Dear all:
I hope someone can give me some insight to this problem.
Considering that I have a micro size metallic particle. Lets say a symmetric Gold cubic particle, about 6 micrometers of lenght.
What physical framework do I have to use if I want to find the Electric Field distribution coming out from this particle ?
Will Maxwell Equations be enough for this case ? Ohm's Law ? some consideration of some Quantum Mechanical Effects ? Consideration of Plasmons ?
I'am posing this problem cuz I'm attempting to solve an Octahedral Gold particle of these dimensions in COMSOL Multiphysics, where I need to compute the E field distribution. Therefore I continue thinking if is as straightforward as to set the system of linear PDEs using just Maxwell Equations and Ohm's Law, and obtain physical representative results.
Please, any insight will be helpful
Regards ! :)
Dear Franklin Uriel Parás Hernández ,
"electric field" sounds like a static problem. If indeed you consider an electrically charged resting particle then you need only one of the Maxwell equations: div D = rho, and an additional condition is that the electric field has to be perpendicular to the surface of the particle.
Consequently, the highest charge density and the highest field strength will be at the corners of the particle. Beyond some distance from the particle, I guess about 5 times the length of the edges, the electric flux density will be virtually radial, its strength independent of the direction, and proportional to 1/r2, with r denoting the distance from the center of the particle.
If the particle is moving, as it probably will due to thermal energy, then it is surrounded by an electromagnetic field, you will need indeed the complete set of Maxwell equations, Ohm's law, and Newton's laws of motion, and the problem is more difficult due to the fact that the electromagnetic field acts back on the particle, in particular on its charge distribution. Generally, if there is an external electric, magnetic, or electromagnetic field present then the moving particle will generate an electromagnetic field even if it is electrically neutral.
Additionally, if there are other particles present, the gold particle and the other particles will be coupled electromagnetically, and the whole system will radiate (and probably receive) (thermal) energy. However I think if you consider this last case then you can treat all particles as point particles.
I don't know enough about plasmons to make a reliable statement. Hope someone else can fill this gap.
Dear Joerg Fricke :
Indeed what I look for is for the simplified electrostatic charge problem. So for this case the only equation I have to solve is Gauss Law? Is it right? What about the Continuity Eq. in EM theory ?: Div(J)=0 ? However, writing this down, I quickly realized that as long as I don't have any Electric Currents this equation has no use so ever. Am I right?
I want to solve an alike problem with COMSOL Multiphysics approaching it with this first approximation: Zero time-dependent Electric charge/distribution and no Electric Currents.
So, solving for the Gauss Law should be enough to know the distribution of the Electric Field around the particle ?
On the other hand, the actual geometry of my particle is not spherical, but octahedral, as in the image below.
Thank you for you valuable help sir ! :)
Regards !
Dear Franklin Uriel Parás Hernández ,
as far as I can see there is no closed-form expression that describes the surface of the particle (in contrast to an ellipsoid, for example). So I feel there is as well no closed-form expression for the charge density on the surface. Instead, you'll have to run a simulation to get the charge distribution.
If you want results fast, you should use commercial software (I don't know the features and limitations of COMSOL).
If you want to have fun, you could write a simulation yourself. It boils down to:
1. Split up the surface in small "tiles", triangles should work fine, of course in compliance with the symmetry of the particle (12 equal resp. mirrored parts).
2. Assign a charge to each tile; for the sake of simplicity to each tile the same amount. In the following step, treat each charge as a point charge located in the middle of the tile.
3. Calculate the electric field for the midpoint of each borderline between tiles. Due to symmetry, limit the calculation to one of the 12 parts mentioned above. But you have to take the charge on the whole particle into account, and add each contribution vectorially. This is where Gauss law comes into play, because the 1/r2 dependence is due to it.
4. Calculate the component of each E field vector that lies in the plane perpendicular to the border between the adjacent tiles. If this component splits the angle between the tiles in half, left the amount of charge in both tiles unchanged. If not, transfer a small amount of charge between the tiles, according to the E field vector. Apply the same transfer to the corresponding tiles on each of the 11 other surface parts.
5. If there were charge transfers (above a lower limit) during step 4, continue with step 3. Otherwise, the calculation of the charge distribution is complete.
Now, based on this charge distribution, you can calculate the E field at arbitrary locations in space.
Dear Joerg:
Thank you sir for this valuable knowledge,
Regards ! :)
Paweł Kwaśny :
So I have to compute the surface charge density distribution of my geometry, which I guess this will be the difficult part. I am planing to use COMSOL Multiphysics to model this hollow octahedral microstructure. I hope I can get the charge density distribution using this software and next solve for the E field.
Regards !
An example would be the distribution of electrical charges in an electrolytic capacitor according to electrostatics (Gauss law) at a given electrical voltage on the capacitor.
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