Tensor of electromagnetic field

[Petar Mali]

$$F_{\mu\nu}=\frac{\partial A_{\nu}}{\partial x^{\mu}}-\frac{\partial A_{\mu}}{\partial x^{\nu}}$$

$$F_{\mu\nu}=-F_{\nu\mu}$$

$$F_{ii}\equiv 0$$

$$F_{11}=F_{22}=F_{33}=F_{44}=0$$
where

$$A_{\mu}=(-\vec{A},\frac{1}{c}\varphi)$$$$(F_{\mu\nu})=\left(\begin{array}{cccc} 0& -B_z&B_y& -\frac{1}{c}E_x\\ B_z&0&-B_x& -\frac{1}{c}E_y \\ -B_y&B_x&0&-\frac{1}{c}E_z\\ \frac{1}{c}E_x& \frac{1}{c}E_y & \frac{1}{c}E_z & 0\\ \end{array} \right)$$$$F^{\mu\nu}=g^{\mu\rho}g^{\nu\sigma}F_{\rho\sigma}$$$$(F^{\mu\nu})=\left(\begin{array}{cccc} 0& -B_z&B_y& \frac{1}{c}E_x\\ B_z&0&-B_x& \frac{1}{c}E_y \\ -B_y&B_x&0&\frac{1}{c}E_z\\ -\frac{1}{c}E_x& -\frac{1}{c}E_y & -\frac{1}{c}E_z & 0\\ \end{array} \right)$$

How do I know that $$rotA_{\mu}=\frac{\partial A_{\nu}}{\partial x^{\mu}}-\frac{\partial A_{\mu}}{\partial x^{\nu}}$$ is electromagnetic field tensor?

 

[tiny-tim]

Hi Petar!

How do I know that $$rotA_{\mu}=\frac{\partial A_{\nu}}{\partial x^{\mu}}-\frac{\partial A_{\mu}}{\partial x^{\nu}}$$ is electromagnetic field tensor?

I don't understand your question …

isn't A_µ defined as the potential of the electromagnetic field tensor (in which case that has to be rotA)?

 

[Petar Mali]

Well its all ok for me except why I say that $$F_{\mu\nu}=\frac{\partial A_{\nu}}{\partial x^{\mu}}-\frac{\partial A_{\mu}}{\partial x^{\nu}}$$

is EM field tensor. Why not

$$F_{\mu\nu}=\frac{\partial A_{\mu}}{\partial x^{\nu}}-\frac{\partial A_{\nu}}{\partial x^{\mu}}$$

for example?

Or some other functions?

It's like a postulate. EM field tensor is $$F_{\mu\nu}=\frac{\partial A_{\nu}}{\partial x^{\mu}}-\frac{\partial A_{\mu}}{\partial x^{\nu}}$$

Let's form a matrix. No problem. Components of that matrix are electric field components and magnetic field components. Ok. That have sence. But how I know to start with $$F_{\mu\nu}$$. Do you know perhaps history of this problem.

 

[tiny-tim]

Why not

$$F_{\mu\nu}=\frac{\partial A_{\mu}}{\partial x^{\nu}}-\frac{\partial A_{\nu}}{\partial x^{\mu}}$$

for example?

That's minus the electromagnetic field tensor, so yes, it'll also be an electromagnetic field tensor.

Let's form a matrix. No problem. Components of that matrix are electric field components and magnetic field components. Ok. That have sence. But how I know to start with $$F_{\mu\nu}$$. Do you know perhaps history of this problem.

As you say, E and B are the 6 components of F.

F has to be a tensor because experiment tells us that is the way E and B transform in different frames.

 

[sardar]

As a scientist, you may know that the electromagnetic field is described by the electromagnetic field tensor, which is a mathematical object that describes the electromagnetic field in terms of its components. The equation given, F_{\mu\nu}=\frac{\partial A_{\nu}}{\partial x^{\mu}}-\frac{\partial A_{\mu}}{\partial x^{\nu}}, is the definition of the electromagnetic field tensor, where A_{\mu} is the electromagnetic potential and \frac{\partial A_{\nu}}{\partial x^{\mu}} is the derivative of the potential with respect to space-time coordinates.

This equation shows that the electromagnetic field tensor is a symmetric tensor, as F_{\mu\nu}=F_{\nu\mu}. This symmetry is important because it represents the fundamental symmetry of the electromagnetic field, known as the Maxwell's equations.

The components of the electromagnetic field tensor also follow certain patterns, such as F_{ii}=0, which means that the diagonal components of the tensor are all equal to zero. This is a consequence of the fact that the electric and magnetic fields are perpendicular to each other, and the diagonal components represent the interaction between these fields.

Furthermore, the electromagnetic field tensor is invariant under Lorentz transformations, which means that it has the same form in all inertial frames of reference. This is a fundamental property of the electromagnetic field, as it allows for the consistent description of electromagnetic phenomena in different frames of reference.

In conclusion, the equation given is the definition of the electromagnetic field tensor, and the properties and patterns observed in its components confirm that it is indeed the mathematical object that describes the electromagnetic field.