About the final oral exam

Chapter 1

• Differentiation, $\myv \grad$, $\myv \grad\cdot$, $\myv \grad\times$.
• Line, surface, volume integrals.
• Fundamental theorem for gradients (Relation of $\myv E$ to $V$).
• Fundamental theorem for divergence (basis of Gauss' law).
• Fundamental theorem for curl (basis of Ampere's law).
• Cartesian and non-Cartesian coordinate systems.
• Dirac delta functions.

Chapter 2 - Electrostatics

• Coulomb's law, the electric field, principle of superposition.
• Integrating over a continuous charge distribution to get $\myv E$.
• Electric field lines (qualitative), conventions for drawing
• Divergence of the electric field $\myv \grad\cdot E=\frac{1}{\epsilon_0}\rho$. Gauss' law: meaning and application. Fields in high-symmetry situations (planes of charge (including, often, the field between flat capacitor plates), spheres, cylinders). Gauss' law will also turn out to be useful for the "displacement" field, since its divergence is the free charge density.
• $\myv \grad \times \myv E=0$ (curlless field) implies that there's a potential $V$ such that $\myv E=-\myv \grad V$. Integrating $\myv E$ to get potential differences.
• Griffith's figure 2.35 is a summary of relations between $\myv E$, $V$, and $\rho$.
• Work / energy of charge distributions
• Conductors
• Capacitance

Chapter 3 - Potentials

• Laplace's equation, Uniqueness of solutions
• Method of images
• Spherical harmonics as solutions to Laplace's equations in S-P coordinates
• Multi-pole expansion of the potential, far from an isolated charge distribution.
• Calculating the monopole moment, and the dipole moment of a charge distribution, or of a collection of point charges.
• Field of a dipole.

Chapter 4 - Dielectrics

• Induced and permanent dipoles
• Polarization density, $\myv P$.
• Bound charge: $\sigma_b=\myv P\cdot \uv n$ and $\rho_b=-\myv \grad\cdot \myv P$.
• The electric displacement field $\myv D=\epsilon_0\myv E+\myv P$.
• Relationship of displacement to "free" charge, $\myv \grad\cdot\myv D=\rho_f$. Applying Gauss' law to the displacement implies $\oint\myv D\cdot d\myv a=(Q_f)_\text{enc}$.
• Sketching $\myv D$: Field lines only start or end on free charges. And what is the meaning of "free" charge?
• Linear dielectrics: $$\myv D \equiv \epsilon_0 \myv E + \myv P = \epsilon_0 \myv E + \epsilon_0 \chi_e \myv E = \epsilon \myv E,$$ where $$\epsilon=\epsilon_0(1+\chi_e)=\epsilon_0\epsilon_r$$ and $\epsilon_r$ is the "dielectric constant (a dimensionless number).
• The work needed to charge up a capacitor can be found by integrating $$W = \frac{1}{2}\int \myv D \cdot \myv E d \tau$$

Chapter 5 - Magnetic fields

• Right-hand rule for finding the direction of $\myv B$ from a current segment
• The Lorenz force on moving charges in a magnetic field $\myv F_\text{mag}=Q\myv v\times \myv B$.
• Cyclotron motion / cyclotron frequency
• Current and current density
• The Biot-Savart law: Calculating the magnetic field by integrating over the current.
• Magnetic field: divergence, $\myv \grad\cdot\myv B=0$; and curl $\myv \grad \times \myv B=\mu_0\myv J$.
• Applying Stokes' law to the curl gives Ampere's law: $$\oint \myv B \cdot d \myv l = \mu_0 \int \myv J \cdot d \myv a.$$ The right hand side is equal to $\mu_0$ times "$I_\text{enc}$", the total current "piercing" any area bounded by the closed path in the integral on the left hand side.
• Using Ampere's law to find the field by using an imaginary "Amperian loop" and Ampere's law.
• Magnetic moment or magnetic dipole moment, and the field of a magnetic dipole.

Chapter 6 - Magnetism / magnetization

• Paramagnetism, diamagnetism, ferromagnetism