Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/56

placed at the middle of the axis experiences a force $4\pi I$ in the direction of magnetization arising from the superficial magnetism on the circular surfaces of the disk^[1]. Since the components of I are A, B and C, the components of this force are 4πA, 4πB and 4πC. This must be compounded with the force whose components are α, β, γ.

400.] Let the actual force on the unit pole be denoted by the vector ${\mathfrak {B}}$, and its components by a, b and c, then

${\begin{matrix}a=\alpha +4\pi A,\\b=\beta +4\pi B,\\c=\gamma +4\pi C.\end{matrix}}$

(6)

We shall define the force within a hollow disk, whose plane sides are normal to the direction of magnetization, as the Magnetic Induction within the magnet. Sir William Thomson has called this the Electromagnetic definition of magnetic force.

The three vectors, the magnetization ${\mathfrak {J}}$, the magnetic force ${\mathfrak {H}}$, and the magnetic induction ${\mathfrak {B}}$ are connected by the vector equation

${\mathfrak {B}}={\mathfrak {H}}+4\pi {\mathfrak {J}}.$

(7)

Line-Integral of Magnetic Force.

401.] Since the magnetic force, as defined in Art. 398, is that due to the distribution of free magnetism on the surface and through the interior of the magnet, and is not affected by the surface-magnetism of the cavity, it may be derived directly from the general expression for the potential of the magnet, and the line-integral of the magnetic force taken along any curve from the point A to the point B is

$\int _{A}^{B}{\left(\alpha {\frac {dx}{ds}}+\beta {\frac {dy}{ds}}+\gamma {\frac {dz}{ds}}\right)ds}=V_{a}-V_{B},$

(8)

where V_A and V_b denote the potentials at A and B respectively.

1.  

   On the force within cavities of other forms.

   1. Any narrow crevasse. The force arising from the surface-magnetism is $4\pi I\cos \epsilon$ in the direction of the normal to the plane of the crevasse, where ε is the angle between this normal and the direction of magnetization. When the crevasse is parallel to the direction of magnetization the force is the magnetic force ${\mathfrak {H}}$; when the crevasse is perpendicular to the direction of magnetization the force is the magnetic induction ${\mathfrak {B}}$.

   2. In an elongated cylinder, the axis of which makes an angle ε with the direction of magnetization, the force arising from the surface-magnetism is $2\pi I\sin \epsilon$, perpendicular to the axis in the plane containing the axis and the direction of magnetization.

   3. In a sphere the force arising from surface-magnetism is ${\frac {4}{3}}\pi I$ in the direction of magnetization.