In physics, the electric dipole moment is a measure of the separation of
positive and negative electrical charges in a system of electric charges, that
is, a measure of the charge system's overall polarity. The SI units are
Coulomb-meter. This article is limited to static phenomena, and does not
describe time-dependent or dynamic polarization. The magnitude of dipole
moment determines the electric field strength.
Elementary definition In the simple case of two point charges,
one with charge +q and the other one with charge −q, the electric dipole
moment p is: where d is the displacement vector
pointing from the negative charge to the positive charge. Thus, the electric
dipole moment vector p points from the negative charge to the positive charge.
An idealization of this two-charge system is the electrical point dipole
consisting of two charges only infinitesimally separated, but with a
finite p. Torque
An object with an electric dipole moment is subject to a torque τ when placed in
an external electric field. The torque tends to align the dipole with the
field. A dipole aligned parallel to an electric field has lower potential
energy than a dipole making some angle with it. For a spatially uniform
electric field E, the torque is given by:
where p is the dipole moment, and the symbol "×" refers to the vector cross
product. The field vector and the dipole vector define a plane, and the torque is
directed normal to that plane with the direction given by the right-hand rule.
A dipole placed parallel or anti-parallel to the direction in which
a non-uniform electric field is increasing will not experience a torque,
only a force in the direction of its dipole moment. It can be shown that this
force will always be parallel to the dipole moment regardless of parallel or
anti-parallel orientation of the dipole. Expression
More generally, for a continuous distribution of charge confined to a
volume V, the corresponding expression for the dipole moment is:
where r locates the point of observation and d3r0 denotes an elementary volume in
V. For an array of point charges, the charge density becomes a sum of Dirac
delta functions: where each ri is a vector from some
reference point to the charge qi. Substitution into the above integration
formula provides: This expression is equivalent to the
previous expression in the case of charge neutrality and N = 2. For two
opposite charges, denoting the location of the positive charge of the pair as r+
and the location of the negative charge as r− :
showing that the dipole moment vector is directed from the negative charge to the
positive charge because the position vector of a point is directed outward
from the origin to that point. The dipole moment is most easily
understood when the system has an overall neutral charge; for example, a
pair of opposite charges, or a neutral conductor in a uniform electric field.
For a system of charges with no net charge, visualized as an array of paired
opposite charges, the relation for electric dipole moment is:
which is the vector sum of the individual dipole moments of the neutral
charge pairs. Thus, the value of p is independent of the choice of reference
point, provided the overall charge of the system is zero.
When discussing the dipole moment of a non-neutral system, such as the dipole
moment of the proton, a dependence on the choice of reference point arises. In
such cases it is conventional to choose the reference point to be the center of
mass of the system, not some arbitrary origin. For a charged molecule the
center of charge should be the reference point instead of the center of mass. For
neutral systems the references point is not important. The dipole moment is an
intrinsic property of the system. Potential and field of an electric
dipole An ideal dipole consists of two opposite
charges with infinitesimal separation. The potential and field of such an ideal
dipole are found next as a limiting case of an example of two opposite charges at
non-zero separation. Two closely spaced opposite charges have
a potential of the form: with charge separation, d, defined as
The position relative to their center of mass, R, and the unit vector in the
direction of R are given by: Taylor expansion in d/R allows this
potential to be expressed as a series. where higher order terms in the series
are vanishing at large distances, R, compared to d. Here, the electric dipole
moment p is, as above: The result for the dipole potential also
can be expressed as: which relates the dipole potential to
that of a point charge. A key point is that the potential of the dipole falls
off faster with distance R than that of the point charge.
The electric field of the dipole is the negative gradient of the potential,
leading to: Thus, although two closely spaced
opposite charges are not quite an ideal electric dipole, at distances much
larger than their separation, their dipole moment p appears directly in
their potential and field. As the two charges are brought closer
together, the dipole term in the multipole expansion based on the ratio
d/R becomes the only significant term at ever closer distances R, and in the
limit of infinitesimal separation the dipole term in this expansion is all
that matters. As d is made infinitesimal, however, the dipole
charge must be made to increase to hold p constant. This limiting process
results in a "point dipole". Dipole moment density and polarization
density The dipole moment of an array of
charges, determines the degree of polarity of the
array, but for a neutral array it is simply a vector property of the array
with no information about the array's absolute location. The dipole moment
density of the array p(r) contains both the location of the array and its dipole
moment. When it comes time to calculate the electric field in some region
containing the array, Maxwell's equations are solved, and the
information about the charge array is contained in the polarization density
P(r) of Maxwell's equations. Depending upon how fine-grained an assessment of
the electric field is required, more or less information about the charge array
will have to be expressed by P(r). As explained below, sometimes it is
sufficiently accurate to take P(r) = p(r). Sometimes a more detailed
description is needed and sometimes even more elaborate versions of P(r) are
necessary. It now is explored just in what way the
polarization density P(r) that enters Maxwell's equations is related to the
dipole moment p of an overall neutral array of charges, and also to the dipole
moment density p(r). Only static situations are considered in what
follows, so P(r) has no time dependence, and there is no displacement current.
First is some discussion of the polarization density P(r). That
discussion is followed with several particular examples.
A formulation of Maxwell's equations based upon division of charges and
currents into "free" and "bound" charges and currents leads to introduction of
the D- and P-fields: where P is called the polarization
density. In this formulation, the divergence of this equation yields:
and as the divergence term in E is the total charge, and ρf is "free charge",
we are left with the relation: with ρb as the bound charge, by which is
meant the difference between the total and the free charge densities.
As an aside, in the absence of magnetic effects, Maxwell's equations specify
that which implies
Applying Helmholtz decomposition: for some scalar potential φ, and:
Suppose the charges are divided into free and bound, and the potential is
divided into Satisfaction of the boundary conditions
upon φ may be divided arbitrarily between φf and φb because only the sum φ
must satisfy these conditions. It follows that P is simply proportional to
the electric field due to the charges selected as bound, with boundary
conditions that prove convenient. In particular, when no free charge is
present, one possible choice is P = ε0 E.
Next is discussed how several different dipole-moment descriptions of a medium
relate to the polarization entering Maxwell's equations.
= Medium with charge and dipole densities=
As described next, a model for polarization moment density p(r) results
in a polarization restricted to the same model. For a
smoothly varying dipole moment distribution p(r), the corresponding
bound charge density is simply as we will establish shortly via
integration by parts. However, if p(r) exhibits an abrupt step in dipole moment
at a boundary between two regions, ∇•p(r) results in a surface charge
component of bound charge. This surface charge can be treated through a surface
integral, or by using discontinuity conditions at the boundary, as
illustrated in the various examples below.
As a first example relating dipole moment to polarization, consider a
medium made up of a continuous charge density ρ(r) and a continuous dipole
moment distribution p(r). The potential at a position r is:
where ρ(r) is the unpaired charge density, and p(r) is the dipole moment
density. Using an identity: the polarization integral can be
transformed: The first term can be transformed to an
integral over the surface bounding the volume of integration, and contributes a
surface charge density, discussed later. Putting this result back into the
potential, and ignoring the surface charge for now:
where the volume integration extends only up to the bounding surface, and
does not include this surface. The potential is determined by the total
charge, which the above shows consists of:
showing that: In short, the dipole moment density p(r)
plays the role of the polarization density P for this medium. Notice, p(r)
has a non-zero divergence equal to the bound charge density.
It may be noted that this approach can be extended to include all the
multipoles: dipole, quadrupole, etc. Using the relation:
the polarization density is found to be: where the added terms are meant to
indicate contributions from higher multipoles. Evidently, inclusion of
higher multipoles signifies that the polarization density P no longer is
determined by a dipole moment density p alone. For example, in considering
scattering from a charge array, different multipoles scatter an
electromagnetic wave differently and independently, requiring a
representation of the charges that goes beyond the dipole approximation.
Surface charge Above, discussion was deferred for the
first term in the expression for the potential due to the dipoles.
Integrating the divergence results in a surface charge. The figure at the right
provides an intuitive idea of why a surface charge arises. The figure shows
a uniform array of identical dipoles between two surfaces. Internally, the
heads and tails of dipoles are adjacent and cancel. At the bounding surfaces,
however, no cancellation occurs. Instead, on one surface the dipole heads
create a positive surface charge, while at the opposite surface the dipole tails
create a negative surface charge. These two opposite surface charges create a
net electric field in a direction opposite to the direction of the
dipoles. This idea is given mathematical form
using the potential expression above. The potential is:
Using the divergence theorem, the divergence term transforms into the
surface integral: with dA0 an element of surface area of
the volume. In the event that p(r) is a constant, only the surface term
survives: with dA0 an elementary area of the
surface bounding the charges. In words, the potential due to a constant p inside
the surface is equivalent to that of a surface charge
which is positive for surface elements with a component in the direction of p
and negative for surface elements pointed oppositely.
If the bounding surface is a sphere, and the point of observation is at the
center of this sphere, the integration over the surface of the sphere is zero:
the positive and negative surface charge contributions to the potential cancel.
If the point of observation is off-center, however, a net potential can
result because the positive and negative charges are at different distances from
the point of observation. The field due to the surface charge is:
which, at the center of a spherical bounding surface is not zero but is
instead : If we suppose the polarization of the
dipoles was induced by an external field, the polarization field opposes
the applied field and sometimes is called a depolarization field. In the
case when the polarization is outside a spherical cavity, the field in the
cavity due to the surrounding dipoles is in the same direction as the
polarization. In particular, if the electric
susceptibility is introduced through the approximation:
where E, in this case and in the following, represent the external field
which induces the polarization. Then:
Whenever χ(r) is used to model a step discontinuity at the boundary between
two regions, the step produces a surface charge layer. For example, integrating
along a normal to the bounding surface from a point just interior to one
surface to another point just exterior: where An, Ωn indicate the area and
volume of an elementary region straddling the boundary between the
regions, and a unit normal to the surface. The right side vanishes as the
volume shrinks, inasmuch as ρb is finite, indicating a discontinuity in E,
and therefore a surface charge. That is, where the modeled medium includes a step
in permittivity, the polarization density corresponding to the dipole
moment density necessarily includes the contribution of
a surface charge. A physically more realistic modeling of
p(r) would have the dipole moment density drop off rapidly, but smoothly
to zero at the boundary of the confining region, rather than making a sudden step
to zero density. Then the surface charge will not concentrate in an infinitely
thin surface, but instead, being the divergence of a smoothly varying dipole
moment density, will distribute itself throughout a thin, but finite transition
layer. Dielectric sphere in uniform external
electric field The above general remarks about surface
charge are made more concrete by considering the example of a dielectric
sphere in a uniform electric field. The sphere is found to adopt a surface
charge related to the dipole moment of its interior.
A uniform external electric field is supposed to point in the z-direction,
and spherical-polar coordinates are introduced so the potential created by
this field is: The sphere is assumed to be described by
a dielectric constant κ, that is, and inside the sphere the potential
satisfies Laplace's equation. Skipping a few details, the solution inside the
sphere is: while outside the sphere:
At large distances, φ> → φ∞ so B = -E∞ . Continuity of potential and of the
radial component of displacement D = κε0E determine the other two constants.
Supposing the radius of the sphere is R, As a consequence, the potential is:
which is the potential due to applied field and, in addition, a dipole in the
direction of the applied field of dipole moment:
or, per unit volume: The factor/(κ+2) is called the
Clausius-Mossotti factor and shows that the induced polarization flips sign if κ