# How do point charges in a conductor move and stop?

• I
feynman1
Let's speak in the classical context (non quantum). We assume that point charges move in a conductor following Newtonian mechanics. How do point charges move along the boundary of the conductor and how do they stop (equilibrium) in the end?

Gold Member
Non-quantum charge transport in a conductor is not a thing; it is an inherently quantum mechanical process.
We can of course use semi-classical models such as the Drude model, but you can't (and shouldn't) use these to think about individual charges.

It is possible to create systems where you have "true" single electron transport (usually in semiconducting systems); but this is an exception and if you look into details about what is actually moving you will find that it is a wave packet; not something which can be described by Newtonian mechanics.

feynman1 and anorlunda
Mentor
2021 Award
you can't (and shouldn't) use these to think about individual charges
Well said. Classically, the sources in Maxwell’s equations are ##\rho## and ##\vec J## which are continuous. As far as I know, all of the paradoxes and inconsistencies in classical EM are based on using Dirac delta functions as the sources.

anorlunda and feynman1
feynman1
We can of course use semi-classical models such as the Drude model, but you can't (and shouldn't) use these to think about individual charges.
Can we only think of them as an ideal gas, thus a continuous medium? What result will go wrong when thinking about individual charges?

feynman1
all of the paradoxes and inconsistencies in classical EM are based on using Dirac delta functions as the sources.
What's the problem with using direct Delta function at its source?

Mentor
2021 Award
What's the problem with using direct Delta function at its source?
There are several. The most obvious one is that the field energy is infinite around a point charge. This would lead to point charges being infinitely massive, an infinite amount of energy release as two opposite charges collide, etc.

Another common issue is the radiation self-reaction. This can lead to runaway acceleration.

I am sure there are others.

feynman1
Gold Member
How do point charges move along the boundary of the conductor and how do they stop (equilibrium) in the end?
They respond to an electric field.

feynman1
They respond to an electric field.
If there is only an electrostatic field, particles can't stop, can they

Homework Helper
Gold Member
If i understand correctly you are asking why the free electrons are not being ejected from the conductor when an external electrostatic field is applied, instead they seem to stop at the surface. This cannot be explained by classical physics, from what I know according to quantum physics the free electrons must be accelerated to a very high velocity in order to be ejected from the conductor.

feynman1
If i understand correctly you are asking why the free electrons are not being ejected from the conductor when an external electrostatic field is applied, instead they seem to stop at the surface. This cannot be explained by classical physics, from what I know according to quantum physics the free electrons must be accelerated to a very high velocity in order to be ejected from the conductor.
No, I wasn't asking this question. 2 questions below. I wonder what happens when charges hit the boundary. Do they bounce several times on the boundary or do they just stick to the boundary upon colliding it? How do charges move along the boundary and finally grind to a halt?

Homework Helper
Gold Member
Do they bounce several times on the boundary or do they just stick to the boundary upon colliding it
I think you have some sort of mechanical model in your mind (the electrons being some sort of balls that are bouncing against a solid wall-the boundary) which i don't think is entirely correct. But maybe a more knowledgeable member than me should answer here, maybe @vanhees71 .

feynman1
feynman1
I think you have some sort of mechanical model in your mind (the electrons being some sort of balls that are bouncing against a solid wall-the boundary) which i don't think is entirely correct. But maybe a more knowledgeable member than me should answer here, maybe @vanhees71 .
Exactly. I need (wish) a mechanical model with electrons thought of as balls bouncing around, as suggested by the original post (classical).

sophiecentaur, weirdoguy, Motore and 1 other person
Gold Member
Exactly. I need (wish) a mechanical model with electrons thought of as balls bouncing around, as suggested by the original post (classical).

As was stated above: This is simply not possible, because electrons are NOT "small balls" meaning a "mechanical" model simply won't work.
For the electrons close to a surface the wiki on Friedel oscillations might be relevant.

The simplest model treats the free electrons in a Fermi gas with a plane wave-like behaviour
See wiki
https://en.wikipedia.org/wiki/Friedel_oscillations

The reason this sort-of works is because it takes the wave-like behaviour into account which means you can get interference.

Last edited:
Staff Emeritus
2021 Award
Exactly. I need (wish) a mechanical model with electrons thought of as balls bouncing around, as suggested by the original post (classical).
Why would you want such a thing?

You have Classical E&M, which works well in its domain of validity. Now you want to take quantum objects, electrons, and treat them like classical billiard balls. Why would you think this would be helpful?

In 1900, there was something called the Drude model, which attempted to do this. It got Ohm's Law as an output (sort of) and pretty much got everything else wrong.

sophiecentaur, feynman1, Twigg and 1 other person
Mentor
2021 Award
I wonder what happens when charges hit the boundary. Do they bounce several times on the boundary or do they just stick to the boundary upon colliding it? How do charges move along the boundary and finally grind to a halt?
Classically, on a conductor at DC there is a surface charge density and a volume current density. Both of these are continuous charge distributions classically, not point charges, for the reasons I pointed out above.

I wonder what happens when charges hit the boundary. Do they bounce several times on the boundary or do they just stick to the boundary upon colliding it? How do charges move along the boundary and finally grind to a halt?
The surface charge density produces an internal field such that the current density near the surface has no component perpendicular to the surface. Thus there is no “hitting the boundary” etc.

feynman1 and vanhees71
Gold Member
2021 Award
The classical charged point particle is a contradiction in itself. It simply doesn't make fully sense. After subtracting an infinite mass from the also infinite bare mass you get a 3rd-order equation of motion, the Lorentz-Abraham Dirac equation, which has big problems being acausal and/or exploding ("runaway and preacceleration solutions") of you make a perturbative approximation, leading to the Landau-Lifshitz equation, which has not the deficits of the LAD equation but violates (to some extent) energy conservation.

With quantum (field) theory you are much better! There you can, after mass, charge, and wave function renormalization, calculate things at any order of perturbation theory and for some quantities like (g-2) of the electron or the Lamb shift of the hydrogen atom you get a description which is correct for 12 or more significant digits.

On top in the many-body version you can also describe material properties correctly. The quantum (statistical) version of the above mentioned Drude model for electric (as well as heat) conductivity for metals was among the first quantum many body systems which have been solved in the early days of quantum theory (Sommerfeld and Bethe wrote an early textbook on it). This was the beginning of the huge and tremendously successful field of condensed-matter theory, and it's all quantum. There's no way to understand the behavior of matter (including the usual matter surrounding us in everyday life) within any classical model. It already starts with the observation that there is stable matter at all. With classical mechanics you can't even understand the most simple atom, the hydrogen atom!

sophiecentaur, feynman1, Dale and 1 other person
Gold Member
If there is only an electrostatic field, particles can't stop, can they
If the direction of the electric field is opposite to the direction of motion.

feynman1
The surface charge density produces an internal field such that the current density near the surface has no component perpendicular to the surface. Thus there is no “hitting the boundary” etc.
Isn't the work function rather than the surface charge itself that inhibits current density from going perpendicular to the surface?

feynman1
If the direction of the electric field is opposite to the direction of motion.
Even that doesn't mean charges will stop. But they can still bounce around.

Mentor
2021 Award
Isn't the work function rather than the surface charge itself that inhibits current density from going perpendicular to the surface?
No, it is the surface charge. The work function keeps the surface charge from leaving, but it only affects the surface charge density. The surface charges, in contrast, affect the current density throughout the conductor.

feynman1
feynman1
Classical physics has a lot of issues indeed. But all I want is just a classical model describing the motion of point charges in a crude way much like the classical model of the atom with electrons orbiting the nucleus.

feynman1
The surface charge density produces an internal field such that the current density near the surface has no component perpendicular to the surface. Thus there is no “hitting the boundary” etc.
Are you speaking under the context of electrostatics or electrodynamics, with current or without?

Mentor
2021 Award
Are you speaking under the context of electrostatics or electrodynamics, with current or without?
I started that post with “on a conductor at DC”. I didn’t want to deal with the messy details of electromagnetic waves and skin depth etc. But of course electrostatics is just DC with 0 current density, so what I said applies there too.

feynman1
Homework Helper
Gold Member
The surface charge density produces an internal field such that the current density near the surface has no component perpendicular to the surface. Thus there is no “hitting the boundary” etc
Since the surface charge density interacts with the free electrons, so free electrons apply forces to the surface charges, what keeps the surface charges at their place, that is at the surface. Why they do not move to the outside of the conductor or to the interior?

feynman1
Since the surface charge density interacts with the free electrons, so free electrons apply forces to the surface charges, what keeps the surface charges at their place, that is at the surface. Why they do not move to the outside of the conductor or to the interior?
I know why they don't move outside of the conductor. That is due to a work function. But I don't know why they can't move back to the interior.

Delta2
Mentor
2021 Award
Since the surface charge density interacts with the free electrons, so free electrons apply forces to the surface charges, what keeps the surface charges at their place, that is at the surface. Why they do not move to the outside of the conductor or to the interior?
The bulk of the wire is neutral so there is no force on the surface charge density from the current density.

Delta2
Gold Member
No, it is the surface charge. The work function keeps the surface charge from leaving, but it only affects the surface charge density. The surface charges, in contrast, affect the current density throughout the conductor.

I know why they don't move outside of the conductor. That is due to a work function. But I don't know why they can't move back to the interior.
It may be a bit confusing to try to use Energy and Field in the same argument. You can do it all with Energy by saying the energy to move from place to place within the conductor is a lot lower than the energy to remove charge from it. Water moves easily from place to place in a bowl but it needs a lot of energy to climb up the sides. You can tilt the bowl slightly to displace the water so it can flow into another bowl through a hole just above the surface, which is like a connection to another conductor (If you really want a mechanical approach / analogy.)

feynman1
It may be a bit confusing to try to use Energy and Field in the same argument. You can do it all with Energy by saying the energy to move from place to place within the conductor is a lot lower than the energy to remove charge from it. Water moves easily from place to place in a bowl but it needs a lot of energy to climb up the sides. You can tilt the bowl slightly to displace the water so it can flow into another bowl through a hole just above the surface, which is like a connection to another conductor (If you really want a mechanical approach / analogy.)
who mentioned energy?

Gold Member
who mentioned energy?
Work Function?

Gold Member
How do point charges move along the boundary of the conductor and how do they stop (equilibrium) in the end?
Equilibrium doesn’t mean that mobile electrons ever “stop” in a conductor in a literal sense. In case a dynamical equilibrium is established – the electrochemical potential of the electrons becomes constant throughout the conductor –, currents in one direction are – so to speak –“cancelled” by currents in the opposite direction.

feynman1
Work Function?
that's not a kind of energy

davenn
Gold Member
that's not a kind of energy
Yes it is, it is the minimum amount of energy needed to move an electron from a solid to the vacuum,

feynman1
Yes it is, it is the minimum amount of energy needed to move an electron from a solid to the vacuum,
Yes, sorry I didn't mean to deny it being an energy. I just meant that wasn't 'work' in the usual sense of introductory physics.

Motore