Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/82

Hence, eliminating ${\mathfrak {J}}$, we find

${\mathfrak {B}}=(1+4\pi \kappa ){\mathfrak {J}}$

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as the relation between the magnetic induction and the magnetic force in substances whose magnetization is induced by magnetic force.

In the most general case κ may be a function, not only of the position of the point in the substance, but of the direction of the vector ${\mathfrak {H}}$, but in the case which we are now considering κ is a numerical quantity.

If we next write

$\mu =1+4\pi \kappa ,$

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we may define μ as the ratio of the magnetic induction to the magnetic force, and we may call this ratio the magnetic inductive capacity of the substance, thus distinguishing it from κ, the co efficient of induced magnetization.

If we write U for the total magnetic potential compounded of V, the potential due to external causes, and Ω for that due to the induced magnetization, we may express a, b, c, the components of magnetic induction, and α, β, γ, the components of magnetic force, as follows:

${\begin{aligned}&a=\mu \alpha =-\mu {\frac {dU}{dx}},\\&b=\mu \beta =-\mu {\frac {dU}{dy}},\\&c=\mu \gamma =-\mu {\frac {dU}{dz}}\end{aligned}}$

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The components a, b, c satisfy the solenoidal condition

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

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Hence, the potential U must satisfy Laplace's equation

${\frac {d^{2}U}{dx^{2}}}+{\frac {d^{2}U}{dy^{2}}}+{\frac {d^{2}U}{dz^{2}}}=0$

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at every point where μ is constant, that is, at every point within the homogeneous substance, or in empty space.

At the surface itself, if ν is a normal drawn towards the magnetic substance, and ν' one drawn outwards, and if the symbols of quantities outside the substance are distinguished by accents, the condition of continuity of the magnetic induction is

$a{\frac {d\nu }{dx}}+b{\frac {d\nu }{dy}}+c{\frac {d\nu }{dz}}+a'{\frac {d\nu }{dx}}+b'{\frac {d\nu }{dy}}+c'{\frac {d\nu }{dz}}=0;$

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