Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/260

line-integral of ${\mathfrak {A}}$ into the surface-integral of another vector, ${\mathfrak {B}}$, whose components are a, b, c, and we found that the phenomena of induction due to motion of a conductor, and those of electromagnetic force can be expressed in terms of ${\mathfrak {B}}$. We gave to ${\mathfrak {B}}$ the name of the Magnetic induction, since its properties are identical with those of the lines of magnetic induction as investigated by Faraday.

We also established three sets of equations : the first set, (A), are those of magnetic induction, expressing it in terms of the electromagnetic momentum. The second set, (B), are those of electromotive force, expressing it in terms of the motion of the conductor across the lines of magnetic induction, and of the rate of variation of the electromagnetic momentum. The third set, (C), are the equations of electromagnetic force, expressing it in terms of the current and the magnetic induction.

The current in all these cases is to be understood as the actual current, which includes not only the current of conduction, but the current due to variation of the electric displacement.

The magnetic induction ${\mathfrak {B}}$ is the quantity which we have already considered in Art. 400. In an unmagnetized body it is identical with the force on a unit magnetic pole, but if the body is magnetized, either permanently or by induction, it is the force which would be exerted on a unit pole, if placed in a narrow crevasse in the body, the walls of which are perpendicular to the direction of magnetization. The components of ${\mathfrak {B}}$ are a, b, c.

It follows from the equations (A), by which a, b, c are defined, that

${\frac {da}{dx}}+{\frac {db}{dt}}+{\frac {dc}{dz}}=0.$

This was shewn at Art. 403 to be a property of the magnetic induction.

605.] We have defined the magnetic force within a magnet, as distinguished from the magnetic induction, to be the force on a unit pole placed in a narrow crevasse cut parallel to the direction of magnetization. This quantity is denoted by ${\mathfrak {H}}$, and its components by α, β, γ. See Art. 398.

If ${\mathfrak {J}}$ is the intensity of magnetization, and A, B, C its components, then, by Art. 400,

$\left.{\begin{aligned}a=\alpha +4\pi A,\\b=\beta +4\pi B,\\c=\gamma +4\pi C.\end{aligned}}\right\}{\text{(Equations of Magnetization.)}}$

(D)