Potential due to a charged plate using the dipole approximation

In summary, the problem involves finding the potential of an infinite plane with changing charge density according to the law σ = σ° sin(αx) sin (βy). The suggested approach of using the dipole approximation is not applicable since the plane is infinite. Instead, the potential can be calculated using the formula Φ = ∫ (ρ×dV)/r, with the introduction of additional constants. The textbook being used is based on the CGS system and the problem does not specify any particular approximation to be used. The potential is to be calculated at any arbitrary point in space.
  • #1
sid0123
49
4

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


[/B]
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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  • #2
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.
 
  • #3
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?
 
  • #4
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.
 
  • #5
Orodruin said:
What would be the potential of a general charge distribution ##\rho(\vec x)##?

This one?
Φ = (ρ×dV)/r
 
  • #6
With the introduction of some additional constants, yes.
 
  • #7
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.
 

1. What is the dipole approximation method?

The dipole approximation method is a technique used in physics to simplify the calculation of the potential due to a charged plate. It assumes that the plate is made up of two opposite charges, which cancel each other out, and the resulting potential is calculated using the distance between these two charges.

2. How does the dipole approximation affect the accuracy of the potential calculation?

The dipole approximation method is a good approximation for calculating the potential due to a charged plate as long as the distance between the charges is small compared to the distance from the plate. However, for larger distances, the accuracy of the calculation decreases.

3. Can the dipole approximation method be used for any shape of charged plate?

No, the dipole approximation method is only applicable for plates with a uniform charge distribution and a negligible thickness. If the plate has a non-uniform charge distribution or is not thin, this method will not yield accurate results.

4. How is the dipole moment related to the potential due to a charged plate?

The dipole moment is a measure of the separation and magnitude of the two opposite charges that make up a dipole. In the dipole approximation method, the potential is calculated using the dipole moment of the charged plate.

5. Are there any other methods for calculating the potential due to a charged plate?

Yes, there are other methods such as the method of images, which takes into account the boundary conditions at the edges of the plate, and the method of moments, which uses the charge distribution on the plate to calculate the potential. These methods may provide more accurate results, but they are more complex and require more advanced mathematical techniques.

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