Lorenz gauge

The Lorenz gauge is an electrodynamic gauge for Maxwell's equations where we set the Divergence of the magnetic vector potential to

∇⋅A=−μ0ε0∂V∂t=−1c2∂V∂t\nabla\cdot \mathbf{A}=-\mu_{0}\varepsilon_{0}\frac{ \partial V }{ \partial t }=- \frac{1}{c^{2}}\frac{ \partial V }{ \partial t }

The benefit of this specific choice is that it cancels out two terms in the potential formulation of Maxwell's equations. The differential equation for A\mathbf{A} then is

∇2A−μ0ε0∂2A∂t2=−μ0J\nabla ^{2}\mathbf{A}-\mu_{0}\varepsilon_{0}\frac{ \partial ^{2}\mathbf{A} }{ \partial t^{2} } =-\mu_{0}\mathbf{J}

and the one for VV is

∇2V−μ0ε0∂2V∂t2=−ρε0\nabla ^{2}V-\mu_{0}\varepsilon_{0}\frac{ \partial ^{2}V }{ \partial t^{2} } =- \frac{\rho}{\varepsilon_{0}}

The benefit of the Lorenz gauge is that it makes the equations for VV and A\mathbf{A} virtually identical; this can be rendered obvious by using the d'Alembertian operator □2\square ^{2}:

□2V=−ρε0,□2A=−μ0J\square ^{2}V=- \frac{\rho}{\varepsilon_{0}},\qquad\square ^{2}\mathbf{A}=-\mu_{0}\mathbf{J}

These two are inhomogeneous wave equations □2ψ=f\square ^{2} \psi=f. ff can be interpreted as a source term that's responsible for generating the electromagnetic wave, and in this gauge, the entirety of electrodynamics can be regarded as just the solution to these two equations (plus boundary conditions and the Lorentz force); everything else follows.

They also make the connection to electro/magnetostatics very clear: if we take away time dependence, the d'Alembertian becomes the Laplacian and the our two equations become the usual

∇2V=−ρε0,∇2A=−μ0J\nabla ^{2}V=- \frac{\rho}{\varepsilon_{0}},\qquad \nabla ^{2}\mathbf{A}=-\mu_{0}\mathbf{J}