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.