Maxwell Equations: Deriving 1 Equation from 2

[ghery]

Hi:

In electromagnetism, Maxwell equations originally were 6, with the aid of vector analysis, these equations were simplified and they became 4, after that with the aid of special relativity and tensor analysis (for the electromagnetic tensor) they became 2.

Now I have seen (I don't remember where) that these two equations became just one without any loss of information, Does anybody know how to derive these equation?, What is that equation?, and by the way what other mathematical tools do I need in order to derive it?

Thanks for your support

 

[ismaili]

Hi:

In electromagnetism, Maxwell equations originally were 6, with the aid of vector analysis, these equations were simplified and they became 4, after that with the aid of special relativity and tensor analysis (for the electromagnetic tensor) they became 2.

Now I have seen (I don't remember where) that these two equations became just one without any loss of information, Does anybody know how to derive these equation?, What is that equation?, and by the way what other mathematical tools do I need in order to derive it?

Thanks for your support

I guess what you want is Maxwell equations written in terms of differential forms. However, it takes two. You may first try
http://en.wikipedia.org/wiki/Maxwell's_equations
As for the basic introduction of forms, you can read Ryder's QFT book.

 

[astrokreng]

,

I can provide some insight into the process of deriving one equation from two Maxwell equations. The two equations in question are known as Maxwell's equations of electromagnetism and are fundamental laws that describe the behavior of electric and magnetic fields.

To derive one equation from two, we need to use the mathematical tools of vector calculus and special relativity. Vector calculus is a mathematical method used to describe and analyze vector fields, which are quantities that have both magnitude and direction. Special relativity is a theory that explains the relationship between space and time, and how they are affected by the speed of an object.

The first step in deriving one equation from two Maxwell equations is to combine the electric and magnetic fields into a single electromagnetic field. This can be done using the Lorentz force law, which describes the force on a charged particle in an electromagnetic field.

Next, we use the concept of electromagnetic potential, which is a mathematical construct that helps us describe the behavior of the electromagnetic field. By integrating the electromagnetic potential over space and time, we can derive the single equation known as the Maxwell-Faraday equation.

This equation states that the curl of the electric field is equal to the negative rate of change of the magnetic field over time. In simpler terms, it describes how changing magnetic fields can induce electric fields.

In summary, the process of deriving one equation from two Maxwell equations involves combining the electric and magnetic fields into a single electromagnetic field and using mathematical tools such as vector calculus and special relativity. The resulting equation, known as the Maxwell-Faraday equation, captures all the information contained in the two original equations.