Potential in parallel plate system from charge density?

[smileandbehappy]

Homework Statement

Note - I have used an image here. For the simple reason I don't want to confuse anyone by missing out subscripts. But if that is not acceptable let me know and I will attempt to write it out.

Homework Equations

I have considered using Poissons equation - but couldn't get that to work.

I have also attempted to consider this as a parallel plate capacitor, so: u = 0.5e(o)E^2

The Attempt at a Solution

I am getting stuck with this. Annoyingly I think it's probably quite simple - but it's confusing me.

I have drawn a diagram here to understand what is going on, but I don't think I am allowed to post too many photo's. And have no idea how to draw that on a computer.

So I thought I could find the E field magnitude. To do this I could use the boundary conditions. But I am not going from air into a dialectic medium. So I got a bit confused there. I don't think I can say the E=V/d because of the dialectic material - however I do think I can say: E = eV/d

Subsitituting this into the above equation I get:

u = 0.5e(0)(eV/d)^2

Now the change in V is going to be V(0) so:

u = 0.5e(0)e^2V(o)^2/d^2

And I know that charge density = charge/volume

therefore: charge/volume = p(0)/d

But here I get stuck... I am at a bit of a loss. As I've said I think it's probably simpler than I am making it. But some hints would be amazing.

Thanks

Sam

 

[Orodruin]

I have considered using Poissons equation - but couldn't get that to work.

What were your thoughts on this exactly?

 

[smileandbehappy]

Well as the plates are infinitely long - you are only dealing with d^2x/dx^2 = -p/e

So I tried to integrate this twice to solve the equation - and then applied boundary condition are V(x=0)=0 and V(x=d) = V(o)

But I still have no idea where to go after this...

 

[Orodruin]

If you have done things correctly, you then have the potential and there should not be anywhere else to go ...

 

[smileandbehappy]

If you have done things correctly, you then have the potential and there should not be anywhere else to go ...

Here is my problem - I am not doing it properly, or not understanding the question then!

I've posted my work below. I can't type that out, as I don't know how to put the differentials in etc... Or the integral signs. But I am not sure how to do get rid of the x. I am failing to understand this badly.

Basically - I am doing something awfully wrong. I though x=d, but when I do that all I get is V = V(o) which is clearly not what they are after.

 

[Orodruin]

Why do you want to put x = d? The coordinate x is perfectly fine as a parameter in your answer as it describes the potential at the point x. Since the problem asks for the potential at any point, the result will generally be a function of the point, in this case x. You have essentially solved the problem without realising that you already have the answer.

 

[Orodruin]

Also, I recommend reading this https://www.physicsforums.com/help/latexhelp/ for typesetting issues.

 

[smileandbehappy]

Thanks on both making me see how stupid I am being - and pointing out how I can use latex. I think I am trying to make a lot of this stuff harder than it has to be... Either that or I'm just a bit thick. But I appreciate you helping me out. So all I need to do now is substitute in the expression for the charge density and I'm good to do.

Thanks again.

 

[Orodruin]

I think I am trying to make a lot of this stuff harder than it has to be...

Some of the most difficult problems are those whose answers are staring us in the face ...

So all I need to do now is substitute in the expression for the charge density and I'm good to do.

Yes.