Topics covered: Electrodynamics II

(Guest lectures in Relativistic Electrodynamics)

• Lecture 1 :
  • Invariance of Maxwell's equations leading to Lorentz transformations
  • Constancy of speed of light leading to Lorentz transformations
• Lecture 2:
  • Velocity addition in relativity
  • Momentum conservation leading to a consistent definition of momentum (using elastic collision of identical particles)
  • Energy conservation leading to a consistent definition of relativistic energy (modulo an additive constant, which is provided by experiments)
  • (E,p) form a 4-vector
  • Relativistic Doppler shift: longitudinal Doppler shift, aberration
  • Definition of force and acceleration: acceleration need not be in the same direction as force
  • (rho, J) form a 4-vector
  • (phi, A) are solutions of a covariant equation (wave equation with Lorentz gauge) hence form a 4-vector
• Lecture 3:
  • "radius vector", Lorentz transformations, invariant (length)^2
  • Covariant and contravariant components
  • Transformations of covariant components
  • Definition of a 4-vector, covariant-contravariant components, invariants: length, scalar product
  • Differentiation opeator as a 4-vector
  • definition of a 4-tensor, covariant-contravariant-mixed components
  • Special symmetric 4-tensors: delta, g(ij), use for raising or lowering indices
  • Antisymmetric 4-tensor: components as "polar 3-vectors" and "axial 3-vectors"
  • Completely antisymmetric 4th order (pseudo)tensor epsilon, use for forming "dual" tensors
  • Definition of line element, area, (3-d) volume/ hypersurface, 4-volume
  • Gauss's theorem, Stokes' theorem, "unnamed" theorem
• Lecture 4:
  • 4-velocity, 4-accelation, their orthogonality
  • Lagrangian formulation of electrodynamics in steps:
  • Free particle: action [integral m ds], Lagrangian, momentum, energy in 3-d formalism (Euler-Lagrange equations of motion)
  • Free particle: equation of motion in 4-d formalism (deltaS=0)
  • Add matter-field interaction term [integral A.dx], Lagrangian, momentum, energy in 3-d formalism, Euler-Lagrange equation of motion: Lotentz force
• Lecture 5:
  • Equations of motion from the action in 4-d form (deltaS=0) gives 4-d analog of Lorentz force
  • Electromagnetic field tensor the only physical part of A
  • Gauge invariance in A
  • Multiple charges take [integral A.dx] to [integral A.J dOmega]
  • Action term for the field: [integral F_ij F^ij dOmega]
  • Lorentz invariant quantities (B^2 -E^2) and (E.B), physical interpretation
  • Maxwell's equations [delta_l Ftilde^lm = 0] from antisymmetry of Fij
  • Maxwell's equations [delta_k F^ik = -4 pi J^i] from [delta S =0]
  • Continuity equation as trivial consequence
  • The complete forlumation of classical ED: action, equations of motion (Lorentz, Maxwell)
• Lecture 6:
  • Particle in a uniform electric field: the trajectory is a cosh curve, not a parabola
  • Particle in a uniform magnetic field: Larmor frequency [e B/ E], not [e B / m]
  • Field due to a moving charge: Electric field "contracted" in the direction of motion and expanded perpendicular to it.
  • Particle in a coulomb field: qualitative differences from the Newtonian case