Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/250

and η the angle which it makes with the positive direction of the normal to the parallelogram AA'B'B.

We may represent the result geometrically by the volume of the parallelepiped, whose base is the parallelogram AA'B'B, and one of whose edges is the line AM, which represents in direction and magnitude the magnetic induction ${\mathfrak {B}}$. If the parallelogram is in the plane of the paper, and if AM is drawn upwards from the paper, the volume of the parallelepiped is to be taken positively, or more generally, if the directions of the circuit AB, of the magnetic induction AM, and of the displacement AA', form a right-handed system when taken in this cyclical order.

The volume of this parallelepiped represents the increment of the value of p for the secondary circuit due to the displacement of the sliding piece from AB to A'B'.

Electromotive Force acting on the Sliding Piece.

595.] The electromotive force produced in the secondary circuit by the motion of the sliding piece is, by Art. 579,

$E=-{\frac {dp}{dt}}.$

(14)

If we suppose AA' to be the displacement in unit of time, then AA' will represent the velocity, and the parallelepiped will represent ${\frac {dp}{dt}}$, and therefore, by equation (14), the electromotive force in the negative direction BA.

Hence, the electromotive force acting on the sliding piece AB, in consequence of its motion through the magnetic field, is represented by the volume of the parallelepiped, whose edges represent in direction and magnitude the velocity, the magnetic induction, and the sliding piece itself, and is positive when these three directions are in right-handed cyclical order.

Electromagnetic Force acting on the Sliding Piece.

596.] Let i₂ denote the current in the secondary circuit in the positive direction ABC, then the work done by the electromagnetic force on AB while it slides from the position AB to the position A'B' is (M' – M)i_li₂, where M and M' are the values of M₁₂ in the initial and final positions of AB. But (M' – M)i₁ is equal to p' – p, and this is represented by the volume of the parallelepiped on AB, AM, and AA'. Hence, if we draw a line parallel to AB