# Doubt On Maxwell's Stress Tensor

I was reading Introduction to Electrodynamics By D.J. Griffiths Chapter 8 Conservation Laws , Maxwell's Stress Tensor.The starting lines are the following

Let's calculate the total electromagnetic force on the charges in volume $$\textit{V}$$ :

$$\mathbf{F}=\int _\mathit{V} ( \mathbf{E} + \mathbf{v} \times \mathbf{B} )\rho d\tau$$

$$=\int _\mathit{V} ( \rho \mathbf{E} + \mathbf{J} \times \mathbf{B} )d\tau .$$

My doubt is that '' Is $$\mathbf{E} \ \mathit{and} \ \mathbf{B}$$ are external fields or internal fields? If internal due to charge distribution of you are applying lorentz force? If external then How they are using the Maxwell's Equations where As I know $$\mathbf{E} \ \mathit{and} \ \mathbf{B}$$ are not the source terms but produced due to $$\mathit{J} \ \textit{and} \ \rho$$. Correct me if I am going in wrong direction.

• This is the Lorentz force, not the stress tensor. There is an error in the last expression. Jan 20, 2020 at 19:59
• Sorry! Now it's coreect. Jan 21, 2020 at 5:57
• What do you mean by internal/external fields? Jan 21, 2020 at 6:01
• I think you've got it the wrong way around: In the case of external fields, B and E are given by other factors than $\rho$ and J. For internal fields they are given by $\rho$ and J and it should be possible to solve this. Jan 21, 2020 at 10:59
• In my opinion, the electric and magnetic fields refer to all the fields, including the one produced by the charge distribution under consideration. Jul 30, 2021 at 2:33

Indeed ,E means electric field to to both free and bound charges . 1.how we are applying lorentz force - we are taking one electron (charged particle) as a system and calculating the force on it. And we are just applying the summation over to all electron .remember here individual electron is the system and not the whole body so you can apply lorentz force. Similarly you can apply for B (as explicitly B has been written in the book). So basically we dealing with interaction among themselves at the fundamental level. One can refer to Edward M purcell Chapter 1, art-no 1.14, to better understand "what is the electrical force experienced by the charges that make up distribution?"

• More or less what you are doing is to find the force on a electric due to it's own electric field. Jan 28, 2020 at 13:18
• I read in an older book there was some controversy (could also have been D instead of E) but it was settled in experiment. Unfortunately the experiment was not cited. Do you know a citation? Jan 30, 2020 at 7:32
• It's very unclear what you are asking. E and B are the E and B at the location of each charge.
– CWPP
Jul 30, 2022 at 15:57

The expression

$$\int _\mathit{V} ( \mathbf{E} + \mathbf{v} \times \mathbf{B} )\rho d\tau$$ is not the same as, and does not necessarily imply the expression

$$\int _\mathit{V} ( \rho \mathbf{E} + \mathbf{J} \times \mathbf{B} )d\tau$$

This is because current density $$\mathbf J$$ is not, in general, equal to $$\rho\mathbf v$$, where $$\mathbf v$$ is some velocity. We can have non-zero current density even at places where $$\rho=0$$, for example, when DC current flows through metallic wire - there is no single velocity, as positive charges are at rest, and only the negative charges move and form the current.

Derivation of the law of local conservation of momentum in EM theory, and of the "Maxwell stress tensor", works with total fields, and is based on the second expression above. Thus $$\rho,\mathbf J$$ are total charge density and total current density, and $$\mathbf E,\mathbf B$$ are total electric and total magnetic field.