Potential due to a charged plate using the dipole approximation

AI Thread Summary
The discussion revolves around calculating the potential of a charged plane with a periodic charge density given by σ = σ° sin(αx) sin(βy). The user initially attempts to apply the dipole approximation but receives feedback that this method is inappropriate due to the infinite nature of the plane. Suggestions are made to consider alternative approaches for calculating the potential, including the use of a general charge distribution formula. The conversation also touches on the need for correct units and the distinction between finite and infinite plates. Ultimately, the user is encouraged to find the potential at any arbitrary point in space using an appropriate method.
sid0123
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Homework Statement


A plane z=0 is charged with density, changing periodically according to the law:
σ = σ° sin(αx) sin (βy)
where, σ°, α and β are constants.
We have to find the potential of this system of charges.

Homework Equations

The Attempt at a Solution


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I considered it as a plate of dimension l/2 X l/2 whose centre lies at the origin (0,0,0)
Then, I used the formula for the dipole:

d = ∫∫ (σ° sin(αx) sin (βy)(xy) dx dy and I put the limits -l/2 to l/2 for both x and y.

I feel I may have used the wrong formula for calculating the dipole. I feel kind of stuck. Help anyone?
 
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sid0123 said:
I considered it as a plate of dimension l/2 X l/2 whose centre lies at the origin (0,0,0)
Then, I used the formula for the dipole:
The question suggests that the plane is infinite. This makes the dipole approximation a bad one. I believe you are barking up the wrong tree here and you might want to consider a different approach.
 
Orodruin said:
The question suggests that the plane is infinite. This makes the dipole approximation a bad one. I believe you are barking up the wrong tree here and you might want to consider a different approach.

Well, in what another way can I approach this problem?
And if the plate was finite, is my formula right for that case?
 
sid0123 said:
Well, in what another way can I approach this problem?
What would be the potential of a general charge distribution ##\rho(\vec x)##?

sid0123 said:
And if the plate was finite, is my formula right for that case?
No. It has the wrong units and is not a vector (the dipole moment is a vector). The potential would also only be correct far away from the plate.
 
Orodruin said:
What would be the potential of a general charge distribution ##\rho(\vec x)##?

This one?
Φ = (ρ×dV)/r
 
With the introduction of some additional constants, yes.
 
Orodruin said:
With the introduction of some additional constants, yes.

Addition of which additional constant?
And also, in my case, will the potential be calculated as Φ = (σ(x,y)/ r) ds
 
  • #10
sid0123 said:
We have to find the potential of this system of charges.
Potential at any point in space or at some particular point?
Are you told to use a dipole approximation, or just some approximation, or does it not mention approximations at all?
 
  • #11
haruspex said:
Potential at any point in space or at some particular point?
Are you told to use a dipole approximation, or just some approximation, or does it not mention approximations at all?
I just confirmed it and it says by any method.
And potential at any arbitrary point in the space.
 
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