Answer to Question #112890 in Atomic and Nuclear Physics for DEEPAK SINGH

Question

Prove Lorentz conoriance of the first equation of the electromagnetic field.
divB=0, and curlE= -db/dt

Answer 1

We have:

$\text{div}\textbf{B}=0,\\ \text{curl}\textbf{ E}=-\frac{\partial \textbf{B}}{\partial t}$

The first equation does not have time derivatives, we can satisfy this constraint by introducing a vector potential A which will comply with the condition

$\textbf{B}=\text{curl}\textbf{ A}.$

When we set

$\text{curl}\textbf{ E}=-\frac{\partial \textbf{A}}{\partial t},$

it implies a basic law of electromagnetism, Faraday's law:

$\text{curl}\textbf{ E}=-\frac{\partial \textbf{B}}{\partial t}.$

Reference

Michael Stone, Paul Goldbart (2009). Mathematics for Physics: A Guided Tour for Graduate Students. Publisher: Science.