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PH504 – Part 4
1. Electrostatics: The electric dipole
Many physical systems are electrically neutral (
)
but still produce an electric field and are affected when
placed in an electric field (e.g. molecules).
This arises because the positive and negative
charges are physically separated. The simplest system
is the electric dipole.
The electric dipole – definition:
Consists of two equal and opposite charges Q separated
by a distance s.
In an ideal dipole s is very small compared to the
distances to any other charges and also to any points
where we wish to find the resultant electric potential or Efield.
However Q is sufficiently large such that the product Qs is
finite.
Potential and E-field of an ideal dipole
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Find potential V and electric field E at point P due to
charges Q.
Assume V = 0 at .
(A)
Use the cosine rule (or Pythagoras)
or
.
(B)
Using (B) to eliminate the term (r1-r2) from (A)
This result is exact for any dipole. However for a perfect
dipole s<<r and hence r1r2r and 2. Hence
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(C)
Potential of dipole falls as inverse square of distance.
In contrast to a single point charge the potential falls
off as r -2 (not r--1) .
Qs is defined as the electric dipole moment, symbol p.
This is actually a vector p=Qs where s is defined as
pointing from
-Q to +Q.
Equation (C) can hence be written in the form
or because
along r
where
is the unit vector
To find E we use E = -V but for convenience we use the
spherical form of .
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The dipole moment is aligned with the z-axis so that we
have 2 components (no azimuthal component):

The total E-field has a magnitude given by
and the angle  with respect to r is given by
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In contrast to a single point charge E falls off as r--3 for an
electric dipole (not r--2).
In Vector form:
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2. Dipole in a uniform E-field
Forces acting on the two charges (±Q) are +EQ and –
EQ.
So net force is zero. However there is a resultant
torque (T).
Torque = Force x perpendicular distance
But torque is a vector quantity whose direction gives the
axis of rotation which is normal to the plane containing p
and E. Hence
T= pxE
The direction of T is given by the right hand screw rule
(into the page in the present case).
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Force on a dipole in a non-uniform E-field
Simplest case is (a) for a dipole lying along the x-axis with
the field also along the x-axis.
Field at –Q is Ex and at +Q is Ex+dEx
 Fx = (-QEx) + (+Q(Ex+dEx)) = QdEx
but dEx = (dEx/dx).dx and dx=s

When the dipole does not lie along x-axis (Fig. (b)) then
the x component of the field at charge +Q is given by
Ex+(Ex/x)dx+(Ex/y)dy+(Ex/z)dz
So the net force along the x-axis is given by
where Qdx is the component of p along the x-axis etc.
Similar expressions exist for Fy and Fz.
In a non-uniform field the torque is still given by T=pxE unless the
E-field varies significantly over the spatial extent of the dipole (in
this case we would not have an ideal dipole).
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3. Potential energy of a dipole in an E-field
Need to calculate work done in rotating dipole from zero potential
position to new position
Work done = Torque x angle =
Choose zero energy configuration when both charges are on the
same equipotential line. Hence total energy is zero
Hence potential energy (U) of dipole p in a field E and an angle  is
 U = -pEcos = -p.E
3. Higher order poles
We can imagine a system where not only is
but also
(sum of dipole moments equals zero). Some examples
(known as quadrupoles) are
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(b) is a positive linear quadrupole and can be thought of as two
separate dipoles aligned end-to-end but with opposite direction.
In analogy to single charges (monopoles) and dipoles, the
potential due to a quadrupole falls off as r- -3 and the E-field as r- -4.
Dipole-dipole interactions are electrostatic interactions of
permanent dipoles in molecules. These interactions tend to align
the molecules to increase the attraction (reducing potential
energy).
An example of dipole-dipole interactions can be seen in hydrogen
chloride:
Conclusions

Concept and definition of a dipole

Potential due to a dipole

E-field due to a dipole

Torque on a dipole in a uniform E-field

Force on a dipole in a non-uniform E-field

Potential of a dipole in an E-field
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
Quadrupoles
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