--- title: "Gauge Freedom and the Potentials as the Working Variables" description: "Different scalar and vector potentials can give the same electric and magnetic fields. A gauge condition restricts this freedom. Lorenz gauge gives sourced wave equations for the potentials; Coulomb g" canonical: https://lightmysky.com/learn/science/gauge-freedom-and-the-potentials-as-the-working-variables-mt_BovDzq3WZm source: https://lightmysky.com/learn/science/gauge-freedom-and-the-potentials-as-the-working-variables-mt_BovDzq3WZm.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`. # Gauge Freedom and the Potentials as the Working Variables Different scalar and vector potentials can give the same electric and magnetic fields. A gauge condition restricts this freedom. Lorenz gauge gives sourced wave equations for the potentials; Coulomb gauge gives a Poisson equation for the scalar potential. Subject: Science · Area: Electricity & Magnetism · Ages 22 to 24 Page: https://lightmysky.com/learn/science/gauge-freedom-and-the-potentials-as-the-working-variables-mt_BovDzq3WZm ## Ready when they can - Shows that a gauge transformation leaves the measured fields unchanged - Picks a gauge condition and states what it simplifies - Writes the sourced wave equation for the potentials in a chosen gauge ## Lesson: Many potentials, same fields For classical electromagnetism in vacuum with sources, use SI units: E = -grad phi - ∂t A and B = curl A. phi is the scalar potential and A the vector potential; ∂t is a time derivative. Different potentials can describe the same fields. E and B, and measurable predictions made from them, are gauge independent. For a smooth function chi(x,t), set A' = A + grad chi and phi' = phi - ∂t chi. The prime labels a new potential description in the same frame, not a Lorentz boost. The two shifts must be matched. An identity transformation may leave the potentials unchanged. *(drawing: Here dt chi means ∂t chi. A gauge change leaves field components unchanged in the same observer frame.)* The curl of a gradient is zero, so B' = curl(A + grad chi) = B. Also E' = -grad(phi - ∂t chi) - ∂t(A + grad chi) = E: grad ∂t chi cancels ∂t grad chi because smooth mixed derivatives commute. A spatially uniform vector potential has zero curl. The Lorenz condition is div A + (1/c²)∂t phi = 0. Define L = ∇² - (1/c²)∂t², where ∇² is the spatial Laplacian. Maxwell's equations give L phi = -rho/eps0 and L A = -mu0 j. Here rho and j are charge and current densities; eps0 and mu0 are vacuum constants, with 1/c² = mu0 eps0. These are sourced wave equations. *(drawing: L is the wave operator. Charge density sources the scalar equation; current density sources the vector equation.)* A gauge condition need not select unique potentials. Let G = div A + (1/c²)∂t phi. A matched gauge shift gives G' = G + L chi. Thus L chi = 0 preserves Lorenz gauge. These are residual gauge transformations; initial and boundary conditions can restrict them further. The Coulomb condition is div A = 0. Since div E = -∇²phi - ∂t(div A), Gauss's law gives ∇²phi = -rho/eps0. This Poisson equation applies to static or time-varying sources. It gives a direct route to phi for static charges, although Lorenz gauge is valid for statics too. With time-varying sources, its scalar solution can look instantaneous. *(drawing: These are changes in the two contributions to E. Neither contribution alone is gauge independent.)* For localized source changes with causal initial and boundary data, physical field changes reach distant points no faster than c. In Coulomb gauge, both -grad phi and -∂t A contribute to E. Their sum has the same causal response as in Lorenz gauge. The scalar potential alone is not the full time-dependent electric field. A physical answer can be a measured force or an energy transfer, as well as E or B. Such predictions must be gauge independent. Potential values may change with gauge; an absolute potential-energy value can also depend on its reference. Matching E and B from two gauges checks their agreement, not correctness. Check Maxwell's equations for the given sources and the initial and boundary conditions too. **Recap.** Matched potential shifts leave E and B unchanged. Lorenz gauge gives sourced wave equations. Coulomb gauge gives a scalar Poisson equation, not faster physical signals. *Lesson, questions, worked solutions and flashcards written for this dataset.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Covariant Electrodynamics and the Field Tensor](https://lightmysky.com/learn/science/covariant-electrodynamics-and-the-field-tensor-mt_RoxM9z6DfK) ## Optional connections - [Calculating electric potential from a charge distribution](https://lightmysky.com/learn/science/calculating-electric-potential-from-a-charge-distribution-mt_B4WiVkpH4j) ## Opens up - [Radiation from an Accelerating Charge](https://lightmysky.com/learn/science/radiation-from-an-accelerating-charge-mt_Qr0rMEOYAg)