FYS236 Electrodynamics Spring 2025 Lecture Notes Summary

Coulomb’s Law and Electric Field

• Coulomb's Law: $F = \frac{1}{4\pi\epsilon_0} \frac{qQ}{r^2} \hat{r}$

• Electric Field: $E(r) = \frac{1}{4\pi\epsilon<em>0} \sum</em>{i=1}^{n} \frac{q<em>i}{r</em>i^2} \hat{r}_i$

• Electric Field and Force: $F = QE$

• Permittivity of free space: $\epsilon_0 = 8.854 \times 10^{-12} \text{ F/m}$

Continuous Charge Distributions

• Electric field for continuous charge distributions:

  • Line charge: $E(r) = \frac{1}{4\pi\epsilon_0} \int \frac{\lambda(r')}{r^2} \hat{r} dl'$

  • Surface charge: $E(r) = \frac{1}{4\pi\epsilon_0} \int \frac{\sigma(r')}{r^2} \hat{r} da'$

  • Volume charge: $E(r) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(r')}{r^2} \hat{r} d\tau'$

Gauss’s Law

• Integral Form: $\oint<em>S E \cdot da = \frac{Q</em>{enc}}{\epsilon_0}$

• Differential Form: $\nabla \cdot E = \frac{\rho}{\epsilon_0}$

• Electric flux: $\Phi<em>E = \int</em>S E \cdot da$

Curl of E and Electric Potential V

• Curl of E for a point charge: $\nabla \times E = 0$

• Electric Potential: $V(r) = -\int_O^r E \cdot dl$

• Electric Field as Gradient of Potential: $E = -\nabla V$

• Potential Difference: $V(b) - V(a) = -\int_a^b E \cdot dl$

Electric Potential, Boundary Conditions, Energy

• Poisson's Equation: $\nabla^2 V = -\frac{\rho}{\epsilon_0}$

• Laplace's Equation (in charge-free regions): $\nabla^2 V = 0$

• Boundary Condition (tangential): $E^{\text{above}}<em>{\parallel} = E^{\text{below}}</em>{\parallel}$

• Boundary Condition (normal): $E^{\text{above}}<em>{\perp} - E^{\text{below}}</em>{\perp} = \frac{\sigma}{\epsilon_0} \hat{n}$

• Energy density in electrostatic fields: $W = \frac{\epsilon<em>0}{2} \int</em>{\text{all space}} E^2 d\tau$

Conductors

• Electrostatic properties:

  • E = 0 inside a conductor.

  • ρ = 0 inside a conductor.

  • Conductor is an equipotential body.

  • E is perpendicular to the surface.

• Capacitance: $C = \frac{Q}{V}$

Dipoles, Polarization, Electric Fields in Matter

• Dipole moment: $p = qd$

• Potential of a dipole: $V(r) = \frac{p \cdot \hat{r}}{4 \pi \epsilon_0 r^2}$

• The field of a polarized object:

  • Bound surface charge: $\sigma_b = P \cdot \hat{n}$

  • Bound volume charge: $\rho_b = -\nabla \cdot P$

• Gauss’s law with dielectric: $\nabla \cdot D = \rho<em>f$ with $D = \epsilon</em>0 E + P$

• Linear dielectric materials: $P = \epsilon<em>0 \chi</em>e E$ and $D = \epsilon E$

Magnetostatics

• Lorentz law: $F = q(E + v \times B)$

Biot-Savart Law and Maxwell Equations

• Biot-Savart Law: $B(r) = \frac{\mu<em>0}{4 \pi} \int \frac{I dl' \times \hat{r}}{r^2} = \frac{\mu</em>0}{4 \pi} \int \frac{J(r') \times \hat{r}}{r^2} d\tau'$

Ampere’s Law

• Ampère’s Law (integral form): $\oint B \cdot dl = \mu<em>0 I</em>{enc}$

• Magnetic Vector Potential: $B = \nabla \times A$

• Vector potential of a current loop: $A(r) = \frac{\mu_0}{4 \pi} \frac{m \times \hat{r}}{r^2}$

Magnetic Fields in Matter

• The auxiliary field H: $H = \frac{1}{\mu_0} B - M$

• Magnetic susceptibility and permeability:

  • $\vec{M} = \chi_m \vec{H}$

Electromotive Force

• Ohm’s law: $J = \sigma E$

• Resistance: $R = \frac{V}{I}$

• Electromotive force: $E = \oint f \cdot dl$

• Faraday's Law: $E = - \frac{d \Phi_B}{dt}$

Inductance, Displacement Current, Maxwell Equations

• Inductance: $L = \frac{\Phi}{I}$
  Maxwell’s repair of Ampère’s Law:
  *Ampère-Maxwell equation:
  $\nabla \times B = \mu<em>0 J + \mu</em>0 \epsilon<em>0 \frac{\partial E}{\partial t}$ *Displacement current: $\epsilon</em>0 \frac{\partial E}{\partial t}$