Classical Electrodynamics: Explaining the Lorentz Gauge Condition

[nrjsingh413]

what is physical meaning of Lorentz gauge condition in classical electrodynamics??

 

[Jano L.]

Actually the constraint

$$
\frac{1}{c^2}\frac{\partial\phi}{\partial t} + \nabla \cdot \mathbf A = 0
$$

is due to Lorenz (Lorenz and Lorentz are easily confused):

https://en.wikipedia.org/wiki/Lorenz_gauge_condition

This equation is sometimes used because it leads to simple and symmetric wave equations for the scalar and vector potential, which are then easily solved for known charge and current distribution and initial conditions on the field.

The potentials are auxiliary functions without direct physical meaning. The meaning of the constraint is really just simplification of the relativistic equations so they become nice and simple.

 

[vanhees71]

You confuse me a bit with the speed-of-light factor. In relativistically covariant notation, it's
$$\partial_{\mu} A^{\mu}=0.$$
Split into temporal and spatial components this reads
$$\partial_0 A^0+\vec{\nabla} \cdot \vec{A}=\frac{1}{c} \partial_t \Phi + \vec{\nabla} \cdot \vec{A}.$$
This is, of course, in Heaviside-Lorentz units.

The good thing with this particular gauge, which should indeed be named after the Danish physicists Ludvig Lorenz instead of the Dutch physicist Hendrik Antoon Lorentz, because Lorenz was the first, using this gauge condition.

The physical merit of this particular gauge is clear: It's a Poincare invariant condition, leading to Poincare invariant equations of motion for the four-potential that at the same time separate into the components. This makes it particularly nice for radiation problems.

For other problems like the description of bound states in quantum mechanics other gauges are more convenient. In this case the Coulomb gauge is good.

It always depends on the physical problem you want to solve, what's the most appropriate gauge constraint. Choosing a gauge is an art comparable to the one to find the most convenient set of coordinates to solve a problem.

 

[andrien]

what is physical meaning of Lorentz gauge condition in classical electrodynamics??

Can you see that maxwell eqn are total 8 in numbers but there are only 6 quantities to determine.

 

[PJC01]

The Lorentz gauge condition is a mathematical constraint in classical electrodynamics that ensures the consistency of Maxwell's equations. It states that the divergence of the vector potential must be equal to the negative of the time derivative of the scalar potential, multiplied by the speed of light squared. This condition is named after the Dutch physicist Hendrik Lorentz, who first proposed it in the late 19th century.

The physical meaning of this condition can be understood by considering the nature of electric and magnetic fields. In classical electrodynamics, the electric and magnetic fields are described by the vector potential and scalar potential, respectively. These potentials are related to the electric and magnetic fields through Maxwell's equations.

The Lorentz gauge condition imposes a constraint on the vector potential, which in turn affects the behavior of the electric and magnetic fields. This constraint ensures that the fields are well-behaved and do not exhibit any unphysical behavior, such as infinite values or singularities. It also ensures that the fields satisfy the principle of causality, meaning that their values at a given point in space and time are determined only by their values at previous points in space and time.

In essence, the Lorentz gauge condition serves as a mathematical tool to maintain the consistency and physical validity of Maxwell's equations. Without this condition, the equations would not accurately describe the behavior of electric and magnetic fields, and the predictions made by classical electrodynamics would not match experimental observations. Therefore, the Lorentz gauge condition is a crucial aspect of classical electrodynamics that helps us understand and explain the fundamental principles of electromagnetism.