Electric Dipoles

For two equal, but opposite charges $+q,-q$ separated by a small distance $d$, dipole moment is:

$$\vec{p}=q\vec{d}$$

Note

$\vec{d}$ goes from the negative charge to the positive.

$$\begin{array}{rl}& V_p\approx k\frac{\vec{p}\cdot {\vec{u}}_r}{r^2}\\ & ⟹{\vec{E}}_p=k\frac{3(\vec{p}\cdot {\vec{u}}_r){\vec{u}}_r-\vec{p}}{r^3}\end{array}$$

Dipole in an Electric Field

A torque is applied since there are two opposite charges.

$$\begin{array}{rl}& {\vec{F}}_{\text{NET}}=\vec{0}\\ & \vec{\tau }=\vec{p}\times \vec{E}\end{array}$$

The torque aligns the charges with the field.

Polarization

Definition

When a neutral atom is in a magnetic field electrons and protons are forced in opposite directions, creating a dipole moment $\vec{p}$ which depends on field strength and atom properties $\alpha$. The total dipole moment of all atoms in a solid is called polarization $\mathcal{P}$.

For small intensities of fields we can approximate using Maclaurin Expansion:

$$\begin{array}{rl}& \vec{p}={\epsilon }_0\alpha \vec{E}+o(\vec{E})\\ & [\alpha ]=C^{2}m{N}^{-1}\\ & \mathcal{P}=n\vec{p}={\epsilon }_0\chi \vec{E}\end{array}$$

Where $\chi =n\alpha$ is the material’s electrical susceptibility.

Warning

There is a limit as to how much a dielectric can be polarized. After this limit (the field is greater than the dielectric strength), the atoms break down and become conductive.

When a material is polarized, it has its own field, so it will contribute to the overall field:

$${\vec{E}}_{\text{net}}={\vec{E}}_0+{\vec{E}}_P$$

Since ${\vec{E}}_P$ is opposite to the external field,

$$\begin{array}{rl}{E}_{\text{net}}& =E_0-E_P\\ & =E_0-\frac{{\sigma }_P}{{\epsilon }_0}\\ & =E_0-\frac{Q_P}{S{\epsilon }_0}\end{array}$$

Finally, we can say:

$$E=\frac{E_0}{1+\chi }$$

Where ${\epsilon }_r=\kappa =1+\chi$ is the relative dielectric constant of the material, or relative permittivity.

$$\epsilon ={\epsilon }_0{\epsilon }_r$$

Electrical Displacement Vector

Definition

The electrical displacement vector $\mathbf{D}$, is a vector field that represents the effect of free charges within dielectric materials, separating their contribution from bound charge polarization.

$$\begin{array}{rl}& \vec{D}\approx \epsilon \vec{E}\\ & ⟹{\oint }_S\vec{D}\cdot {\vec{u}}_{n}dS=Q_{\text{FREE}}\end{array}$$