Alternative definition of the electric field
The behavior of a dipole in an externally created field leads us
to an alternative definition of the electric field:
The electric field vector,E, at any location in space is defined
by observing the torque exerted on a test dipoleDtplaced there.
The direction of the field is the direction in which the field tends
to align a dipole (from−to +), and the field’s magnitude is|E|=
τ/Dtsinθ. In other words, the field vector is the vector that satisfies
the equationτ=Dt×Efor any test dipoleDtplaced at that point
in space.
The main reason for introducing a second definition for the same
concept is that the magnetic field is most easily defined using a
similar approach.Energy of a dipole in a field
A dipole in an external field has a stable equilibrium orientation
in whichDis parallel toE. Since the vector cross product vanishes
for parallel vectors, the field’s torque on the dipole is zero. (There
is also an unstable equilibrium withDandEantiparallel.) This
fact is probably more familiar from the magnetic context, where a
magnetic dipole such as a compass needle tends to align itself with
an external magnetic field. We will encounter magnetic fields and
dipoles later, and they have many properties analogous to those of
their electric counterparts. Depending on its orientation, the dipole
will have some interaction energyUwhen it interacts with the field.
If the physical size of the dipole is small, the electric field can be
approximated as having a single value throughout. Since energy is
a scalar, and the dot product is the only way (up to a multiplicative
constant) to multiply two vectors to get a scalar (sec. 3.4.5, p. 216),
we must haveU∝D·E. Because the energy is minimized whenD
andEare parallel, the constant of proportionality must be negative,
and one can easily show by considering a concrete example that this
constant equals−1. Therefore we haveU=−D·E. [energy of a pointlike dipole in a field]588 Chapter 10 Fields