Calculate E Field between 4 cylinders

[strokebow]

Hi,

I have four cylinder with circular cross section. Each has the same potential (+V) applied to it. They are all the same size, length, diameter etc. and each are spaced an equal distance apart so that they all touch on to an imaginary circle with a distance x from the origin.

How to calculate the E field in between these cylinders?

 

[mfb]

Are the cylinders very long compared to their diameter? Then the field will be negligible. If not, you probably need a three-dimensional numerical approximation.

 

[strokebow]

Thank you mfb for ur reply. yes - they are very long compared to the diameter. but i was thinking that an analytical solution may b possible. if one were to treat this as say a 2d dimensional problem. so in say an x-y plane (assuming we are at the mid-point along the cylinders) we have 4 circles each with potential V applied.
any ideas?

 

[mfb]

If something is surrounded by areas of constant voltage, the electric field is zero. You only get an electric field from the regions at the ends of the cylinders, something a two-dimensional analysis (x/y) misses.

 

[strokebow]

If something is surrounded by areas of constant voltage, the electric field is zero. You only get an electric field from the regions at the ends of the cylinders, something a two-dimensional analysis (x/y) misses.

Thanks! I take your point. Nevertheless in the space in between the cylinders there exists an region which wud invoke a force on a unit positive test charge. So if say we assumed dat a unit positive test charge entered the region between the 4 cylinders how wud it behave? if we assumed say constant axial velocity and ignored the effects from the cylinders ends. what sort of motion wud the charge exhibit. can we describe that?

 

[mfb]

As long as the cylinders are kept at their potential, there would be no force on charged objects.

 

[strokebow]

As long as the cylinders are kept at their potential, there would be no force on charged objects.

I get that the field is zero But what if

a charged particle was moving axially in between the cylinders but very close to one of the cylinder surfaces. Do you mean 2 say that it would not be repelled more by that particular cylinder. . . Intuitively I am struggling to grasp this. Can any1 offer some help? Scientific, anecdotal or otherwise pls

Also let's say a charged particle wss again moving axially in between the cylinders but had some angular momentum which directed it towards a cylinder surface. .. surely in this case there must be a force exerted which will push the particle towards the 'central axis'? Any help wud b gladly received

 

[mfb]

If the cylinders have the same potential everywhere, nothing happens. Why should there be a force?

 

[strokebow]

Due to proximity. if i have 2 opposite charges and i try to bring them together the force will increase as they get closer, this is coulombs law as well all know. but i am trying to get a real grasp on this. just because the potential is the same on all the cylinders doesn't mean that a test charge will feel the same force,? is that true. i can grasp your point if u say that the test charge is exactly in the center. but what if that test charge moves closer to one of the cyinders, surely its influence is now greater than that of the other 3.
sorry if this sounds dumb but any sort of illustration you or any1 else can offer may help me
tks

 

[strokebow]

just incase i am not on the same page as u. this is what i was thinking of in the attahed png image. the black dot is the positive test charge. left is 3d view, right is a 2d cross section somewher along the middle of the cylinders
tks

 

[Khashishi]

Voltage isn't the same thing as charge. Charge has an absolute meaning. Voltage is only relative. You need to specify a zero point for your potential.

 

[mfb]

I interpreted post #1 as touching cylinders. If they do not touch, there will be some electric field. Yes, this field will be a bit stronger close to the cylinders, and a positive charge would be repelled by positively charged cylinders.

 

[strokebow]

Thanks. So is there a way I could determine by an equation motion of a charged particle moving axially between the cylinders? On the face it seems to me that there might be. If we assume the test charge has constant axial velocity and the cylinders are all the same with same Volts.

How would I begin to solve this?

And are your previous statements still correct for my drawing?

Tks

 

[mfb]

And are your previous statements still correct for my drawing?

No, with such a setup you have an electric field.
A two-dimensional numerical simulation will work. Make a fine grid, solve for the potential at every point with the Poisson equation, then calculate the electric field as the derivative.

 

[strokebow]

How can u be so sure that no analytical solution exists. How can u tell

Tks

 

[mfb]

I am not sure, but it would surprise me.

 

[strokebow]

ok. i thought maybe u had a way of telling.

so solving this numerically. for sake of making this easier for me to understand let us just say we only want to solve the potential at 1 point on the grid. i know the poisson equation. how do does someone actually use the poisson equation in practise? for my case with the 4 cylinders - or circles for this 2d case in question - can u give me any hints or tips or point me in the right direction.

tks

 

[mfb]

so solving this numerically. for sake of making this easier for me to understand let us just say we only want to solve the potential at 1 point on the grid.

That doesn't help, unfortunately.

For cartesian coordinates with a rectangular grid, the Poisson equation simplifies to "every cell has a potential that is the average of the 4 cells around it". Fix the potential of cells that are in your cylinders, fix the potential "far away" (borders of the simulation, far away from the cylinders), iterate the calculation of averages until the system does not change any more.

Edit: quick and dirty with excel:

 

[strokebow]

very nice how do u do it. how long did it take to calculate? i never thought of using excel.
few questions if u can pls.
what is far away and to what potential do u fix it?
is it possible that u can share the excel formula with the corresponding theory. that wud be helpful to see how u did it.

tks

 

[mfb]

very nice how do u do it.

As described. I fixed some cells (in four roughly circular patterns) to 1 and the boundary to 0, then I let each cell to be the mean of the four surrounding cells and let excel solve it iteratively.

what is far away

Every finite distance will introduce some error, but you cannot calculate infinite grids of course. Something that is far away compared to the size of the cylinder structure will keep the error small.

I attached the file. I changed the central potential values to 9 so the 1-digit display has a bit more details than "1" and "0", Everything changes linearly, so the absolute values are quite meaningless anyway.

 

[strokebow]

As described. I fixed some cells (in four roughly circular patterns) to 1 and the boundary to 0, then I let each cell to be the mean of the four surrounding cells and let excel solve it iteratively.
Every finite distance will introduce some error, but you cannot calculate infinite grids of course. Something that is far away compared to the size of the cylinder structure will keep the error small.

I attached the file. I changed the central potential values to 9 so the 1-digit display has a bit more details than "1" and "0", Everything changes linearly, so the absolute values are quite meaningless anyway.

tks

its a nice idea to do that with excel.
how do u manage to change the cewll colors to follow the magnitude in excel? its nice.

how would u then use the information provided from the grid to determine how a test charge might behave moving in the field between the cylinders. cud that be done in excel also?

TKS

 

[mfb]

I don't know how it is called in English, but probably something like "conditional formatting". There are options for spectra like that.

Excel is not a good tool for precise results, but it is quick to set up something there.

how would u then use the information provided from the grid to determine how a test charge might behave moving in the field between the cylinders. cud that be done in excel also?

Well, the difference between two cells is proportional to the electric field, you can calculate the position and velocity step by step. Excel is probably not the best tool for that, but again you can get some approximation with it.

 

[strokebow]

I don't know how it is called in English, but probably something like "conditional formatting". There are options for spectra like that.

Excel is not a good tool for precise results, but it is quick to set up something there.
Well, the difference between two cells is proportional to the electric field, you can calculate the position and velocity step by step. Excel is probably not the best tool for that, but again you can get some approximation with it.

just thinking that through. i must assume axial position/velocity is constant so each step moves by a certain axial distance and is a specific value of time. from an initial position and velocity <both for x and y> i wud have eE = ma. if i assume some nominal mass also. then at step 1 i have a new x velocity, new x position, new y position and new y velocity.
But for a rough approximation how do i link the field value from the excel sheet to E in the equation?

tks

 

[mfb]

The steps can be anything. You will get some error from the evaluation of the electric field and from the finite time step size. Nothing has to be constant.

But for a rough approximation how do i link the field value from the excel sheet to E in the equation?

For a very rough approximation, take the cell your particle is in, and the upper and left neighbor for the field estimate. For a better estimate, you can take into account more cells, or make the choice of neighbors dependent on the position inside the cell, or various other things.

 

[strokebow]

The steps can be anything. You will get some error from the evaluation of the electric field and from the finite time step size. Nothing has to be constant.
For a very rough approximation, take the cell your particle is in, and the upper and left neighbor for the field estimate. For a better estimate, you can take into account more cells, or make the choice of neighbors dependent on the position inside the cell, or various other things.

for rough approx. can i use u = v + at and s = ut + 0.5 *a*t*t
a = eE/m
where E is from excel collection of cells, m is some constant value.
then fot a 1st time step = T,
s<x> = x0 * T + 0.5*(eE/m) * T * T

or is there a better way? and is there some intuition which can tell u will the motion be of a certain type - for example, simple harmonic

 

[mfb]

Yes, that should work.

There are no stable orbits. Everything subject to this field only and close to the clinders will quickly hit one of the cylinders, or fly away to infinity.
Far away from the cylinders, an orbit around the group of cylinders is possible for negatively charged particles. The larger the distance the better the long-term stability.

 

[strokebow]

TKS

how did u come to that conclusion?

what was it about the arrangement that allowed u realize there will be no stable orbit?

 

[mfb]

Don't use "u" and similar "words" please.

The system is similar to orbital mechanics with gravity, multiple centers won't lead to stable orbits.

 

[rumborak]

This may be way beyond the OP's current capabilities, but this stuff, if it CAN be solved analytically at all, often involves transforming the coordinate system smartly,via conformal mapping.

E.g., like this:
https://www.google.com/url?sa=t&sou...-jkRZFoQN7_Ku3Q6A&sig2=l5_elHV9cl3ay7fRgCwGAg

(the problem there involves two cylinders)