How does the retarded scalar potential satisfy the Lorentz gauge condition?

• kent davidge
In summary, the conversation discusses how to show that the retarded scalar potential satisfies the Lorenz gauge condition and the inhomogenous wave equation. The person speaking suggests that proving one of these assertions automatically proves the other, making it redundant to prove both. However, the importance of fulfilling the gauge condition and the solution to Maxwell's equations is emphasized. It is also noted that the gauge should be called the "Lorenz gauge" rather than the commonly used "Lorentz gauge."
kent davidge
As homework, I shall show that the retarded scalar potential satisfîes the Lorentz gauge condition as well as the inhomogenous wave equation. We saw in class how to do it. But I was thinking about this, and it seems to me that it's redundant to prove both of those things. For, if the scalar potential satisfies the Lorentz gauge condition, it will automatically lead to the inhomogenous wave equation. So if I show that the retarded scalar potential satisfies the inhomogenous wave equation, that automatically implies that it satisfies the Lorentz gauge condition. So why would I have to prove both assertions?

First of all to do historical justice one should call the gauge Lorenz gauge (after the Danish physicist Ludvik Lorenz) rather than Lorentz gauge (after the Dutch physicist Hendrik Antoon Lorentz).

I don't understand what you mean by "the retarded scalar potential satisfies the Loren(t)z gauge condition". A scalar potential alone cannot fulfill a gauge condition (except in the temporal gauge condition, ##\Phi=0##, but than in general you cannot in addition fulfill the Lorenz gauge condition if charges and currents are present).

If you impose the Lorenz gauge condition the four-potential separate component wise to wave equations,
$$\Delta A^{\mu}=\frac{1}{c} j^{\mu},$$
and the usually needed solution is the retarded one, using the corresponding Green's function of the D'Alembert operator. The solution then reads
$$A^{\mu}(x)=\int_{\mathbb{R}^4} \mathrm{d}^4 x' G_{\text{ret}}(x-x') \frac{1}{c} j^{\mu}(x').$$
It's important to check that this solution fulfills the gauge condition,
$$\partial_{\mu} A^{\mu}=0,$$
since otherwise these solutions do not lead to a valid solution of Maxwell's equations.

For details, see my SRT FAQ:

https://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

What is the Lorentz gauge condition?

The Lorentz gauge condition is a mathematical constraint used in the study of electromagnetic fields. It states that the divergence of the vector potential must be equal to the negative of the time derivative of the scalar potential, ensuring that the equations of motion for the electromagnetic fields are satisfied.

Why is the Lorentz gauge condition important?

The Lorentz gauge condition is important because it allows for a more simplified and elegant description of electromagnetic fields, reducing the number of independent variables from four to two. It also helps to ensure that the equations of motion are satisfied, making it a useful tool for theoretical and experimental studies of electromagnetism.

How is the Lorentz gauge condition derived?

The Lorentz gauge condition can be derived from Maxwell's equations, specifically the equation for the conservation of charge. By taking the divergence of both sides of this equation and using the identities for the divergence of a curl and the curl of a gradient, the Lorentz gauge condition can be obtained.

What are the implications of the Lorentz gauge condition?

The Lorentz gauge condition has several implications, including the fact that it helps to simplify the equations of motion for electromagnetic fields, making them easier to solve. It also ensures that the equations of motion are satisfied, providing a more accurate description of electromagnetic phenomena.

How is the Lorentz gauge condition used in practical applications?

The Lorentz gauge condition is used in a variety of practical applications, such as in the study of electromagnetic waves, the design of antennas, and the development of electromagnetic field theories. It is also used in the analysis of experimental data, helping to ensure that the results are consistent with the equations of motion for electromagnetic fields.

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