Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/65

 Rh direction everywhere normal to its surface, the intensity of the magnetization at any place multiplied by the thickness of the sheet at that place is called the Strength of the magnetic shell at that place.

If the strength of a shell is everywhere equal, it is called a Simple magnetic shell ; if it varies from point to point it may be conceived to be made up of a number of simple shells superposed and overlapping each other. It is therefore called a Complex magnetic shell.

Let dS be an element of the surface of the shell at Q, and Φ the strength of the shell, then the potential at any point, P, due to the element of the shell, is where ε is the angle between the vector QP, or r and the normal drawn from the positive side of the shell.

But if dω is the solid angle subtended by dS at the point P and therefore in the case of a simple magnetic shell or, the potential due to a magnetic shell at any point is the product of its strength into the solid angle subtended by its edge at the given point.

410.] The same result may be obtained in a different way by supposing the magnetic shell placed in any field of magnetic force, and determining the potential energy due to the position of the shell.

If V is the potential at the element dS, then the energy due to this element is or, the product of the strength of the shell into the part of the surface-integral of V due to the element dS of the shell.

Hence, integrating with respect to all such elements, the energy due to the position of the shell in the field is equal to the product of the strength of the shell and the surface -integral of the magnetiinduction taken over the surface of the shell.

Since this surface-integral is the same for any two surfaces