An introduction to the mathematical theory of attraction ...

Examples. 99
sum of the magnetic moments of the particles whose centres
lie on the surface element dS divided by that element is
called the strength of the shell, and is supposed to be finite
and continuous. If J denote the strength of the shell at any
point, JdS may be regarded as the moment of a magnetic
particle whose axis is the normal to dS.
If r denote the distance of dS from an external point 0,
at which V is the potential of the shell, E the angle which r
makes with the normal to dS, and dw the element of solid
angle which dS subtends at O, we have
=\J - dS cos E { \2d Jdw.
Jr2 J= r= Jd.
If the shell be uniform, J is constant, and
V= J2, (30)
where a2 is the solid angle which the shell, or its bounding
curve, subtends at O.
In a magnetic shell whose surface is S, that side of S at
which north poles of the magnetic particles are situated is
regarded as the positive side.
EXAMPLES.
1. Prove that the potential at a point P of a uniform magnetic shell, whose
strength is J, is increased by 47rJ as P passes from the negative to the positive
side of the shell.
JdS cos e
The potential at P due to an element of the shell is r, but this is
the expression for the normal force at P due to an element of surface dS whose
density is J. As P passes through the element in the direction of the force, this
force increases by 47rJ, Arts. 16, 29; therefore so also does the magnetic potential
due to the same surface element of the shell; the rest of the magnetic potential
varies continuously; hence, on the whole, the magnetic potential of the shell
is increased by 47rJ as P passes from the negative to the positive side of the
shell.
2. Find the potential energy due to the mutual action of two small linear
magnets. (See Art. 17.)
Take the centre of the first magnet for origin, and let hl and h2 be lines
parallel to the two magnetic axes, drawn through any point P, dhl and dh2
being displacements in these directions; then, if e be a coordinate of P measured
in any direction, d is the cosine of the angle between the directions of e and h;
dr dhi\
also - is the cosine of the angle between r and hl. Similar results hold good
dh\
for h2.
H 2