--- title: "Covariant Electrodynamics and the Field Tensor" description: "Collecting the fields into one antisymmetric tensor lets us write Maxwell's four equations as two. The split into electric and magnetic fields depends on the frame. Charge density and current become a" canonical: https://lightmysky.com/learn/science/covariant-electrodynamics-and-the-field-tensor-mt_RoxM9z6DfK source: https://lightmysky.com/learn/science/covariant-electrodynamics-and-the-field-tensor-mt_RoxM9z6DfK.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Covariant Electrodynamics and the Field Tensor Collecting the fields into one antisymmetric tensor lets us write Maxwell's four equations as two. The split into electric and magnetic fields depends on the frame. Charge density and current become a single four-vector. Subject: Science · Area: Electricity & Magnetism · Ages 22 to 24 Page: https://lightmysky.com/learn/science/covariant-electrodynamics-and-the-field-tensor-mt_RoxM9z6DfK ## Ready when they can - Assembles the field tensor from the electric and magnetic components - Shows how a purely electric field in one frame appears as a mixture in another - Writes Maxwell's equations in four-vector form and identifies which pair each one covers ## Lesson: One tensor for E and B Maxwell's laws extend steady-current loops to changing fields. We now describe E and B together using Lorentz transformations. Use SI units, flat spacetime, coordinates (ct,x,y,z), and metric eta = diag(1,-1,-1,-1). Greek indices run over 0,x,y,z; repeated indices are summed. The field tensor obeys F^{mu nu} = -F^{nu mu}, so its diagonal is zero. An antisymmetric n by n tensor has n(n-1)/2 independent entries: each off-diagonal pair supplies one value. With rows and columns ordered (0,x,y,z), set diagonal entries to zero. Use F^{0i} = -Ei/c and F^{i0} = Ei/c, with 1=x, 2=y and 3=z. Set F^{12} = -Bz, F^{23} = -Bx and F^{31} = -By. Fill each reversed-index entry with the opposite sign. These rules assemble F from the three electric and three magnetic components. *(drawing: Each cell lists one row of F, in order 0, x, y, z. The entries inside each cell follow that same column order.)* Let S' move with velocity v relative to S, with parallel spatial axes. If B=0 at an event in S, then E'_parallel = E_parallel, E'_perp = gamma E_perp, and B' = -gamma(v cross E)/c². Here gamma = 1/sqrt(1-|v|²/c²). Parallel and perpendicular refer to v. A nonzero transverse E produces B'; if v is parallel to E, B' remains zero. *(drawing: The minus sign in B' = -gamma(v cross E)/c² reverses the cross-product direction.)* The four-current is J^mu = (c rho, jx, jy, jz). Here rho is charge density and j is current density. The factor c gives all four entries the same units. Under a Lorentz transformation Lambda, J'^mu = Lambda^mu_nu J^nu. Charge-density and current-density components can mix between frames. In vacuum with charge and current sources, ∂_mu F^{mu nu} = mu0 J^nu. Here ∂_mu differentiates with respect to x^mu, so ∂_0 = (1/c)∂t. Vacuum constants obey 1/c² = mu0 eps0. The nu=0 component gives div E = rho/eps0. The spatial components give curl B - (1/c²)∂t E = mu0 j: Ampere-Maxwell's law. Lower indices using F_{mu nu} = eta_{mu alpha} eta_{nu beta} F^{alpha beta}. The homogeneous equation is ∂_lambda F_{mu nu} + ∂_mu F_{nu lambda} + ∂_nu F_{lambda mu} = 0. Its spatial-index part gives div B = 0. Its time-space parts give curl E + ∂t B = 0. This pair has no source term, even where electric charges or currents are present. *(drawing: Each tensor equation contains a pair of Maxwell laws. Homogeneous means that the equation has no explicit source term.)* Transform both indices: F'^{mu nu} = Lambda^mu_alpha Lambda^nu_beta F^{alpha beta}, or F' = Lambda F Lambda^T in matrix form. Read E'i = -c F'^{0i}, B'x = -F'^{23}, B'y = -F'^{31}, and B'z = -F'^{12}. Transform J too when sources are needed. Direct E and B transformation formulas are also valid when their boost and sign conventions match. **Recap.** F has six independent entries. Lorentz boosts can mix E and B. Its sourced and homogeneous equations contain the four Maxwell laws. *Lesson, questions, worked solutions and flashcards written for this dataset.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Spacetime Intervals and Four-Vectors](https://lightmysky.com/learn/science/spacetime-intervals-and-four-vectors-mt_CYdi7Qpvrf) - [Displacement Current and Maxwell's Equations](https://lightmysky.com/learn/science/displacement-current-and-maxwells-equations-mt_YR8m8TNfyy) ## Opens up - [Gauge Freedom and the Potentials as the Working Variables](https://lightmysky.com/learn/science/gauge-freedom-and-the-potentials-as-the-working-variables-mt_BovDzq3WZm)