Maxwell's Equations from EM field tensor

[Amentia]

Hello,

I have derived two Maxwell's equations from the electromagnetic field tensor but I have a problem understanding the second formula, which is:

$$\partial_{\lambda} F_{\mu\nu} + \partial_{\mu} F_{\nu\lambda}+\partial_{\nu} F_{\lambda\mu} =0$$

I have a few questions to help me start:
1) Is the summation convention used which means it is a single equation, or is it a set of equations?
2) Where does this formula come from? Can we derive it from something simple?
3) How should I start to get the Maxwell's equation? For my other calculation: $$\partial_{\mu}F^{\mu\nu}=J^{\nu}$$ it was clear to me that there was a summation and 4 equations but here I don't know if it is 1, 2, ..., 20 equations?

I have started to replace all the indices by all the possible values but it looks like horrible and I assume there must be some simple method. Since I will never have any correction elsewhere, I hope you can help me there to do that properly.

Thank you!

 

[TSny]

Hello,

I have derived two Maxwell's equations from the electromagnetic field tensor but I have a problem understanding the second formula, which is:

$$\partial_{\lambda} F_{\mu\nu} + \partial_{\mu} F_{\nu\lambda}+\partial_{\nu} F_{\lambda\mu} =0$$

I have a few questions to help me start:
1) Is the summation convention used which means it is a single equation, or is it a set of equations?

Since none of the indices are repeated in any term, there is no Einstein summation going on here. What you have is a set of equations, one equation for each choice of $\mu, \nu, \lambda$. That seems like a lot of equations! However, you should be able to show that no information results when any two of the indices take on the same value. (You just get 0 = 0.) Also, you should be able to see that permuting the indices does not give a different equation. ($\mu = 1, \nu = 2, \lambda = 3$ gives the same equation as $\mu = 2, \nu = 3, \lambda = 1$, say.) So, the number of independent equations is actually fairly small.

2) Where does this formula come from? Can we derive it from something simple?

I'm not sure how to answer this. If you already know Maxwell's equations in the usual form, then you can validate this formula by carrying out your exercise. Or, you could take this formula as an expression of a law of Nature which is just postulated.

3) How should I start to get the Maxwell's equation?

I have started to replace all the indices by all the possible values but it looks like horrible.

It's not that bad. After showing that you only need to consider cases where the three indices are different and that permutations of the indices give the same result, there are not that many possible values to look at.

 

[Amentia]

Since none of the indices are repeated in any term, there is no Einstein summation going on here. What you have is a set of equations, one equation for each choice of $\mu, \nu, \lambda$. That seems like a lot of equations! However, you should be able to show that no information results when any two of the indices take on the same value. (You just get 0 = 0.) Also, you should be able to see that permuting the indices does not give a different equation. ($\mu = 1, \nu = 2, \lambda = 3$ gives the same equation as $\mu = 2, \nu = 3, \lambda = 1$, say.) So, the number of independent equations is actually fairly small.

It's not that bad. After showing that you only need to consider cases where the three indices are different and that permutations of the indices give the same result, there are not that many possible values to look at.

Thank you, I was not sure about the convention, if the indices had to be repeated on each term or only several times in the equation. It makes more sense to me that it is actually a set of equations since I have to find two equations with this... And I have seen that some equations are 0=0 but I was getting lost with the permutations giving me several identical equations. I will try to do that carefully and eliminate the redundancies.

I'm not sure how to answer this. If you already know Maxwell's equations in the usual form, then you can validate this formula by carrying out your exercise. Or, you could take this formula as an expression of a law of Nature which is just postulated.

Of course, but what I ask is: imagine I am asked to find an equation that unifies two Maxwell's equations without knowing it. Is that doable in a few lines? I like to try thinking like the first physicists who developped a theory to understand better the topic.

 

[TSny]

what I ask is: imagine I am asked to find an equation that unifies two Maxwell's equations without knowing it. Is that doable in a few lines? I like to try thinking like the first physicists who developped a theory to understand better the topic.

I'm afraid I'm not good with this type of question. (My fault, not yours!)

 

[Amentia]

Ok no problem, it is more a side question, the main question was about the method for deriving the equations.