Lorentz Force Equation: Coercion & Maxwell Stress Tensor

[goran d]

When deriving the Maxwell Stress tensor, the Lorentz formula is converted from point particle:
F=qE+qv x B
Into current and charge density:
F=ρ E + j x B
However an argument can be made that we can't "fieldify" both q and E at one step, and thus, a "coercion" of the field to a value is necessary.
F=ρ VAL (E) + j x VAL (B)
Where VAL(x) is the "coercion" function, which can't be differentiated with respect to space coordinates (the differentiation would give zero).
Under these conditions, the Maxwell Stress tensor is no longer conserving impulse.

 

[Dale]

However an argument can be made that we can't "fieldify" both q and E at one step

I have not seen this argument made in the professional scientific literature. Have you?

 

[PeterDonis]

When deriving the Maxwell Stress tensor, the Lorentz formula is converted from point particle:
F=qE+qv x B
Into current and charge density:
F=ρ E + j x B

Where are you getting this from?

 

[Dale]

Where are you getting this from?

Those are pretty standard, except that the second $F$ is a force density instead of a force, so it really should have a different variable than the first one which is a force.

https://en.wikipedia.org/wiki/Lorentz_force

Although I think it seems pretty bizarre to talk about "fieldifying" E which is already a field. Anyway, I think that you and I both agree that this topic needs some references.

 

[PeterDonis]

Those are pretty standard

I wasn't asking about the "Lorentz force" part. I was asking about the "when deriving the Maxwell stress tensor" part. The Maxwell stress tensor is derived from the EM field tensor, not from the Lorentz force.

 

[Dale]

I wasn't asking about the "Lorentz force" part. I was asking about the "when deriving the Maxwell stress tensor" part. The Maxwell stress tensor is derived from the EM field tensor, not from the Lorentz force.

Ah, yes, my mistake.

 

[PAllen]

I wasn't asking about the "Lorentz force" part. I was asking about the "when deriving the Maxwell stress tensor" part. The Maxwell stress tensor is derived from the EM field tensor, not from the Lorentz force.

While this is what I've seen, the wikipedia link @Dale provided contains the following in the section on Lorentz force for continuous charge distribution:

" By eliminating ρ {\displaystyle \rho }

and J {\displaystyle \mathbf {J} }

, using Maxwell's equations, and manipulating using the theorems of vector calculus, this form of the equation can be used to derive the Maxwell stress tensor σ {\displaystyle {\boldsymbol {\sigma }}}

, in turn this can be combined with the Poynting vector S {\displaystyle \mathbf {S} }

to obtain the electromagnetic stress–energy tensor T used in general relativity.[9] "

The source referenced is Griffith's electrodynamics, which I don't have a copy of.

 

[PeterDonis]

the following in the section on Lorentz force for continuous charge distribution

Eliminating $\rho$ and $\mathbf{J}$ using Maxwell's equations means writing everything in terms of the fields. Deriving the Maxwell stress tensor that way is basically the same as deriving it in terms of the fields.

In fact, in the Wikipedia article on the Maxwell stress tensor, the derivation actually doesn't even derive the stress tensor from the force; it derives a formula for the force in terms of the fields, and then introduces the Maxwell stress tensor (and the Poynting vector) in terms of the fields and uses that to simplify the force expression.

 

[PAllen]

Eliminating $\rho$ and $\mathbf{J}$ using Maxwell's equations means writing everything in terms of the fields. Deriving the Maxwell stress tensor that way is basically the same as deriving it in terms of the fields.

In fact, in the Wikipedia article on the Maxwell stress tensor, the derivation actually doesn't even derive the stress tensor from the force; it derives a formula for the force in terms of the fields, and then introduces the Maxwell stress tensor (and the Poynting vector) in terms of the fields and uses that to simplify the force expression.

All I was trying to point out is the the OP might have seen a source that did start from the Lorentz force forumula, and ultimately arrive at the EM stress energy tensor, then thinking that this was the normal way to do it.

 

[Dale]

In any case, it seems backwards to talk about "fieldifying" the Lorentz force. The continuous expression is the fundamental one. If you look at Maxwell's equations they are in terms of $\rho$ and $\vec J$, so the compatible form of the Lorentz force law is the one to start with. You then "particleize" that one by setting $\rho=q \ \delta(\vec r - \vec r_0)$ and $\vec J = q \ \vec v \ \delta(\vec r - \vec r_0)$ and then integrating over all $\vec r$

 

[PeterDonis]

the OP might have seen a source

Possibly, but if so, I would like the OP to tell us what source, as I have already asked. No response as yet.

 

[vanhees71]

Indeed, in electrodynamics the continuum-mechanical version to treat matter is conceptually much simpler. The real (only partially solved) problems start when trying to describe point particles!