# Lorentz force and Maxwell Equations

1. Jun 26, 2009

### sokrates

I am trying to bridge a gap between Maxwell equations and Lorentz force. I know that they are not independent and in theory, one could be derived from the other but I cannot see that.

More physics oriented people prefer the Lorentz force because it describes the effect of B and E as a final force on the particle, which is good.

But all that should also be captured by Maxwell's set. And in classical EM theory, they don't even mention Lorentz force.

Could anyone show a theoretical derivation or provide some remarks on this issue?

many thanks,

2. Jun 26, 2009

### Born2bwire

Oh, one nice way you can derive the Lorentz force is by taking the extremum of the action, just use good old Lagrangian physics to find the path of a charge through an electric and magnetic field. It should come out nicely, Feynman has a good derivation of it in his Quantum Mechanics and Path Integrals book if you are lucky enough to find a copy.

3. Jun 28, 2009

### Andrew Mason

Wikipedia has a good article on this: http://en.wikipedia.org/wiki/Lorentz_force

Someone has done a paper to show that the Lorentz force can be derived from Maxwell's equations: http://arxiv.org/abs/physics/0206022

AM

4. Jun 28, 2009

### Peeter

I was pointed to the stress energy tensor, which relates the two up to a point. For example, if you take the time derivative of the field energy density

$$U = \frac{\epsilon_0 (\mathbf{E}^2 + c^2 \mathbf{B}^2)}{2}$$

you'll can get after application of Maxwell's equations and some messy algebra

$$\frac{\partial}{\partial t}\frac{\epsilon_0 (\mathbf{E}^2 + c^2 \mathbf{B}^2)}{2} + c^2 \epsilon_0 \nabla \cdot ( \mathbf{E} \times \mathbf{B} ) = - \mathbf{E} \cdot \mathbf{j}$$

The $- \mathbf{E} \cdot \mathbf{j}$ term on the RHS is the energy term of the Lorentz force equation. If you do the same thing, taking time derivatives of the momentum density terms (the Poynting vector components $\mathbf{E} \times \mathbf{B}$), then the RHS terms will be of the form

$$\rho \mathbf{E} + \mathbf{j} \times \mathbf{B}$$

These are the momentum parts of the Lorentz force equation per unit volume. So, only considering derivatives of fields, one is able to see that there is a fundamental relationship between the Maxwell's equations and something that is like the Lorentz force, but without doing something else one still needs a way to relate this to the force on a mass.

5. Jun 28, 2009

### marcusl

The Lorentz force equation contains both electric and a magnetic force terms. It is possible to show that they are both manifestations of the same electric force, as seen from stationary and moving reference frames according to the laws of special relativity. The derivation is given in a simple way in Nobel laureate E.M. Purcell's first year undergraduate book ("Electricity and Magnetism").

Maxwell's equations may be derived in the same way, starting from Coulomb's law and using special relativity and symmetry arguments. The upper level undergraduate book M. Schwartz, Principles of Electrodynamics, by another Nobel laureate, lays it out in the first few chapters. Once he derives the Lorentz force, Schwartz needs only a few more pages to derive the vector potential, magnetic field, and then all of the Maxwell equations in vacuuo.

The unity of electricity and magnetism, and their relation through relativity, is one of the great beauties in physics. It is accessible to undergrads through the treatments mentioned above.

6. Jun 28, 2009

### cabraham

But the electric force can also be viewed as a manifestation of the magnetic force according to Einstein in his 1905 paper "On The Electrodynamics Of Moving Bodies". He stated that neither the electric nor magnetic force is the "seat", seat meaning root, or basis.

Electric and magnetic forces are inclusive. Either can be viewed as a manifestation of the other with regard to relativity. E is not more basic than H, nor less so.

Claude

#### Attached Files:

• ###### Einstein.pdf
File size:
203.4 KB
Views:
163
7. Jun 28, 2009

### marcusl

Fair enough. Having said that, however, there is considerable merit to starting with the electric force and not going the other way around. Coulomb's law is taught in high-school physics, and, being a central force, is intuitively reasonable. I venture that few beginners in physics are sufficiently comfortable with magnetism, with its curly behavior and its dependence on currents, to be able to start from the magnetic force and then move to the electric.

Last edited: Jun 29, 2009
8. Jun 28, 2009

### atyy

The Lorentz force cannot be derived from Maxwell's equations.

9. Jun 28, 2009

### cabraham

And your point is ----- ?

Claude

10. Jun 29, 2009

### clem

I think his point is an answer to the original question:

"I am trying to bridge a gap between Maxwell equations and Lorentz force. I know that they are not independent and in theory, one could be derived from the other but I cannot see that."

11. Jun 29, 2009

### marcusl

This paper seems to show just the opposite, that an additional postulate is required.

12. Jun 29, 2009

### gel

True. You could linearly rescale the Lorenz force and it would still be consistent with Maxwell's equations + conservation of energy/momentum.

13. Jun 29, 2009

### atyy

I wasn't thinking of linear rescalings. I was thinking that the Lorentz force law can't be derived from Maxwell's equations in the same way that charge conservation can, without additional assumptions. If one adds the assumption of energy/momentum conservation then the Lorentz force law can be derived from Maxwell's equations, as you say.

14. Jun 30, 2009

### clem

How?

15. Jun 30, 2009

### samalkhaiat

16. Jun 30, 2009

### atyy

I don't actually know how, am simply putting blind faith in MTW, who state the result without derivation on p473 http://books.google.com/books?id=w4Gigq3tY1kC&printsec=frontcover&source=gbs_navlinks_s. I thought it reasonable because in the context of GR, the geodesic equation of motion doesn't have to be supplied independently of the Einstein field equations, so perhaps if one has the Einstein and Maxwell field equations, then the Lorentz equation of motion will pop out.

Last edited: Jun 30, 2009
17. Jul 1, 2009

### sokrates

I think it's fair under any circumstances to "assume" the energy and momentum conservations are correct since they are fundamental.

I don't even consider those as "extra" assumptions so is it true that Lorentz force can be derived from Maxwell equations "in consistence with known laws of physics" after all?

18. Jul 1, 2009

### sokrates

19. Jul 1, 2009

### atyy

An introduction to the Lorentz-Dirac equation
Eric Poisson
http://arxiv.org/abs/gr-qc/9912045

20. Jul 1, 2009

### Bob S

The relativistic transformation of a static transverse electric field yields a transverse field that includes both a transverse electric field and a transverse v x B term. The transformation does not have to be relativistic, because the transformed fields are multiplied by γ, not β, and γ(β=0) = 1. So it can be demonstrated in a Crookes tube for example. Apparently Maxwell actually knew of the "Lorentz" force before Lorentz did. If there were eight, rather than four, Maxwell's Equations (and maybe there should be), this thread would have been unnecessary.
History (from Wikipedia)
Hendrik Lorentz introduced this force in 1892.[5] However, the discovery of the Lorentz force was before Lorentz's time. In particular, it can be seen at equation (77) in Maxwell's 1861 paper On Physical Lines of Force. Later, Maxwell listed it as equation "D" of his 1864 paper, A Dynamical Theory of the Electromagnetic Field, as one of the eight original Maxwell's equations.