106
II
INDUCTION IN ROTATING SPHERES
To satisfy the equations of condition we put
ቀያ
=
Απε
$ = &° + $',
1+ 4πε n+1
-
=
W
Xn
ρ
-nzxn
Əz
Απε
1+4πen + 1
W
[(
R\n+2 n
p²
ρ dxn
zxn
n+1 2n+1 Əz
R\" R2 Xn
+
-
2n+1 Əz
p satisfies the partial differential equation which is to
satisfy. is so formed that (1) it satisfies the equation
V200 = 0,
(2) at the surface of the sphere it is equal to 4. That the
first condition is satisfied is seen when we notice that the
expressions under the straight lines are spherical surface
harmonics of degrees (n+1) and (n − 1), as is easily proved.
Substituting 4º+p' in the equations for 4, we get for 'these
equations
-
V² = 0 everywhere, p' continuous,
when p = R
i
аф оф офо
(1+4π€)
др
др
Earth in
dielectric
space.
to satisfy which is not difficult, as we have already expressed
pe as a series of spherical harmonics.
A case of especial interest is that in which a spherical
magnet rotates in a surrounding dielectric. For the earth is
a rotating magnet, and according to many physicists inter-
planetary space is a dielectric. To determine the electric
potential in this case we must remember that the earth is a
conductor; hence in it a distribution will form which will
react on the dielectric and make the potential constant at the
earth's surface.
If x = Ex is the earth's potential the problem reduces to
this:-