# Is the Lorentz force conservative?

1. May 20, 2014

### Dash-IQ

When a wire has current I in a magnetic field B, there is the Lorentz force is it considered a conservative force or not? Please do explain as to why it is.

2. May 20, 2014

### UltrafastPED

The force on the wire is given here: http://en.wikipedia.org/wiki/Lorentz_force#Force_on_a_current-carrying_wire

If the electric field is static then it's curl is zero - and the electric field is conservative.

But when we consider the magnetic field Maxwell's equations tell us that it's divergence is always zero -so if it's curl were also zero we would have B=0. Thus the magnetic field is not conservative, and the exact situation shouldn't matter.

3. May 20, 2014

### Dash-IQ

If the magnetic field is not conservative, yet the electric field is... the Lorentz force shouldn't be conservative? Why does it not matter?

4. May 20, 2014

### UltrafastPED

It doesn't matter what the situation is: the magnetic field is not conservative is what I meant to say; the electric force is conservative only when static.

5. May 21, 2014

### Dash-IQ

Ah, so all in examples the demonstrate the Lorentz force are always nonconservative?

6. May 21, 2014

### Robert_G

No, It's not, there is an easy way to see if a force is conservative, or not. If the force is only depends on the position, or in other word.
$\mathbf{F}=\mathbf{f(\mathbf{r})}$,
then, is conservative. because, only if the force only depends on $\mathbf{r}$, A plane with the equal potential energy can be introduced.

now for the Lorentz force, this force actually has something to do with the velocity of the charged particles. Different velocity means Different force, that plane will never be introduced. So it's not conservative.

7. May 21, 2014

### DrStupid

Lorentz force is conservative because the work done between two points is independent from the path.

8. May 21, 2014

### Dash-IQ

What about the case of the wire? F = IL x B?
Different current values ?

9. May 21, 2014

### Dash-IQ

Hm, what about the statements of the rest?

10. May 21, 2014

### DrStupid

I can't speak for the rest but for myself only. I just checked if Lorentz force meets the condition for a conservative force and due to
$d\vec E = \vec F \cdot d\vec s = q \cdot \left( {\vec v \times \vec B} \right) \cdot \vec v \cdot dt = 0$
it does.

11. May 21, 2014

### UltrafastPED

12. May 21, 2014

### DrStupid

I conclude that we need to distinguish between Lorentz force and magnetic field. As Lorentz force has no force field the corresponding formalisms does not apply. Thus there is only one condition left: conservative forces conserve mechanical energy.

13. May 21, 2014

### BruceW

yeah... I think there are several possible definitions of a conservative force. 1) does work done by the force on a particle depend on the path, or just the end points? 2) Does the force depend only on the position of the particle?

These two ways to define conservative force are similar, but not the same, so the answer will depend on which definition you choose. Also, why can't the Lorentz force be a force field?

edit: ah, OK, definition 2) defines a force field. I just looked this up, since I didn't know the standard definition of a force field.

14. May 22, 2014

### Dash-IQ

Im confused, potential energy can't be predicted here? How can it still be conservative?

15. May 22, 2014

### BruceW

It looks like there's more than one possible definition for 'conservative force'. One possible definition is that there must be an associated potential. But another definition (as DrStupid is saying) would be that the work done on the particle does not depend on the path, but only on the endpoints. Anyway, look at the equation for Lorentz force:
$$\vec{F} = q \vec{E} + q \vec{v} \wedge \vec{B}$$
The bit due to the electric field is 'nice', and the bit due to the magnetic field is not so nice. But what happens when we integrate this force, over the path of the particle? What happens to the term due to the magnetic field?

16. May 22, 2014

### Dash-IQ

No idea...

17. May 22, 2014

### DrStupid

See my equation above. The integral of the magnetic term is always zero.

18. May 22, 2014

### Meir Achuz

The physical meaning of a force being conservative is that any work done by the force can be retrieved.
The Lorentz force on the current in a wire does no work. Since energy conservation is a statement about the work done by a force, the concept of conservative force is not relevant to the Lorentz force, except in the trivial sense that 0 = 0.

19. May 22, 2014

### Meir Achuz

That is wrong.

20. May 22, 2014

### Meir Achuz

The Coulomb force has zero divergence and curl.

21. May 22, 2014

### BruceW

why is that wrong? It is a different definition of conservative force. But I've seen more than one definition used. For example, on the wikipedia page, they seem to use at least two different definitions.

22. May 22, 2014

### Khashishi

All fundamental forces are conservative. The Lorentz force is just the electromagnetic force, which is conservative.

23. May 22, 2014

### DrStupid

Counterexample: F = [x-z,y,0]

24. May 22, 2014

### Meir Achuz

To clarify my post #18:

I was referring to only the magnetic part of the Lorentz force. Of course the electric part is conservative.

Another case is the magnetic force on a contained current distribution, such as a current loop. That can be described in terms of the magnetic moment of the current distribution as
$${\bf F}=\nabla(\mu\cdot{\bf B})$$. This is a conservative force, but I would not call it the 'Lorentz force'.

Last edited: May 22, 2014
25. May 22, 2014

### Meir Achuz

It is the two onlys that make that statement wrong. The equation given is for a conservative force, but there are many other examples of conservative forces. One example, among many, is the force in my previous post.