# Conservation of momentum in an electric field

1. Nov 20, 2013

### Voltz

My question is about what I believed to be one of the stronger conservation laws - the conservation of momentum. I heard that electrons propagating through a wire travel at a speed similar to the flow of honey and electrons having mass naturally led me to conclude that they had a finite momentum. But then I started thinking about electrons being accelerated in the first place and I started to wonder about how this momentum would be cancelled out, I have looked online but cannot find a satisfying answer

2. Nov 20, 2013

### Staff: Mentor

The electrons lose energy and momentum through collisions with the ions and through EM interactions with the circuit as a whole. Think of the back EMF generated in an inductor. The electrons transfer energy (and momentum?) into the magnetic field, which is why it takes time for the current to build up and reach its max value.

3. Nov 20, 2013

### WannabeNewton

Conservation of energy-momentum for the electromagnetic field, including interactions with charges, is given by $\vec{\nabla}\cdot \vec{S} + \frac{\partial \mathcal{E}}{\partial t} = -\vec{j}\cdot\vec{E}$ where $\vec{S}$ is the 3-momentum density of the electromagnetic field (Poynting vector), $\mathcal{E}$ is its energy density, and $\vec{j}$ is the 3-current density of the interacting charges (hence $-\vec{j}\cdot\vec{E}$ is nothing more than Joule heating).

This is a direct consequence of Maxwell's equations.

4. Nov 20, 2013

### Voltz

I am not familiar with the concept of a field having momentum. how is this quantified when a field has no mass?
also just to clarify the question I understand that the electrons will collide with the conductor ions but I am talking about the 'recoil' of the original acceleration of the electron (which I guess is directed in the opposite direction to electron flow)

5. Nov 20, 2013

### Voltz

so what physical entity cancels out the momentum given to the electron?

6. Nov 20, 2013

### WannabeNewton

The electromagnetic field loses momentum and the electrons gain momentum.

7. Nov 20, 2013

### jartsa

Constant current:

There's an electric field pushing the electrons, and there's friction pushing the electrons to the opposite dirction. The two forces are opposite ones to each other.

Positive charges feel the same electric field and the same friction as the electrons.

Increasing current:

Electrons feel an electric field and a friction force that is "too small" compared to the force caused by the electric field.

Positive charges feel the same electric field and the same friction force as the electrons, the direction of the forces are opposite ones compared to electrons.

The imbalances balance each other.

8. Nov 21, 2013

### jartsa

A superconducting (no resistance) ring is floating on superfluid (no viscosity) liquid.

A change of magnetic field induces a current into the ring.

The total current consists of two currents:

Free electrons flow into one direction, other parts of the ring spin into other direction.

The ratio of the two currents is: mass of free electrons / (mass of ring - mass of free electrons)

9. Nov 21, 2013

### Voltz

What I am getting at is that before the application of an electric field, i.e. a voltage the drift velocity of the electrons is zero and therefore their momentum is zero, then after the application of the voltage the drift velocity of the electrons is non zero and therefore there is an increase in momentum in the direction they begin moving, it is my understanding that the positive lattice ions are effectively fixed and therefore cannot be accelerated in the opposite direction to the electrons (even though I'm aware that due to the vast differences in mass they would only need a small velocity to cancel out the electrons forward momentum) - the only positive things that I though moved are electron holes and having no mass since they are pseudo-entities surely they can't carry momentum??

10. Nov 21, 2013

### Crazymechanic

no before an applied electric field the drift velocity of electrons is not zero , it's just random , because as one electron moves forward two other are moving say backwards, to measure a current and voltage at the ends of a wire you need the electrons to join up in a chain like kids in kindergarden and move to either one direction or to move to one direction and then back with a given frequency(rate of change of net electron movement per second) which is what AC is.

electrons momentum is not zero before you applied the electric field.It can get higher after you apply the field that's for sure.

Also since the electric field travels with the speed of light in vacuum and close to that in other mediums you can pretty much safely assume that once you apply the field /voltage at one end of a wire the electrons will align all together over the whole course of the wire.If it would be much slower you would actually have to wait some time before the electrons that got accelerated at one end switching on voltage transfer the energy to the ones at the other end of a wire assuming a really long wire.

Ions are heavier than electrons so from such a point of view the lightest have to move first if the force exerted on both of them is the same the lighter ones will give first.
But there are other reasons why ions stay but electrons move.I am sure some more experienced folks will comment on that one.

11. Nov 22, 2013

### jartsa

Your universe has a hole into which momentum can be dumped, so that the momentum disappears.

A fixed point is the said momentum sink.

12. Nov 22, 2013

### jartsa

Yes, I understand.

I gave a correct answer, other people have given answers that deal with irrelevant things, like momentum of electromagnetic field. That's how it is.

13. Nov 22, 2013

### xAxis

Exactly what jartsa sad. In other words, momentum is conserved in closed system, which conductor is not. Think of the battery as the outside energy input to the circuit.

14. Nov 22, 2013

### Khashishi

You have to include the source of the electromagnetic field in your system. This is analogous to gravity. The momentum of a falling mass is not conserved, but the momentum of the falling mass and oppositely falling planet is.

An electric field with no magnetic field cannot propagate, so it cannot exist independently of a source. The static electric field has no momentum, but the source does have momentum, and must be taken into account.

(An electromagnetic wave consists of both electric and magnetic fields. This can exist independently of a source, and it does indeed carry momentum and energy. But this is a different situation.)

15. Nov 23, 2013

### Voltz

I understand the whole closed system concept and I appreciate the gravitational example, the momentum gain of the falling object is cancelled by the planet 'falling' towards the object with zero net momentum change. So if I consider the electrons and the E field source as a closed system how does the source 'fall' so to speak to cancel out the electrons?

16. Nov 23, 2013

### jartsa

Let's consider a wire in which the lattice of positive ions has velocity 0 and the conducting electrons have velocity 1 mm/s. From experience we know that people will point at the wire and say "that wire is at rest".

Next we consider a wire in which the lattice of positive ions has velocity 2 mm/s and conducting electrons have velocity 0. From experience we know that people will point at the wire and say "that wire is moving".

Now we imagine ourselves to be pylons, holding up power lines were AC current is flowing. We are trying to keep the wires at rest, where "rest" means that kind of thing that we usually call rest.

I have explained earlier that forces of same magnitude push the elctrons and the lattice of positive ions in a wire where elctrons are accelarated.

Our job as pylons is to resist the force that is pushing the lattice of positive ions, because when lattice of positive ions is at rest, then we feel that the wire is at rest.

17. Dec 4, 2013

### Voltz

so the pylons would be pushed backwards a minute amount?

18. Dec 4, 2013

### jartsa

Yes. Correct.

(If a pylon does work against that force, then we have a generator)

19. Dec 5, 2013

### Voltz

thank you very much for your explanation. I'm sorry such a reductionist argument was required for me to understand