Poincaré: On the Partial Differential Equations of Mathematical Physics – Laplace equation for magnets

Suppose one or more permanent magnets placed in the presence of a perfectly magnetically soft body M. It is a question of finding a function V (the magnetic potential) which satisfies the Laplace equation in all the portion of space which is not not occupied by permanent magnets and which is further subject to the following conditions. At the various points where there is permanent magnetism, AV is not zero, but can be regarded as given. The function V is continuous in all space; its derivatives are continuous inside the body M and outside this body, but they are discontinuous at the surface of the body M. In the neighborhood of this surface, will therefore have two different values depending on whether we place ourselves inside or outside the body M; but the ratio of these two values will be a given constant.

Here again, we find the same differential equation, with similar but different boundary conditions:

dV/dn = α dV/dx + β dV/dy + γ dV/dz

for a surface element, and α, β, γ the three direction cosines of the normal to this element.

Source: Henri Poincaré, Sur les Equations aux Dérivées Partielles de la Physique Mathématique, American Journal of Mathematics, Mar., 1890, Vol. 12, No. 3 (Mar., 1890), pp. 211-294. Translation by Nicolae Sfetcu. © 2022 Nicolae Sfetcu