Covariant Electrodynamics

Special relativity demands that Maxwell's equations take the same form in all inertial frames. The covariant formulation uses the four-vector potential Aμ and the field tensor Fμν to express all of electrodynamics in manifestly Lorentz-covariant form.

Key Concepts

Four-vector potential: Aμ = (φ/c, A)
Electromagnetic field tensor: Fμν = ∂μAν - ∂νAμ
Covariant Maxwell: ∂νFμν = μ₀Jμ
Lorentz force in covariant form: dpμ/dτ = qFμνuν
Field transformation under boosts: E and B mix

Key Equations

Four-potential

A^\mu = (\phi/c,\, \mathbf{A})

Field tensor

F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu

Covariant Maxwell

\partial_\nu F^{\mu\nu} = \mu_0 J^\mu

E'·B' invariant

\mathbf{E}\cdot\mathbf{B} = \text{invariant},\quad E^2 - c^2B^2 = \text{invariant}

Worked Example

Example Problem

Problem

A frame S has E=1000 V/m x̂, B=0. Frame S' moves at v=0.6c along x. Find E'_x and B'_y.

Solution

E'_x = E_x = 1000 V/m (parallel component unchanged). B'_y = γ(-v/c²)E_x = -γ×0.6c/c²×1000. γ=1.25. B'_y = -1.25×0.6×1000/3e8... wait: E'_⊥ = γ(E+v×B)_⊥. B'_y = -γvE_x/c² = -1.25×0.6c×1000/c² = -750/c = -2.5×10⁻⁶ T.

Practice

Exercises

7 problems

1 of 7

The Lorentz invariant E²-c²B² for E=500 V/m, B=0 in a frame. Find the invariant in V²/m².

Your answer

V²/m²

2 of 7

In frame S: E=0, B=0.01 T x̂. Frame S' moves at v=0.8c along x. Find E'_y (= -γvB_x... wait E'_⊥ = γ(E+v×B)_⊥). What is E'_z if E_z=0 and B_x=0.01, v along x: (v×B)_z = v_x B_y - v_y B_x = 0. Hmm. Let me use: B along x, motion along x → E'_y = γ(E_y - vB_z) and E'_z = γ(E_z + vB_y). B only has x-component → E'_y and E'_z unchanged: 0. Find B'_x in T.

Your answer

3 of 7

In frame S: E=0, B=B₀ŷ=0.01 T. Boost along x at v=0.6c (γ=1.25). Find |E'_z| in V/m.

Unlock Exercise 3