Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/57

Surface-Integral of Magnetic Induction.

402.] The magnetic induction through the surface S is defined as the value of the integral

$Q=\iint {{\mathfrak {B}}\cos \epsilon \,dS},$

(9)

where ${\mathfrak {B}}$ denotes the magnitude of the magnetic induction at the element of surface dS, and ε the angle between the direction of the induction and the normal to the element of surface, and the integration is to be extended over the whole surface, which may be either closed or bounded by a closed curve.

If a, b, c denote the components of the magnetic induction, and l, m, n the direction-cosines of the normal, the surface-integral may be written

$Q=\iint {(la+mb+nc)dS}.$

(10)

If we substitute for the components of the magnetic induction their values in terms of those of the magnetic force, and the magnetization as given in Art. 400, we find

$Q=\iint {(l\alpha +m\beta +n\gamma )dS}+4\pi \iint {(lA+mB+nC)dS}.$

(11)

We shall now suppose that the surface over which the integration extends is a closed one, and we shall investigate the value of the two terms on the right-hand side of this equation.

Since the mathematical form of the relation between magnetic force and free magnetism is the same as that between electric force and free electricity, we may apply the result given in Art. 77 to the first term in the value of Q by substituting α, β, γ, the components of magnetic force, for X, Y, Z, the components of electric force in Art. 77, and M, the algebraic sum of the free magnetism within the closed surface, for e, the algebraic sum of the free electricity.

We thus obtain the equation

$\iint {(l\alpha +m\beta +n\gamma )dS}=4\pi M.$

(12)

Since every magnetic particle has two poles, which are equal in numerical magnitude but of opposite signs, the algebraic sum of the magnetism of the particle is zero. Hence, those particles which are entirely within the closed surface S can contribute nothing to the algebraic sum of the magnetism within S. The