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PH504
Electrostatics: The electric dipole
Introduction
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 E-field. However Q is
sufficiently large such that the product Qs is finite.
Potential and E-field of an ideal dipole
Wish to 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
(C)
In contrast to a single point charge the potential falls off as r
(not r-1)
-2
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
where
is the unit vector along r
To find E we use E = -V but for convenience we use the spherical
form of .
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
In contrast to a single point charge E falls off as r-3 for an electric
dipole (not r-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).
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).
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
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
(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.
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

Quadrupoles