Dipole

The dipole is the second term (\(n=1\)) of the multipole expansion. This is useful when at large \(r\) when the total charge at large distance is zero.

\[\begin{split} V(\boldsymbol{r}) = \frac{1}{4\pi\epsilon_0}\frac{1}{r^2}\cdot\int{(r'\cos\alpha)\rho(\boldsymbol{r'}) d\mathcal V}\\ V(\boldsymbol{r}) = \frac{1}{4\pi\epsilon_0}\frac{\boldsymbol{\hat r}}{r^2}\cdot\int{\boldsymbol{r'}\rho(\boldsymbol{r'}) d\mathcal V} \end{split}\]

The integral is called the dipole moment,

\[ \boxed{\boldsymbol{p} \equiv \int{\boldsymbol{r'} d\mathcal q} = \int{\boldsymbol{r'}\rho(\boldsymbol{r'}) d\mathcal V}} \]

resulting in the equation of the dipole potential,

\[ \boxed{V(\boldsymbol{r}) = \frac{1}{4\pi\epsilon_0} \frac{\boldsymbol{p} \cdot \boldsymbol{\hat r}}{r^2}} \]

Physical Dipole

The physical dipole is a simplest dipole of two point charges opposite charge with mangitude \(q\) and distance vector \(\boldsymbol d = \boldsymbol r'_+ - \boldsymbol r'_-\), the dipole moment is given by,

\[ \boxed{\boldsymbol p = q \boldsymbol d} \]

Torque

For the dipole, if a uniform electric field was applied a torque may be experience with,

\[ \boxed{\boldsymbol \tau = \boldsymbol{p} \times \boldsymbol{E}} \]

This behavior often describes what if an electric field is applied to a molecule with some polarity

Polarization

A polarized object has a quantity called the polarization defined as,

\[ \boxed{\boldsymbol P \equiv \frac{\mathrm d \boldsymbol{p}}{\mathrm d \mathcal V}} \]

However, the polarization is also a cross-sectional area charge density since \(\mathrm d \boldsymbol p = \boldsymbol r\ \mathrm d q\),

\[\begin{split} \mathrm d \boldsymbol \sigma = \boldsymbol r \mathrm d \mathcal V\\ \boldsymbol P = \boldsymbol r \frac{\mathrm d q}{\mathrm d \mathcal V} = \frac{\mathrm d q}{d \boldsymbol \sigma} \end{split}\]

Bound Charges

For a distriubtion of dipole we may integrate the potential of the dipole along the infinitestimal geometry. Let’s use a 3D object with a 2D surface with infinitestimal \(\mathrm d \mathcal V\) and \(\mathrm d A\) respectively. Let the object be centered at the origin and the potential point be at distance \(d\) away from the origin such that \(d \gg r'\) for every point on the distribution. The integral form of the dipole potential is,

\[\begin{split} \mathrm dV(\boldsymbol{r}) = \frac{1}{4\pi\epsilon_0} \frac{\boldsymbol{\hat r} \cdot \mathrm d \boldsymbol{p}}{d^2} \\ V(\boldsymbol{r}) = \frac{1}{4\pi\epsilon_0} \int \frac{\boldsymbol{\hat r} \cdot \boldsymbol{P}(\boldsymbol r')}{d^2} \mathrm d \mathcal V \end{split}\]

The potential can be derived into two terms,

\[\begin{split} \sigma_b \equiv \boldsymbol{P} \cdot \boldsymbol{\hat n}\\ \rho_b \equiv -\nabla \cdot \boldsymbol{P}\\ V(\boldsymbol{r}) = \frac{1}{4\pi\epsilon_0}\Bigg[\oint_{S} \frac{\sigma_b}{d}\mathrm d A + \int_\mathcal{V}{\frac{\rho_b}{d}\mathrm d \mathcal V}\Bigg] \end{split}\]

• \(\sigma_b\) : Charge density of the surface

• \(\rho_b\) : Charge density of the volume