Geometry and Surface Density of Electric Charge

[dg]

I have never seen an explicit formula to describe the relation between charge density and geometry (e.g. local curvatures) of the surface of a conductor.

We all know that charge tends to accumulate at edges (high curvature) but can we be more quantitative?

Hope someone else will find this a little intriguing...

Dario

No new theory please just regular old-fashioned electrostatics and (possibly differential) geometry

 

[eNtRopY]

Originally posted by dg
I have never seen an explicit formula to describe the relation between charge density and geometry (e.g. local curvatures) of the surface of a conductor.

I don't think charge does accumulate at the edges of a conductor unless you've applied an external electric field. In that case you are dealing with a drift transport problem. I believe this becomes an electromagnetics problem rather than a materials issue. So, use Maxell's equations to solve your problem.

I could be wrong. There may be some relation between geometry and density of states. Why don't you find out? Get a book on surface science and look it up.

eNtRopY

 

[climbhi]

From dg:We all know that charge tends to accumulate at edges (high curvature)

I thought charge didn't build up on points... I know that my high school physics teacher used to always approach van de graff balls with a screw or something pointy in his hand to make it so that he didn't get shocked as he approached the ball. He said this was because charge couldn't accumulate at points and corners. Either way I would also be interested to see some equation or whatever relating areas of high curvature to how much charge can or cannot build up there.

 

[chroot]

I can't say it better than can Richard P. Feynman, so I'll just quote him. The following pages are from The Feynman Lectures on Physics, Volume II. The relevant material occurs in section 6-11, page 6-13 to 6-14. This material is posted without permission; it is copyrighted by Addison-Wesley. It is for educational fair use only.

http://users.vnet.net/warrenc/V2_Ch06_1962-10-18_Electric_Field_in_Var_Circumstances.pdf

- Warren

 

[cala]

Thank you chroot, the reading of your document has bring me new arguments to see my system on "electrostatic paradox ?" topic will work.

As you can see, the charge tends to distribute to surface of conductors and edges.

And also, if you try to introduce charge into the cavity of a rounded conductor, it will have no opposition, because induced positive charges will appear to make the E=0 on this rounded conductor. In fact, the more charge you introduce, the more possibilities you can obtain a breaking current from the inner sphere to the rounded conductor. (The more negative charges in the cavity, the more ATTRACTION to the induced positive charges).

 

[dg]

Originally posted by eNtRopY
I don't think charge does accumulate at the edges of a conductor unless you've applied an external electric field. In that case you are dealing with a drift transport problem. I believe this becomes an electromagnetics problem rather than a materials issue. So, use Maxell's equations to solve your problem.

I could be wrong. There may be some relation between geometry and density of states. Why don't you find out? Get a book on surface science and look it up.

eNtRopY

Charge does accumulate on surface of conductors without any external field! (According to classical electrostatics at least).
Hence no drift here has to be considered (even though I agree that it interesting to think of the possibility of some sort of dynamic equilibrium but this is another story!). Everything, including electrostatics, can be solved with Maxwell equation once you have the proper boundary conditions. The problem remains: is there a general result relating surface geometry with local charge density of a conductor?
Density of states is a QM concept so forget it here.
Surface science is a recent offspring of solid state physics. Both rely on very poorly settled formalism (feel free to dig into that, I can suggest the concept of effective mass to begin with) and are usually dealing with semiclassical models everytime they "talk charge": I have never found anything related to differential geometry in any surface science book. That is why I am asking if anybody has seen anything out there...

 

[dg]

Originally posted by climbhi
I thought charge didn't build up on points... I know that my high school physics teacher used to always approach van de graff balls with a screw or something pointy in his hand to make it so that he didn't get shocked as he approached the ball. He said this was because charge couldn't accumulate at points and corners. Either way I would also be interested to see some equation or whatever relating areas of high curvature to how much charge can or cannot build up there.

For your teacher's sake, I hope you are not remembering well since charge strongly accumulate on edges and sharp points (think spark plugs and actually Van der Graff generators as well exploit this effect). The only reason you might want to approach a charged Van der Graff ball with a sharp (conductive) object is so that you can prevent discharge to happen anywhere else: spark (dielectric breakdown) will occur where the field is higher, and the field will be higher between the ball and the sharp object because of the distribution of induced charge on the sharp object.
People do the same approaching their car with their keys first if they do not want to get one of those little shocks when windy dry weather promotes accumulation of static charge on the car.

 

[dg]

Originally posted by chroot
I can't say it better than can Richard P. Feynman, so I'll just quote him. The following pages are from The Feynman Lectures on Physics, Volume II. The relevant material occurs in section 6-11, page 6-13 to 6-14. This material is posted without permission; it is copyrighted by Addison-Wesley. It is for educational fair use only.

http://users.vnet.net/warrenc/V2_Ch06_1962-10-18_Electric_Field_in_Var_Circumstances.pdf

- Warren

Well this is not big news and I cannot find an answer to my original question here: usual stuff on method of images, metalization of equipotential surfaces, etc. but still not a general relation between geometry and surface charge density. What I understand from Feynman's book is that it is a difficult mathematical problem... is it this what you implied?

The question is pretty much related to calculation of the capacitance of a conductor which can be proven to be a function of surface geometry only so also such a formula would do...

 

[eNtRopY]

Originally posted by dg
The question is pretty much related to calculation of the capacitance of a conductor which can be proven to be a function of surface geometry only so also such a formula would do...

Like I said, that would require an externally applied field. Clearly if you take a rectangular box shaped slab of copper there will be an even distribution of charge on the surface if there is no externally applied field.

eNtRopY

 

[Hurkyl]

That's not so clear...

Suppose we start off with a box-shaped slab of copper with equally distributed surplus negative charge.

Electrostatic repulsion would cause electrons to be repelled from the center of the box, and there is no external force pushing back, so electrons would tend to accumulate on the faces of the box only leaving a few in the interior.


So we have 6 faces each with equally distributed surplus negative charge. Again, for each face, electrostatic repulsion would push electrons away from the middle. Here there is an external force pushing back (the adjacent faces), but that external force is zero in the middle of the slab, so again we see that electrons would tend to vacate the center in favor of accumulating somewhat near the edges.


So then we have 12 edges each with equally distributed surplus negative charge. Same idea applies again and electrons would cluster on the corners.


So we have:

A whole lot of electrons clustered on the corners.
A lot of electrons clustered on the edges.
Some electrons clustered on the faces.
Few electrons in the interior.

Hurkyl

 

[dg]

Originally posted by eNtRopY
Like I said, that would require an externally applied field. Clearly if you take a rectangular box shaped slab of copper there will be an even distribution of charge on the surface if there is no externally applied field.

eNtRopY

Well if you want to make explicit where you see an applied field...

As far as the copper slab example, Hurkyl has already made the point pretty clear!

You should really consider some reviewing of basic electromagnetic theory before you keep on going along this thread!

 

[eNtRopY]

Originally posted by dg
Well if you want to make explicit where you see an applied field...

As far as the copper slab example, Hurkyl has already made the point pretty clear!

You should really consider some reviewing of basic electromagnetic theory before you keep on going along this thread!

Whether I'm right or wrong, you should learn some manners and treat people with respect... if you want to be taken seriously on these boards. I would like to add that the only reason I doubt my assertion is because Hurkyl gave a good argument... but I still have a doubt... and here it is.

If I am wrong, and surface charge is not evenly distributed on the surface of a rectangular slab or copper, then there would exist a charge gradient on the surface of this object. If such a gradient exists then there should be a voltage drop from the edge of the slab to the center of a face due to the Poisson Equation:

Laplacian V = rho / epsilon, where
V = electric potential,
rho = charge density, and
epsilon = permittivity.

Also, there should be electric field lines outside of the conductor.

E = - Grad V, where
E = electric field intensity.

So my question is, why don't we observe the effects of such field lines?

eNtRopY

 

[dg]

Originally posted by eNtRopY
Whether I'm right or wrong, you should learn some manners and treat people with respect... if you want to be taken seriously on these boards. I would like to add that the only reason I doubt my assertion is because Hurkyl gave a good argument... but I still have a doubt... and here it is.

If I am wrong, and surface charge is not evenly distributed on the surface of a rectangular slab or copper, then there would exist a charge gradient on the surface of this object. If such a gradient exists then there should be a voltage drop from the edge of the slab to the center of a face due to the Poisson Equation:

Laplacian V = rho / epsilon, where
V = electric potential,
rho = charge density, and
epsilon = permittivity.

Also, there should be electric field lines outside of the conductor.

E = - Grad V, where
E = electric field intensity.

So my question is, why don't we observe the effects of such field lines?

eNtRopY

Well sorry for being so direct in my advice but you sounded as much direct in your first reply
Said that we can keep going without further poking each other

The solution to your doubts goes as follow (remember here we are talking ideal conductors in classical electrostatics):

On a charged conductor in equilibrium, charge distributes so that internal electric field is zero and surface potential is constant. This in turn implies that electric field at the surface is normal to the surface and proportional to charge density.
So it is true that in general there is a gradient of both charge density and potential at the surface; since the potential gradient (electric field) is orthogonal to the surface at every point this does not create any potential difference between points on the surface. As far as the charge gradient is concerned this will be tangent to the surface but this will not create any additional force.

It is true that the charge creates an electric field (potential gradient) outside the conductor and on the surface, but this is not an externally applied field this is simply the field generated by the system itself.

Summarizing, once we deposit charge on a conductor the mutual interaction of charges through their electric fields and the constrain that charges cannot abandon the conductor require that the charge will redistribute until reaching the afore mentioned equilibrium condition: once the electric field is orthogonal to the surface at every point, the force on the charged surface element is compensated by the constrain reaction (surface potential barrier in a real conductor).

I can re-word it or add more if this not clear. Let me know!

Dario

 

[eNtRopY]

Originally posted by dg
Summarizing, once we deposit charge on a conductor the mutual interaction of charges through their electric fields and the constrain that charges cannot abandon the conductor require that the charge will redistribute until reaching the afore mentioned equilibrium condition: once the electric field is orthogonal to the surface at every point, the force on the charged surface element is compensated by the constrain reaction (surface potential barrier in a real conductor).

Okay, I will agree with that. If you have a surplus of uncompensated charge on a conductor, it will redistribute so to achieve an equilibruim. That's pretty basic. I must have misunderstood the statement:

We all know that charge tends to accumulate at edges (high curvature) but can we be more quantitative?

I thought you were trying to convince us that conduction electrons are unevenly distributed over the surface of a charge-balanced conductor.

eNtRopY

 

[dg]

Originally posted by eNtRopY
Okay, I will agree with that. If you have a surplus of uncompensated charge on a conductor, it will redistribute so to achieve an equilibruim. That's pretty basic. I must have misunderstood the statement:

I thought you were trying to convince us that conduction electrons are unevenly distributed over the surface of a charge-balanced conductor.

The presence of a microspic structure in real conductors changes quite a bit of things: it creates a band energy structure and create a sometime definite (if I well remember, in copper this classification depends on the direction of propagation along the lattice) difference between conduction charges and valence/bound charges. Nevertheless the geometry of the surface still influence the shape of local field even changing the lattice structure itself (surface relaxation) or exposing different crystallographic directions: you can bet that charge will be in general unevenly distributed to minimize the potential energy or equalize chemical potential between surface and bulk. The mechanism will be pretty much the same than in a charged conductor but with the complication introduced by the fixed charge distribution.

The question is really interesting but would be much more complicated so for now considering a conductor just as a neutral piece of matter were we can pour charge and this will freely adjust is more than enough.