circuit and S {\displaystyle S} the secondary circuit, then we have a case similar to that of the induction coil. The equations of currents are those marked A {\displaystyle A} and B {\displaystyle B}, and we may here assume L {\displaystyle L}, M {\displaystyle M}, N {\displaystyle N} as constant because there is no motion of the conductors. The equations then become

(13*) To find the total quantity of electricity which passes, we have only to integrate these equations with respect to t {\displaystyle t}; then if x 0 {\displaystyle x_{0}}, y 0 {\displaystyle y_{0}} be the strengths of the currents at time 0 {\displaystyle 0}, and x 1 {\displaystyle x_{1}}, y 1 {\displaystyle y_{1}} at time t {\displaystyle t}, and if X {\displaystyle X}, Y {\displaystyle Y} be the quantities of electricity passed through each circuit during time t

(14*) When the circuit R {\displaystyle R} is completed, then the total currents up to time t {\displaystyle t}, when t {\displaystyle t} is great, are found by making

then

The value of the total counter-current in R {\displaystyle R} is therefore independent of the secondary circuit, and the induction current in the secondary circuit depends only on M {\displaystyle M}, the coefficient of induction between the coils, S {\displaystyle S} the resistance of the secondary coil, and x 1 {\displaystyle x_{1}} the final strength of the current in R {\displaystyle R}. When the electromotive force ξ {\displaystyle \xi } ceases to act, there is an extra current in the primary circuit, and a positive induced current in the secondary circuit, whose values are equal and opposite to those produced on making contact.

(38) All questions relating to the total quantity of transient currents, as measured by the impulse given to the magnet of the galvonometer, may be solved in this way without the necessity of a complete solution of the equations. The heating effect of the current, and the impulse it gives to the suspended coil of Weber’s dynamometer, depend on the square of the current at every instant during the short time it lasts. Hence we must obtain the solution of the equations, and from the solution we may find the effects both on the galvanometer and dynamometer; and we may then make use of the method of Weber for estimating the intensity and duration of a current uniform while it lasts which would produce the same effects.

(39) Let

{\displaystyle n_{1},n_{2}} be the roots of the equation

(16) and let the primary coil be acted on by a constant electromotive force R c {\displaystyle Rc}, so that c {\displaystyle c} is the constant current it could maintain; then the complete solution of the equations for making contact is

(18) From these we obtain for calculating the impulse on the dynamometer,

(19)

(20)

The effects of the current in the secondary coil on the galvanometer and dynamometer are the same as those of a uniform current

for a time

(40) The equation between work and energy may be easily verified. The work done by the electromotive force is

Work done in overcoming resistance and producing heat,

Energy remaining in the system,

(41) If the circuit R is suddenly and completely interrupted while carrying a current c {\displaystyle c}, then the equation of the current in the secondary coil would be

This current begins with a value

{\displaystyle c{\tfrac {M}{N}}}, and gradually disappears.

The total quantity of electricity is

{\displaystyle c{\frac {M}{S}}}, and the value of

The effects on the galvanometer and dynamometer are equal to those of a uniform current

{\displaystyle {\tfrac {1}{2}}c{\tfrac {M}{N}}} for a time

The heating effect is therefore greater than that of the current on making contact.

(42) If an electromotive force of the form

{\displaystyle \xi =E\cos pt} acts on the circuit R, then if the circuit S is removed, the value of x {\displaystyle x} will be

where

and

tan ⁡

The effect of the presence of the circuit S in the neighbourhood is to alter the value of A and α {\displaystyle \alpha }, to that which they would be if R became

and L became

Hence the effect of the presence of the circuit S is to increase the apparent resistance and diminish the apparent self-induction of the circuit R.