# Dipoles and heat

## Main Question or Discussion Point

If we put a neutral conducting sphere in an electric field, it will become polarized. A force per unit area will be experienced on the surface according to:
$\vec{f} = \frac{1}{2\epsilon_0}\sigma^2\hat{n}$

My question is, does this process heat up the conductor? And if so, where is that energy coming from?

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UltrafastPED
Gold Member
The rearrangement time is a few femtoseconds, and only requires small adjustments to the average location of the electrons - a simple way to view this is via the Drude model, which is pre-quantum, but gets many things just about right.

So there is no current in the ordinary sense, hence no measurable Joule heating.

Any energy required - and energy is required to rearrange the electrons - comes from the field; after all, it is a force field.

The rearrangement time is a few femtoseconds, and only requires small adjustments to the average location of the electrons - a simple way to view this is via the Drude model, which is pre-quantum, but gets many things just about right.

So there is no current in the ordinary sense, hence no measurable Joule heating.

Any energy required - and energy is required to rearrange the electrons - comes from the field; after all, it is a force field.
Hm, I wasn't really thinking of the movement of the electrons. More like in comparison to PV=nRT. If adding charge to a given volume in a conductor creates a pressure, then it seems logical that the temperature should also increase, just like the ideal gas atoms bouncing around in a shrinking container.

What's wrong with this line of thinking?

BruceW
Homework Helper
In reality, I'm guessing such an effect is insignificant. The change in electron density due to a 'standard' strength electric field would be pretty small I'd guess.

edit: also, the electrons are not very massive compared to the atoms, so it will make a small difference to the total density, even if the electron density changed by a lot.

In reality, I'm guessing such an effect is insignificant. The change in electron density due to a 'standard' strength electric field would be pretty small I'd guess.

edit: also, the electrons are not very massive compared to the atoms, so it will make a small difference to the total density, even if the electron density changed by a lot.
Ok, interesting. Do I have this right:

PV=nRT does apply similarly with electrons, but this has nothing to do with electrostatic pressure. A considerable force can be preset in the conductor due to charge, but things aren't really bumping into each other all that much, so the change in temperature is negligible.

edit: Or rather, electrons are "bumping" a lot, but the differences in mass make it negligible.

BruceW
Homework Helper
PV=nRT is for a homogeneous substance with all one type of 'particle'. In your case, the substance is definitely not homogeneous and there are the atoms, plus the electrons. You could maybe use PV=nRT over some small volume in your substance, such that these values are almost the same over that volume. But, there is also the flow of heat from the slightly denser side of the sphere to the colder side. and electrons will move across regions also.

I would guess that the difference in temperature between the cold and hot sides (for moderate electric fields) would be pretty small. But I don't know really. The second link that UltrafastPED gave looks pretty good. Wikipedia seems to be not very useful.

UltrafastPED
Gold Member
No, there are many differences - the electrons don't see each other; they scatter off of discontinuities of the material (phonons, grain boundaries, dislocatons, etc). The Drude model gives _some_ insight into how things work, but it has many deficiencies. Among other things, specific heats and heat capacity.

And being inspired by the ideal gas law doesn't mean that the ideal gas law applies here!

Note that the electrons take advantage of their quantum nature and act as waves most of the time while in the metal - they spread out, among other things.

Besides, PV=nRT assumes hydrostatic pressure. I don't see any change to the the hydrostatic pressure applied to the electron gas.

PV=nRT is for a homogeneous substance with all one type of 'particle'. In your case, the substance is definitely not homogeneous and there are the atoms, plus the electrons. You could maybe use PV=nRT over some small volume in your substance, such that these values are almost the same over that volume. But, there is also the flow of heat from the slightly denser side of the sphere to the colder side. and electrons will move across regions also.

I would guess that the difference in temperature between the cold and hot sides (for moderate electric fields) would be pretty small. But I don't know really. The second link that UltrafastPED gave looks pretty good. Wikipedia seems to be not very useful.
Alright, this isn't important for now. It's small enough to not matter, and I'm still just in basic electrostatics.

It's neat that a force like this can exist inside a conductor and (nearly) none of this force is transferred to heat. I feel as if the conductor should be heated by this pressure, and that the electrons should be boiled off. But this simply doesn't happen.

I can't visualize why this is true, but I will just memorize it for now.

Thanks a lot for the help!

edit: I guess this isn't much different to ask:

Why doesn't a book heat the table that it's resting on? If the book was really really heavy, it would tear through the table, just like if an E field was very powerful, it would tear the conductor apart. This still doesn't heat the table nor the conductor.

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UltrafastPED
Gold Member
For the electron motion in the conductor, it requires almost no work because the adjustments required are so small. Very little work means very little heating ... only the wasted work goes into heating.

As for the book on a table ... nothing is moving so no work is being done ... no work means no wasted work means no heating.

So if the book is not moving, what force is opposing gravity?

For the electron motion in the conductor, it requires almost no work because the adjustments required are so small. Very little work means very little heating ... only the wasted work goes into heating.

As for the book on a table ... nothing is moving so no work is being done ... no work means no wasted work means no heating.

So if the book is not moving, what force is opposing gravity?
The normal force is what is the regular answer would be, right? But what is actually opposing gravity? The electromagnetic attraction between molecules? Until the molecules tear apart, then what's under the book will be compressed matter, so again it's electromagnetic force in that the electrons wish to hold their orbit. Then it's the strong force because the neutrons and protons want to hold their bonds. Then it's nothing.

I suppose it makes sense that electromagnetic forces can oppose gravity and no energy be transferred.

In the sphere it's electromagnetic opposing other electromagnetic forces? In the case of a copper sphere, on the inside we have a perfect lattice of copper atoms with covalent bonds. On the surface, we have a nearly infinite supply of free charge. Placed into an E field, the surface charges pull from the poles on the lattice proportionate to the strength of the field. And (nearly) no energy is transferred in this process.

I guess I get it. I don't like it though.

UltrafastPED
Gold Member
In the case of a copper sphere, on the inside we have a perfect lattice of copper atoms with covalent bonds.
The sphere will not be crystalline - metals typically consist of many extremely tiny crystals which form a grain structure within the metal. Otherwise copper would not bend - its ductility would be lost.

Metals do not have covalent bonds - only materials which form molecules have covalent bonds. Salt for example has ionic bonds - you will never find a molecule of NaCl. Nor will you find a molecule of Cu - metals have metallic bonds which consists of the "electron sea" which we have been discussing; the metal of each grain stays neutral due to the to the array of positive charges (the ion cores); the "valance" electrons simply spread out throughout the grain.

Thus when there is an external field the electron sea shifts a bit. How much? Just enough to cancel the electric field that presents itself.

UltrafastPED
Gold Member
The normal force is what is the regular answer would be, right?
Yes, its the normal force; Newton's Third Law in action.

Since materials are held together by electromagetic forces, the rigidity (and elasticity) of those materials is all due to the electromagnetic bonds of that material.

BruceW
Homework Helper
The normal force is what is the regular answer would be, right? But what is actually opposing gravity? The electromagnetic attraction between molecules? Until the molecules tear apart, then what's under the book will be compressed matter, so again it's electromagnetic force in that the electrons wish to hold their orbit. Then it's the strong force because the neutrons and protons want to hold their bonds. Then it's nothing.

I suppose it makes sense that electromagnetic forces can oppose gravity and no energy be transferred.
Yeah. If instead of a book, you had a gas, and could somehow 'switch on' gravity, then the gas will be compressed by gravity, and during this compression, the gas will lose some of its GPE, and this energy will increase the KE of the molecules, or be lost as heat. Now, for a solid book, something similar will happen, but the compression will be much much smaller. In both cases, this initial compression will happen, but then it is compressed no more, so no more heat will be generated.

In reality, we always have gravity 'switched on'. So the air is always compressed by gravity. Maybe if you look into meteorology, they might take into account how the compression of air flows will cause a change in the temperature of that parcel of air. I am not sure. And for the book, when it is in free-fall, it is still under atmospheric pressure. When it is resting on the table, it has the additional pressure due to gravity. The compression of the solid part of this book due to this pressure will be very small I would guess. So any heat generation would also be small (unless you dropped the book from a height). And then once that small compression has occurred, no more heat will be given off.

This kind of physics is useful if you learn about basic models of stars, and how the pressure gradient needs to be enough to prevent the star from collapsing inward on itself due to gravity. And if it does start to collapse inward, it will lose GPE, so the 'astronomical object' will increase its temperature, and start to emit more heat.

UltrafastPED
Gold Member
Compression of the book or of the table is elastic; that is, it will bounce back. For perfect elasticity there is no net work, and no heating. For real substances there is some deformation that is permanent, and this takes energy - and some of that energy will go into heat.

Note that weather is due to:
(1) rotation of the earth which sets up global circulation patterns
(2) insolation (solar heating) which changes the global circulation by changing local air pressure (via temperature)
(3) physical barriers such as mountains which change air flow patterns; note the relationship between mountains and deserts - stands out when the prevailing winds are taken into account
(4) bodies of water have different heat capacity and heating/cooling rates than land

After this the details get finer and finer ... pressure, temperature, relative humidity are the local state variables.