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\epsilon0} \sum{i=1}^{n} \frac{qi}{ri^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: \ointS E \cdot da = \frac{Q{enc}}{\epsilon_0}

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

• Electric flux: \PhiE = \intS 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}}{\parallel} = E^{\text{below}}{\parallel}

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

• Energy density in electrostatic fields: W = \frac{\epsilon0}{2} \int{\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 = \rhof with D = \epsilon0 E + P

• Linear dielectric materials: P = \epsilon0 \chie 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{\mu0}{4 \pi} \int \frac{I dl' \times \hat{r}}{r^2} = \frac{\mu0}{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 = \mu0 I{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 = \mu0 J + \mu0 \epsilon0 \frac{\partial E}{\partial t} *Displacement current: \epsilon0 \frac{\partial E}{\partial t}