Henri Poincaré: Potential of an electric mass distributed on the surface of a sphere

Such is the present state of the question; let us see now what is the goal that I have in view; what would have been most interesting would have been to replace the current calculation methods with other less defective ones. I could not do it, I limited myself to looking for a simpler method of demonstration than those which have been proposed up to now and directly applicable to all cases.

I am going to begin by briefly recalling the main propositions which I shall have to make use of in what follows.

Green’s function, U, relative to a sphere S and a point P is obtained in the following way. Let R be the radius of the sphere.

Let us take on the line OP a length OQ = R²/OP; the function U will be a potential of two masses, one equal to 1 and placed at the point P, the other equal to -√(OQ/OP) and placed at the point Q.

The value of dU/dn corresponding to any element of the sphere S is in inverse proportion to the cube of the distance from this element to the point P. It is equal to

(R² – OP²)/R·1/MP³

M designating the center of gravity of the element considered.

If we consider a sphere S and a point P inside this sphere, the potential of an electric mass equal to 1 and placed at point P will be equal to 1/MP at point M.

Imagine then that this same electric mass equal to 1 is distributed over the surface of the sphere S in such a way that the density on any element of this sphere is in inverse ratio to the cube of the distance of this element from the point P. I say that the potential of this mass thus distributed which we will call W will be equal to 1/MP at any point M exterior to the sphere and smaller than 1/MP if the point M is interior to the sphere.

Indeed let us consider a function which is equal to Green’s function, U, defined above inside the sphere and 0 outside the sphere. This function everywhere satisfies Laplace’s equation; it is continuous except at point P and on the surface of the sphere. At point P the function becomes infinite and its difference with 1/MP remains finite; on the surface of the sphere, the function itself remains continuous, but its derivative d/dn a sudden jump equal to (R² – OP²)/R.MP³ .

We must conclude that this function is equal to the potential of various electric masses distributed as follows:

A mass equal to 1 at point P.
A mass of density -(R² – OP²)/4π.R.MP³ at various points on the surface This second mass is nothing other than the mass defined above and whose potential was W, but changed sign.

So we have:

1/MP – W = 0

outside S and

1/MP – W = U > 0

inside S.

Q.E.D.

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. © 2023 Nicolae Sfetcu