Next: Charge Sheets and Dipole
Up: Electrostatic Fields
Previous: Electrostatic Energy
Electric Dipoles
Consider a charge
located at position vector
, and a charge
located at position vector
.
In the limit that
, but
remains finite, this combination of charges constitutes an
electric dipole, of dipole moment
![$\displaystyle {\bf p} = q\,{\bf d},$](img455.png) |
(195) |
located at position vector
. We have seen that the electric field generated at point
by an electric charge
located
at point
is
![$\displaystyle {\bf E}({\bf r}) = -\nabla\left(\frac{q}{4\pi\,\epsilon_0}\,\frac{1}{\vert{\bf r}-{\bf r}'\vert}\right).$](img456.png) |
(196) |
Hence, the electric field generated at point
by an electric dipole of moment
located at point
is
![$\displaystyle {\bf E}({\bf r}) = -\nabla\left(\frac{q}{4\pi\,\epsilon_0}\,\frac...
...frac{q}{4\pi\,\epsilon_0}\,\frac{1}{\vert{\bf r}-{\bf r}'+{\bf d}\vert}\right).$](img458.png) |
(197) |
However, in the limit that
,
![$\displaystyle \frac{1}{\vert{\bf r}-{\bf r}'+{\bf d}\vert} \simeq \frac{1}{\ver...
...vert}-\frac{{\bf d}\cdot ({\bf r}-{\bf r}')}{\vert{\bf r}-{\bf r}'\vert^{\,3}}.$](img459.png) |
(198) |
Thus, the electric field due to the dipole becomes
![$\displaystyle {\bf E}({\bf r}) =-\nabla\left(\frac{1}{4\pi\,\epsilon_0}\,\frac{{\bf p}\cdot({\bf r}-{\bf r}')}{\vert{\bf r}-{\bf r}'\vert^{\,3}}\right).$](img460.png) |
(199) |
It follows from Equation (154) that the scalar electric potential due to the dipole is
![$\displaystyle \phi({\bf r}) = \frac{1}{4\pi\,\epsilon_0}\,\frac{{\bf p}\cdot({\...
...silon_0}\,{\bf p}\cdot\nabla'\left(\frac{1}{\vert{\bf r}-{\bf r}'\vert}\right).$](img461.png) |
(200) |
(Here,
is a gradient operator expressed in terms of the components of
, but independent of the
components of
.)
Finally, because electric fields are superposable, the electric potential due to a volume distribution of electric dipoles
is
![$\displaystyle \phi({\bf r}) = \frac{1}{4\pi\,\epsilon_0}\int {\bf P}({\bf r'})\cdot\nabla'\left(\frac{1}{\vert{\bf r}-{\bf r}'\vert}\right) dV',$](img463.png) |
(201) |
where
is the electric polarization (i.e., the electric dipole moment per unit volume), and
the integral is over all space.
Next: Charge Sheets and Dipole
Up: Electrostatic Fields
Previous: Electrostatic Energy
Richard Fitzpatrick
2014-06-27