# Why don't high-voltage electrons in a metal wire escape?

sophiecentaur
Gold Member
Well, yes, but then simply explain it.
Twenty posts earlier would have helped. But I still think it was just sloppiness in the question and not ignorance. How else could Work Function come into it?

sophiecentaur
Gold Member
For your info, I investigated field emission, since it is something we were trying to mitigate both in our photocathodes in a photoinjector and in our PMT. So it isn't something that ".. isn't to my taste..". I can talk about it for days!
I'm not going to respond to something that I've never said, implied, and isn't true.
???

anorlunda
Staff Emeritus
The OP has not returned to PF since post #3. Does that influence the back-and-forth?

His wording in post #3, makes it sound like he visualizes a wire as being like a linear particle accelerator where a free electron continuously accelerates in the external E field. If I had seen that earlier, I might have pointed him to The Drude Model.

Sorry I don't understand what you are asking exactly. What I am thinking is that under an electric potential difference, an electron would eventually gain enough kinetic energy to escape from the metal.

Thank you all for the fruitful discussion. Sorry that I didn't go online for the weekend.

I understand that the work function is related to the photoelectric effect. But I read on Wikipedia about the electron emission due to a high temperature - a phenomenon called "Thermionic emission" (https://en.wikipedia.org/wiki/Thermionic_emission). So according to my understanding, as long as an electron gains enough energy, no matter what form of energy (light/kinetic), an emission may occur.

Naively, I think that if an electron travels in an electric field, let's say 1000V over 1km, then after it travels 20m, it will gain 20eV energy, which is sufficient to escape. If the electron does not escape, that means it must have lost energy.

The next thing that comes to my mind is that the electron is not travelling in a vacuum but a metal. The Drude model comes to play. It describes a group of electrons drifting slowly while individual electrons bouncing the stationary ions frequently. That means the electrons may exchange energy with the ions.

Combining the above thoughts, I propose two possibilities where both contradicts to reality.
1. The electrons continually transfer kinetic energy to the ions so they cannot escape from the metal. However, it means that the energy is localised and most of the energy is dissipated as heat which is not the real case as we can transmit our electricity pretty well in the power grid. According to my logic, the maximum energy carried by the electrons cannot exceed the work function, which would mean the eventual voltage arrived at the end-user cannot be bigger than a few volts.
2. The electrons do not lose most of their energy to the ions. Then, in a short time, it shall acquire enough energy to escape from the metal wire.

I understand that I must have made some wrong arguments in the above construction. Sorry that I may not present clearly enough.

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Combining the above thoughts, I propose two possibilities where both contradicts to reality.
1. The electrons continually transfer kinetic energy to the ions so they cannot escape from the metal. However, it means that the energy is localised and most of the energy is dissipated as heat which is not the real case as we can transmit our electricity pretty well in the power grid.
2. The electrons do not lose most of their energy to the ions. Then, in a short time, it shall acquire enough energy to escape from the metal wire.

The main thing is, between which 2 point do you have a 10-100 kV potential difference?
Normally this is between 2 wires. An electron in a vacuum that would move between 2 pieces of metal with a 100 kV difference would get 100 kV of kinetic energy, but it would need to be free of the wire before it could get that 100 kV.

If you have a 10-100 kV potential difference between 2 points with a piece of metal wire between it, you would get a current. Your piece of wire would have to be very long and thin, to get a current that is low enough to not evaporate/melt the wire. The electrons would transfer all the kinetic energy that they get from the field to the ions, and this will heat the wire. The random speeds of the electrons would depend only on the temperature, and not on the electric field in the wire, and the kinetic energies would remain small compared to the work function until about 1000K.
In the power grid you do not get large potential differences across short pieces of wire.

sophiecentaur
The main thing is, between which 2 point do you have a 10-100 kV potential difference?
Normally this is between 2 wires. An electron in a vacuum that would move between 2 pieces of metal with a 100 kV difference would get 100 kV of kinetic energy, but it would need to be free of the wire before it could get that 100 kV.

If you have a 10-100 kV potential difference between 2 points with a piece of metal wire between it, you would get a current. Your piece of wire would have to be very long and thin, to get a current that is low enough to not evaporate/melt the wire. The electrons would transfer all the kinetic energy that they get from the field to the ions, and this will heat the wire. The random speeds of the electrons would depend only on the temperature, and not on the electric field in the wire, and the kinetic energies would remain small compared to the work function until about 1000K.
In the power grid you do not get large potential differences across short pieces of wire.

I understand your point. I also agree with this point of view in the macroscopic world. But if I go down to the microscopic view, I cannot construct a comprehensive picture to comply with the macroscopic observations.

There is not a large potential difference across a short piece of wire. I agree. So the electrons only gain the energy very gently, but they accumulate energy throughout the journey in the wire. My puzzle is that I think an electron cannot carry energy more than a few eV because it would have escaped before it arrives at its destination. But if that's the case, most of the energy would be dissipated which also disagreed with reality.

So the electrons only gain the energy very gently, but they accumulate energy throughout the journey in the wire.
Electrons do not accumulate energy,and they do dissipate all of the energy. The reason that this is possible, is because electric fields in conductors are small.

sophiecentaur
Electrons do not accumulate energy,and they do dissipate all of the energy. The reason that this is possible, is because electric fields in conductors are small.

I kind of get some sense on what you mean. I shall try to self study more on that first. Thanks for the discussion and help!

sophiecentaur
Gold Member
The main thing is, between which 2 point do you have a 10-100 kV potential difference?
Normally this is between 2 wires. An electron in a vacuum that would move between 2 pieces of metal with a 100 kV difference would get 100 kV of kinetic energy, but it would need to be free of the wire before it could get that 100 kV.
It's a great shame that this point was not made explicitly after a couple of posts here. But. as the OP was not on the thread, no harm done.

. So the electrons only gain the energy very gently, but they accumulate energy throughout the journey in the wire.
'very gently' is putting it mildly. They travel through and 'emerge' with average velocities of only around 1mm/s under all conditions. The Resistance of the wire can be looked upon as affecting the Power dissipated in the wire. P = I2R (which comes from P=IV, which we know and love). The Power 'passed on' to the rest of the circuit is actually nothing to do with the Kinetic Energy of the electrons and this is a popular misconception when trying to understand the transfer of Power down a wire.

Perhaps here a comparison between electricity and water helps.

A high voltage wire could be compared to a shallow ditch along the ridge of a high mountain.
The water could gain a lot of energy if it were able to cascade down the mountainside. But in order to do so, it would somehow have to get across or through the banks of the ditch. Shallow as they are compared to the water level inside the ditch, the water inside the ditch does not have free kinetic energy to spill over the banks. The huge potential energy of cascading down the mountainside remains just a potential.

sophiecentaur
Gold Member
Very picturesque but my physical appreciation is a bit overwhelmed with mental images of the flora and fauna in this mountainside canal.

vanhees71
Gold Member
2021 Award
A good toy model for an electron in a metal is a finite potential pot. For simplicity I make a 1D model:
$$V(x)=-V_0 \Theta(0<x<L), \quad V_0>0.$$
The quantum mechanical time-independent Schrödinger equation (determining the energy eigenstates) can be solved analytically.

Now you can consider two plates forming a capacitor and putting a constant voltage on it. The voltage is between the two plates. In order to get an electron out of the (negatively charged) plate you'd need a voltage difference of some eV over a region of the order of a Bohr radius for an electron bound with its nucleus in the cathode. To get this you'll need a huge voltage between the plates. Even with air between the plates, which you can ionize at sufficiently large voltages between the plates, you need quite high voltages to do so and thus getting a conducting medium between the plates due to the partially ionized air (depending on the conditions like humidity, if I remember right, you need several kV voltage to trigger such a "gas discharge").

In contradistinction to that the photoelectric effect is not with a static voltage between plates but by shining an electromagnetic wave at the plate. In a bit hand-waving way you can argue that you only have to absorb one photon from the light source by one bound electron in the metal, and that's why with a sufficiently high frequency (for metals like zinc in the UV range is already sufficient) you can kick out the electron also with light of very low intensity, and this happens within a few nanoseconds after switching on the UV light, which shows that you don't need to wait to accumulate enough energy at the one bound photon located in a sphere of a about one Bohr radius around its nucleus (which would take pretty long compared to the correct quantum description).

anorlunda
Staff Emeritus
Combining the above thoughts, I propose two possibilities where both contradicts to reality.
You seem to think that electrons are little packets of energy that are delivered to the load. That's false. You are jumping all over the place, from thermionic emission to the power grid.

Your strategy for learning about electricity is a bad one. You should go back and learn about simple circuits, about electrostatics, electrodynamics, and Maxwell's Equations, and Poynting's Theorem. Forget that you ever heard the word electron until all that is done. Believe me, electrons first is the most difficult possible way to learn about electricity.

Most of all, you can't get an education by asking questions on an Internet forum. This forum helps students who are studying the traditional way.

sophiecentaur and etotheipi
vanhees71
Gold Member
2021 Award
Well, I think in classical electrodynamics to think about charges and currents a continuum-mechanical picture is of great help. You get very far in establishing the constitutive relations using this classical continuum-mechanical picture of charges and currents. The real problem is only a point-particle picture, which is only in an approximate sense consistent and you need pretty advanced math to establish it, leading to the Landau-Lifshitz approximation of the LAD equation, which is the best you can do in this classical point-particle picture.

etotheipi
sophiecentaur
Gold Member
So the electrons only gain the energy very gently, but they accumulate energy throughout the journey in the wire.
Here's an easy 'mechanical' argument that tells you the Kinetic Energy of electron motion is actually irrelevant. Consider a light but strong bicycle chain. The cyclist could be supplying a couple of hundred Watts of power to the wheels, pedalling once a second. The chain could be, say 0.5kg mass and any part of it would be travelling at say 0.1m/s over the sprocket. The total KE of the chain will be mv2/2=0.025J. So the KE of the chain represents 0.025/200 = 0.125% of the Energy supplied every second the wheels. The remaining 99.875% of the work is being transferred by the tension of the chain and its distance travelled (force times distance) and not the KE of the chain.

Bear in mind that the total mass of the 'mobile' electrons in a wire is 1/(1800X60) *of the mass of the positive copper ion cores and that they are drifting at a mm/s. What is transferring the Power along the wire is the fields 'pushing' the electrons along - the KE of the electrons is much much less of a proportion of the energy transferred than for the bicycle chain links.

*electron mass is about 1/1800 of the mass of a proton and the atomic mass of copper is about 60

kith, vanhees71, anorlunda and 1 other person
Here's an easy 'mechanical' argument that tells you the Kinetic Energy of electron motion is actually irrelevant. Consider a light but strong bicycle chain. The cyclist could be supplying a couple of hundred Watts of power to the wheels, pedalling once a second. The chain could be, say 0.5kg mass and any part of it would be travelling at say 0.1m/s over the sprocket. The total KE of the chain will be mv2/2=0.025J. So the KE of the chain represents 0.025/200 = 0.125% of the Energy supplied every second the wheels. The remaining 99.875% of the work is being transferred by the tension of the chain and its distance travelled (force times distance) and not the KE of the chain.

Bear in mind that the total mass of the 'mobile' electrons in a wire is 1/(1800X60) *of the mass of the positive copper ion cores and that they are drifting at a mm/s. What is transferring the Power along the wire is the fields 'pushing' the electrons along - the KE of the electrons is much much less of a proportion of the energy transferred than for the bicycle chain links.

*electron mass is about 1/1800 of the mass of a proton and the atomic mass of copper is about 60

This is a good one, thanks! I have long been mistakenly understanding the power transmission mechanism of electricity.

anorlunda and sophiecentaur
sophiecentaur
Gold Member
This is a good one, thanks! I have long been mistakenly understanding the power transmission mechanism of electricity.
Along with millions of other people. Glad it helped.

vanhees71