# J. C. Maxwell’s, ‘Dynamical theory of the electromagnetic field’

```                                Case of Two Circuits

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

Rx + L dx/dt + M dy/dy = [xi]
Sy + M dx/dt + N dy/dt = 0 (13)

To find the total quantity of electricity which passes we have
only to integrate these equations with respect to t, then if x<sub>0<\sub> y<sub>0<sub>
be the strengths of the currents at time 0 and x<sub>1<\sub> y<sub>1<\sub> at time t
and if X Y be the quantities of electricity passed through each circuit
during time t

X = 1/R {[xi]t + L (x<sub>0<\sub> - x<sub>1<\sub>) + M(y<sub>0<\sub> - y<sub>1<\sub>)}

Y = 1/S {M(x<sub>0<\sub> - x<sub>1<\sub>) + N(y<sub>0<\sub> - y<sub>1<\sub>)} (14)

When the circuit R is completed, then the total currents up to time t,
when t is great, are found by making x<sub>0<\sub> = 0 x<sub>1<\sub> = [xi]/R y<sub>0<\sub> = 0 y<sub>1<\sub> = 0
then X = x<sub>1<\sub>(t - L/R) Y = -M/S x<sub>1<\sub> (15)

The value of the total counter current in R is therefore independent
of the secondary circuit, and the induction current in S depends
only on M the coefficient of induction between the coils, S the resistance
of the secondary coil and x, the final strength of the current in R

When the electromotive force [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 quantity of transient currents
as measured by the impulse given to the magnet of the galvanometer
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
```
images

## Manuscript details

Author
James Clerk Maxwell
Reference
PT/72/7
Series
PT
Date
1864
IIIF
(What's this?)
This is a link to the IIIF web URL for this item. You can drag and drop the IIIF image link into other compatible viewers

## Cite as

J. C. Maxwell’s, ‘Dynamical theory of the electromagnetic field’, 1864. From The Royal Society, PT/72/7