How to derive Maxwell stress-energy tensor

[aiaiaial]

The problem statement is:

Assuming that we are in vacuum, and that the only work done between mechanical systems and
electricity and magnetism comes from the Lorentz force, give a full, relativistic derivation of the
Maxwell stress-energy tensor.

 

[ZetaOfThree]

Can you show us what you know and what you've done so far?

 

[aiaiaial]

@ZetaOfThree

i know the answer should be this

But the question is "how to derive that", our class have finished Griffiths Electrodynamics chapter 12 (relativity in electrodynamics) and now doing chapter 8 (conservation law). I really don't know where to get started.

@ZetaOfThree

 

[ZetaOfThree]

Well, my hint to you would be read the chapter... there you'll find a lot of help.

 

[paranormal]

The Maxwell stress-energy tensor is a mathematical representation of the energy and momentum associated with electromagnetic fields. It is derived from the fundamental equations of electromagnetism, namely Maxwell's equations, and takes into account the effects of relativity. In order to derive the Maxwell stress-energy tensor, we must first define the Lorentz force and understand its relationship to energy and momentum.

The Lorentz force is the force experienced by a charged particle in an electric and magnetic field. It is given by the equation:

F = q(E + v x B)

where q is the charge of the particle, E is the electric field, v is the velocity of the particle, and B is the magnetic field. This force is responsible for the interactions between electromagnetic fields and mechanical systems.

In vacuum, the only work done between mechanical systems and electricity and magnetism comes from the Lorentz force. This means that the total energy and momentum of the system is conserved, and we can use this to derive the Maxwell stress-energy tensor. To do so, we will use the principle of energy-momentum conservation, which states that the total energy and momentum of a system must remain constant.

We start by considering a small volume element of space, dV, which contains an electric charge q. The electric and magnetic fields within this volume element can be written as:

E = E(x,y,z,t)
B = B(x,y,z,t)

Using the Lorentz force equation, we can calculate the force experienced by the charge q in this volume element as:

dF = q(E + v x B) dV

Since the total energy and momentum of the system is conserved, we can equate this force to the rate of change of energy and momentum within the volume element:

dF = dP/dt + dE/dt

where dP/dt is the rate of change of momentum and dE/dt is the rate of change of energy. We can then use the definitions of energy and momentum in terms of the electric and magnetic fields to write:

dF = (d/dt)(ε0E^2/2 + B^2/2μ0) dV + (1/c)(d/dt)(E x B) dV

where ε0 is the permittivity of free space and μ0 is the permeability of free space. This equation represents the energy and momentum density within the volume element.

To obtain the total energy and momentum