1. Insulators

Dielectric materials have an internal shift of charge, though overall each molecule still has zero net charge. This internal charge separation acts like a small electric dipole or pair of +/- charges.

  • An external electric field will cause forces that further tend to separate the two charges,

  • but the material is in a solid state

  • also, there is no net charge, so no net force tending to move the molecule.

  • This results in a torque that will tend to align the molecule’s dipole along the E-field’s direction or “along the field lines”.

Finally, there is a surface charge that is induced on each side of the material due to this rotation and alignment, the polarization charge \(\rho_v\). Those surface charges set up a polarization field \(\vec{P}\) that is (always?) in the opposite direction of the external \(\vec{E}\) field.

\(\vec{D} = \epsilon_0 \vec{E} + \vec{P}\)

^ what are these units?

1.1. Final version of Gauss’s Law

\[\nabla \bullet \mathbf{\vec{D}} = \frac{\rho_V}{\epsilon_0} \quad \text{Gauss's Law (point)}\label{eq-gauss-point}\]
\[\oint \mathbf{\vec{D}}\bullet\mathbf{\vec{dS}} = \frac{Q_{enc}}{\epsilon_0} \quad \text{Gauss's Law (integral)}\label{eq-gauss-integral}\]

1.2. Constitutive equations

  • Equations which describe the relationship between two related quantities. There will be 3.

\[\begin{align} \vec{D} &= \epsilon \vec{E} \\ &\text{ with } \epsilon = \epsilon_0 \epsilon_r \\ &\text{ and } \epsilon_r = (1 + \chi_e) \end{align}\]
\(\epsilon\) may change with applied field or from other causes. If so, then the equation is non-linear.
\(\epsilon\)

called

\(\epsilon_0\)

called

\(\epsilon_r\)

called

\(\vec{D}\)

called with units

2. Conductors

  • “All conductors are equipotential surfaces” — what does this mean?

  • …​ [the E-field lines] must arrive at the conductor in a perpendicular direction.

Describe how to take one of these statements and show that the other must also be true.