An introduction to the mathematical theory of attraction ...

Examples. 101
This expression for Mshows that it is the sum of the components in the plane of
qp of two couples, one Lh in the plane parallel to the axes of the magnets, and
one Lr in the plane of r and the axis of the second magnet, the magnitudes of
these couples being given by the equations
= sin 012, L =-, cos el sin 62.
r3 n3
The first couple tends to increase 0i2, and the second to diminish 62. The
couple G whose moment is required is the resultant of these two.
4. Find the resultant of the forces exerted by the first magnet on the
second.
Give the second magnet a displacement of translation parallel to a direction
h3; then if H3 be the component of the required force tending to increase h3, the
work done by the forces exerted by the first magnet on the second in the
displacement dhs is H3 dh3, and this must be equal to the loss of potential
dwt
energy, that is to - dh;
dh3
d n d icos 012 r COS E1 r cos E2
hence, S3 = - dh -— 2 -3 -c.
dit3 dh r3
The angle 012 is unaltered by a translation of the second magnet, and
dr d(r cos Ei) d(r cos E2)
-=cos, = COS 013, --- = C 023,
dha dha dh3
where e3,,, 023 are the angles which h3 makes with r, hi, and h2, respectively. Hence, by substitution, we have
H3 = 4 { COS 012 cos E3 + COS 023 COS E1 + COS 031 COS E2 - 5COs E1 cOs E2 COS E3}.
This expression shows that H3 is the sum of the components along h3 of three
forces R, Hi, and H2, in the directions of r, ih and hz, respectively, the
magnitudes of these forces being given by the equations
3R =-1 (cos 12 - 5 cos E1 cos E2),
Hi = ---- COS E2, H2 = ---- COS 61.
r4 r4
The required force F is the resultant of these three.
If we suppose the couple G produced by two equal and opposite forces
applied at the poles of the second magnet, one of these forces is of the order
r
F -, and is therefore very great compared with F.
5. Find the couple and the force acting on the second magnet in each of the
following cases:10. When the axes of the magnets are in the line of centres.
2~. When the axes of the magnets are parallel to each other, and perpendicular to the line of centres.