Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/249

594.] We have next to deduce from dynamical principles the expressions for the electromagnetic force acting on a conductor carrying an electric current through the magnetic field, and for the electromotive force acting on the electricity within a body moving in the magnetic field. The mathematical method which we shall adopt may be compared with the experimental method used by Faraday^[1] in exploring the field by means of a wire, and with what we have already done at Art. 490, by a method founded on experiments. What we have now to do is to determine the effect on the value of p, the electrokinetic momentum of the secondary circuit, due to given alterations of the form of that circuit.

Let AA', BB' be two parallel straight conductors connected by the conducting arc C, which may be of any form, and by a straight conductor AB, which is capable of sliding parallel to itself along the conducting rails AA' and BB'.

Let the circuit thus formed be considered as the secondary circuit, and let the direction ABC be assumed as the positive direction round it.

Let the sliding piece move parallel to itself from the position AB to the position A'B'. We have to determine the variation of p, the electrokinetic momentum of the circuit, due to this displacement of the sliding piece.

The secondary circuit is changed from ABC to A'B'C, hence, by Art, 587,

$p(A'B'C')-p(ABC)=p(A'AB'B).\,$

(13)

We have therefore to determine the value of p for the parallelogram AA'B'B. If this parallelogram is so small that we may neglect the variations of the direction and magnitude of the magnetic induction at different points of its plane, the value of p is, by Art. 591, ${\mathfrak {B}}\cos \eta \cdot AA'B'B$, where ${\mathfrak {B}}$ is the magnetic induction,

1. Exp. Res., 3082, 3087, 3113.