Dipole

The concept of electric - and to a lesser extend - magnetic **dipole** is important for classical electrodynamics and the understanding of interaction of matter with electromagnetic radiation, like for example the understanding of molecular emission spectra.

Let us assume that there is a time independent distribution $\rho$ of electrical charges contained in a sphere $S$ around the origin of a cartesian coordinate system. This charge distribution generates an electric potential

$\Phi(\vec{x}) = \frac{1}{4 \pi \epsilon_0} \int \frac{\rho(\vec{y})}{ \| \vec{x} - \vec{y} \|} d \vec{y}$

When we describe the electric potential as a function of spherical coordinates $\Phi(r, \phi, \theta)$, far away from the sphere $S$, it will decrease proportional to $\frac{1}{r}$, so that we can express it as a series like

$\Phi(r, \phi, \theta) = \sum_{n = 1}^{\infty} f(\phi, \theta) \frac{1}{r^n}$

For an electric charge flying by the sphere $S$, it will be important to know the coefficients of the first summands of this series in order to determine the most important effects that will influence the charge.

When we choose as an orthonormal basis on a sphere the spherical harmonics $Y_{l m}(\phi, \theta)$ , with proper normalization we get what is called the **multipole expansion** of the electric potential:

$\Phi(r, \phi, \theta) = \frac{1}{4 \pi \epsilon_0} \sum_{l = 0}^{\infty} \sum_{m = -l}^{l} \frac{4 \pi}{2 l +1} q_{l m} \frac{Y_{l m}(\phi, \theta)}{r^{l+1}}$

The $l = 0$ term is called the **monopole term**, it is proportional to the electric charge $q$ contained in the sphere $S$. So the first term in the expansion tells us if a charge flying by $S$ will feel a net attractive or repulsive force.

The terms for $l = 1$ form a vector $\vec{p}$ which is called the **dipole moment**. The next terms in the series $Q_ij$ form the **quadrupol** tensor. So, for the expansion of the potential we get

$\Phi(r, \phi, \theta) = \frac{1}{4 \pi \epsilon_0} (\frac{q}{r} + \frac{\vec{p} \cdot \vec{x}}{r^3} + \frac{1}{2} \sum Q_{ij} \frac{x_i x_j}{r^5} + \cdot \cdot \cdot)$

For atoms and molecules the net charge $q$ is zero, so the next relevant term in the series expansion of their electric potential is the dipole moment.

The vector $\vec{p}$ is calculated via

$\vec{p} = \int \vec{x} \rho(\vec{x}) d \vec{x}$

which has a very simple geometric interpretation: It is the vector pointing from the center of negative charge to the center of positive charge. As an example, put a negative elementary charge at the origin represented by the density $-e \delta_0( \vec{x})$, and a positive one outside of the origin represented by $e \delta_{\vec{x}_0}(x)$, where $\delta_{\vec{y}}$ is the Dirac delta function supported at $\vec{y}$. Then we get

$\vec{p} = \int \vec{x} e (\delta_{\vec{x}_0}(x)- \delta_0( \vec{x})) d \vec{x} = \vec{x}_0$

- John David Jackson:
*Classical Electrodynamics*(Wiley; 3 edition (August 10, 1998))