p/An electric dipole spec-
trometer. A beam of randomly
oriented dipoles is shot through a
“croquet hoop” consisting of two
fixed positive charges. Although
the field along the central axis of
symmetry equals zero, the field
is nonuniform, and therefore the
dipoles feel a nonvanishing force,
and are sorted out according to
their orientations. A magnetic
version of this device, described
on p. 743, was used in the his-
toric Stern-Gerlach experiment,
sec. 14.1, p. 957.Force on a dipole in a nonuniform field example 7
If a dipole with zero total charge is placed in a uniform field, it
may experience a torque, but it will not experience anyforce. The
situation changes if the field is nonuniform. The force can be
nonzero, and, perhaps more surprisingly, the force depends only
on the dipole moment, not on the details of the arrangement of
the charges inside the dipole, provided that the dipole is small
enough. This can be shown either by a brute-force calculation
(problem 59, 666) or by the following slightly slicker technique.
The force in thexdirection is
Fx=−∂U
∂x,
where the symbol∂indicates a derivative with respect to thex
coordinate only, holdingyandzconstant. (This is referred to as
a partial derivative.) We then have
Fx=−∂
∂x(D·E) =D·
∂E
∂x,
which depends on the dipole’s properties only throughD. Similar
expressions apply for theyandzcomponents.
This principle can be used as a way of measuring the unknown
dipole moments of a beam of particles, as in figure p. At a more
pedestrian level, this is one way of explaining the fact that we
can use a charged object to pick up uncharged scraps of paper
(p. 478).
Section 10.1 Fields of force 589