Dipole in an
Electric Field
A uniform
electric field exerts a force on negative charges that is in a
direction opposite to the force on positive charges, resulting in
separation of the charges in a material. The equilibrium position is
determined by the competition between the force due to the applied
field, which acts to separate the charges, with the attractive force
between the opposite charges.
As
shown above, a uniform electric field results in a dipole moment. The
work done in separating the two charges, of magnitude q, by a
distance a is given by
so the energy is a linear function of the
applied field and a linear function of the induced dipole moment aq.
Quadrupole Moment in a Field Gradient
When the field is non-uniform in space, the electric
force on a charge depends on its location. This results in an induced
dipole moment that varies from place to place.
In the diagram above, the difference in the
electric field at two nearby points is shown to be quantified by a the
gradient of the electric field in the x-direction. The
resulting dipoles that point in opposite directions is quantified by
the quadrupole moment.
The work done by the electric field
gradient in separating the charges, of magnitude q, by a
distance a is given by
The charges reach their equilibrium
arrangement when the force exerted by the field gradient matches the
internal attractive and repulse force between each pair of charges.
Energy of Multipoles in an Electric Field
We have seen from the description above that an applied
electric leads to a rearrangement of the charges. A uniform electric
field induces an electric dipole moment and a field gradient induces an
electric quadrupole moment. Similarly, the second derivative of an
electric field will induce an octupole moment, and so on.
The work done by the field is then given by
The charge distribution comes into
equilibrium when the work done by the applied field is equal to the
work done by the internal fields.
Properties of the Induced Dipole Moment
A broad range of nonlinear phenomena
originates in the induced dipole moment in response to a time varying
electric field. As such, we will first focus on the dipole
approximation, which assumes that all higher-order moments do not
contribute.
The physical picture of such phenomena can
be framed in the most general terms as follows. An electric field is
applied to a material by illuminating it with a single color or
multiple colors of light (for example using laser beams) in the
presence of static or slowly varying fields, (for example, applied with
electrodes connected to a power supply).
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A Plane Wave in Matter
In response to an electric field, the
charges in a material separate to form a dipole moment, which adds to
the applied electric field.
For illustration, consider a plane
electromagnetic wave that passes through a molecule. The figure below
shows three snapshots at consecutive time intervals. The blue lines are
planes of maximum field amplitude.
As shown in the diagram above, the induced
dipole moment (black arrow) and the resulting dipole field (red
pattern) is maximal where the electric field is at its peak. The
quadrupole field due to the induced quadrupole moment is the strongest
where the field gradient is the largest. In most materials, the
time-averaged quadrupole field is much smaller than the time-averaged
dipole field.
In a linear material, the magnitude of the
induced dipole moment is a function of the strength of the electric
field at that point. The diagram below shows a plane wave traveling
through a material and only shows those dipoles that are induced in the
molecules that are at the location of the peak electric field. Since
the wave moves to the right, the pattern of induced dipole moments and
fields will follow the wave.
The total electric field in the material is
the sum of the electric field of the light beam and the electric field
from all of the induced dipole fields. When all of the fields are added
together, the result is depicted in the diagram below, there the
vertical liens show planes of maximum field amplitude and the material
is represented by the gray-shaded area.
The consequence of the addition of the
fields leads to the following properties:
- The planes of constant electric field inside the
material are closer together (i.e the wavelength is shorter) and the
wave travels more slowly than in vacuum.
- The radiating dipoles at the incident surface lead to
a reflected wave
- In transparent materials, the sum of the energy of
the reflected and transmitted wave equals the incident energy.
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