Electrodynamics based on Maxwell equations (Bachelor level)

Explore essential math for electrodynamics by revisiting complex numbers, multidimensional derivatives, the nabla operator, gradient, curl, divergence, multidimensional integration, and spherical coordinates.

Explore what complex numbers are, introduce the imaginary unit i, and solve quadratics with real and imaginary roots while visualizing them as vectors in the complex plane.

Learn to add and subtract complex numbers and plot them as vectors in the complex plane. Use complex conjugates to extract real and imaginary parts and compute magnitude.

Learn how to multiply and divide complex numbers using polar representation, derive real and imaginary parts in products, and compute quotients with the complex conjugate and modulus squared.

Learn how to express complex numbers in polar form using the exponential representation, modulus and argument, and apply Euler-like relations to multiply and divide with ease.

Explore solving complex-number tasks by adding, multiplying, and dividing complex numbers, using conjugates and Euler form to obtain real and imaginary parts, magnitude, and reciprocal; visualize in the complex plane.

Learn how partial derivatives describe slopes of two-dimensional functions by varying one input, and use the nabla operator and gradient to compute directional derivatives via dot products.

Explore the nabla operator and its uses in vector calculus, including the gradient, divergence, and curl, with three-dimensional and cylindrical examples and physical interpretations.

Practice computing gradient, curl, and divergence in three dimensions, show curl of gradient is zero and divergence of curl is zero, and apply these results to electrostatics and Maxwell-based electrodynamics.

Explore multidimensional integrals by extending one-dimensional integration to two and three dimensions, using density interpretations and boundary parametrization to compute volumes and masses, including a pyramid example.

Explore line integrals in three-dimensional space by parameterizing a path, projecting onto one dimension, and evaluating scalar and vector field integrals with respect to the path, including closed-path cases.

Explore Cartesian, polar, cylindrical, and spherical coordinates, and learn how line, surface, and volume elements enable simple integration and describe vectors in problems like sphere volume.

Derive the volume element in spherical coordinates and use independent integrations to compute the sphere’s volume 4/3 pi r^3 and surface area 4 pi r^2.

Complete this two-hour mathematical section to be ready and understand most of what's coming next quite easily.

Explore the early theories of light and the pre-Maxwell landscape, including whether light is a wave or particle, and review charges, the electric field, Coulomb's law, magnets, and induction.

Explore early electrodynamics, including charges and electric and magnetic fields, tracing Newton's particle light theory to wave theory and Maxwell's equations, explaining reflection, refraction, diffraction, and interference.

Study charge, the electric field, and Coulomb's law, including the inverse-square force, the amber and fur experiments, and that the term electron derives from amber, plus Faraday's lines of force.

Explore the origins of magnetism, magnetic field lines around magnets and current-carrying wires, and how Ampere's law prompts a continuity term in Maxwell equations.

Explore electromagnetic induction: a time-varying magnetic flux induces an electric field and induction voltage, with examples like charging a battery and detecting vehicles via a moving magnet and coil.

Explore early electrodynamics: double-slit wave evidence and particle-like light; Coulomb's law, induction, and the missing term restoring the continuity equation in Maxwell's four electric and magnetic field equations.

Derive the electric fields of a point charge and a dipole via Coulomb's law, and analyze magnetic induction and Ampere's law for moving loops and wires.

Introduce Maxwell's equations, their motivation via symmetry and integral form, and the Lorentz force, establishing these equations as the basis of all electrodynamics.

Introduce Maxwell's equations by formulating charge density and current density as sources across space, showing how time-dependent densities determine the electric and magnetic fields.

Examine Maxwell's four differential equations, showing electric charges as sources, no magnetic monopoles, and how changing electric and magnetic fields induce each other via curl relations with μ0 ε0.

Derive the four integral Maxwell equations from the differential form using Gauss's and Stokes' theorems, linking E, B, rho, and J to volume, surface, and line integrals.

Explore the differential and integrated Maxwell equations, showing charges source electric fields, no magnetic monopoles exist, and changing magnetic flux and currents generate electric and magnetic fields.

Derive Coulomb's law from Maxwell's equations for a spherically symmetric charge, showing a radial field is proportional to 1/r^2 with 1/(4π ε0) and F = Q1 Q2 /(4π ε0 r^2).

Describe how charges and currents interact with electric and magnetic fields through the Lorentz force, and how special relativity yields the electromagnetic fields tensor within Maxwell's equations.

Show how energy conservation arises in electrodynamics from Maxwell equations by deriving the energy density E^2+B^2, the energy current (Poynting vector), and power density -J·E.

Explore Maxwell's equations in differential and integral forms, guided by symmetry. Then study special cases—electromagnetic waves in vacuum, electrostatics, and magnetostatics—before tackling time-dependent problems.

Explore how Maxwell's equations yield electromagnetic waves and illuminate why light behaves as a wave, by examining vacuum conditions where charge and current densities vanish.

Derive the electromagnetic wave equation in vacuum from Maxwell’s equations, showing zero divergence of E and B and zero currents, and obtain E and B plane-wave solutions with speed c.

Derive the dispersion relation for electromagnetic waves, linking frequency Omega to wave vector K via Omega = ± c K, and show how wave packets arise from superposing K modes.

Show how the propagation vector k orients the electric and magnetic fields in a plane wave, with E and B perpendicular to k and to each other.

Explore how a complex electric field with real and imaginary parts produces linear, elliptical, and circular polarization, while E and B stay perpendicular to the wavevector.

Solve exercises on electromagnetic waves by analyzing electric and magnetic field orientations, identifying circular and linear polarization, and deriving the magnetic field from E through the wave vector cross product.

Explore how Maxwell's equations in vacuum yield wave equations for E and B with a dispersion relation and perpendicular fields, and how polarization can be linear, circular, or elliptical.

Explore electrostatics as a static Maxwell regime with charges, introducing the electrostatic potential and solving for fields inside and outside charged spheres, spherical capacitors, dipoles, and boundary conditions.

Examine electrostatics by applying Maxwell's equations with time-independent fields, showing that the electric field's divergence equals charge density and its curl is zero, including the corresponding integral forms.

Derive the Poisson equation from Maxwell's equations in electrostatics by using E = -∇φ and the divergence relation to connect φ with the charge density.

Compute the electrostatic potential for a rotationally symmetric charge distribution by integrating the radial electric field, yielding a 1/r potential with an arbitrary constant set to zero.

Explore how electrostatic potential adds for multiple point charges and general charge distributions, using the sum of potentials and the integral form to compute potentials and introduce dipole analysis.

Derive the electric field and electrostatic potential of a uniformly charged sphere using Maxwell’s integral form and symmetry; obtain outside 1/r^2, inside proportional to r, with continuity at the surface.

Compute electric field and electrostatic potential for a spherical capacitor in three regions. Find the voltage between the metals and the capacitance, with C = 4 pi epsilon zero B/(B-A).

Introduce the electric dipole, two opposite charges separated by distance with dipole moment p, and derive the far-field potential, field, and the zero-potential line between charges via Taylor expansion.

Demonstrates how an electric dipole in a homogeneous field experiences zero net force but a torque that reorients its dipole moment.

Examine how boundary conditions fix electrostatic potentials and field lines, with metals enforcing constant surfaces. Explore mirror charges and how induced dipole-like fields shield nearby charges.

Derive the electrostatic energy from the energy density and the electrostatic potential, reducing to a double integral over charge distribution and addressing surface charges and two point charges.

Derive the electrostatic potential from Maxwell's equations, solve Poisson's equation for charge densities, and compute fields for a charged sphere, a spherical capacitor, and a dipole.

Learn magnetostatics under Maxwell's equations, focusing on magnetic fields from currents, the vector potential, and field formulas, with examples on straight wires, loops, and dipoles, and compare to electrostatics.

Examine Maxwell's equations for magnetostatics by setting time derivatives to zero. Contrast with electrostatics: magnetic fields have zero divergence and nonzero rotation, shaping their solutions.

Introduce the vector potential as a counterpart to the electrostatic potential, and explain gauge freedom by choosing div A = 0 to simplify curl equations for B.

Explore how the Biot-Savart law derives the magnetic field from current density using the vector potential, revealing B = ∇×A and the integral form.

Derive the magnetic field around a long, thin straight wire via the vector potential; B = mu0 I /(2 pi s) circles the wire and is independent of z.

Derives the magnetic dipole moment of a current loop from its vector potential, uses a Taylor expansion to obtain the far-field, and compares it to the electric dipole.

Explore magnetostatic energy by applying the energy density continuity equation to the magnetic field, or use the current distribution and vector potential A via Maxwell's equations to compute the energy.

Assess the force and torque on a magnetic dipole in a static homogeneous field, mirroring the electric case with a current loop and Lorentz force.

Compare magnetostatic results to electrostatics by examining Maxwell equations for magnetic fields, where div B is zero and curl B comes from currents, with a divergence-free vector potential.

Explore time dependent electromagnetism by combining electrostatic and magnetostatic results with retardation via retarded potentials, using the Hartzband dipole as a key example.

This lecture rewrites Maxwell's equations for time-dependent fields using the vector and scalar potentials, introducing the four-potential and Lorentz gauge.

Apply the Lorenz gauge to decouple the scalar and vector potentials. Show that E and B stay invariant under gauge transformations, while charge and current densities determine the potentials.

Demonstrate how retarded potentials solve time-dependent Maxwell equations by expressing the scalar and vector potentials in terms of charge density and current density with retardation, linking electrostatic and magnetostatic limits.

Verify that the electrostatic potential solves Maxwell's equation via retarded potentials, using the delta distribution and careful divergence and gradient calculus; this optional lecture reveals key mathematical tricks.

Derive the retarded potentials for a time-dependent Hertzian dipole using a delta-function current, obtaining the vector and electrostatic potentials under the Lorenz gauge.

Compute Hertzian dipole fields part 2/3 by deriving B from the curl of E with the retardation-aware vector potential, then obtain E via gradient terms, using spherical coordinates for simplification.

Analyze the near-field and far-field behavior of a harmonically oscillating Hertzian dipole by deriving electric and magnetic fields from retarded potentials and comparing with electrostatic results.

Explore how Maxwell's equations handle time-dependent cases by relating them to the electrostatic limit, using potentials, verifying solutions, and solving two independent problems with computational tools.

Learn how Maxwell's equations extend from vacuum to matter by separating the matter's influence and introducing polarization and magnetization, yielding Maxwell's equations in matter that closely resemble the vacuum form.

Explore how matter polarizes under fields, creating polarization charges that keep material neutral, and relate charge density changes to current via the continuity equation and divergence of electric dipole density.

Identify how polarization and magnetization separate charges and currents, introduce polarization and circular currents, and express circular currents as the curl of a magnetization vector, defining magnetic dipole density.

Introduce the displacement field D and the magnetizing field H to rewrite Maxwell's equations in matter, separating free rho0 and J0 from polarization charges and currents.

Celebrate completing the course and apply your new confidence to electrostatic and magnetostatic problems, polarization of light, and time-dependent Maxwell equations in matter.