Potential due to a charged plate using the dipole approximation

[sid0123]

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?

 

[Orodruin]

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.

 

[sid0123]

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?

 

[Orodruin]

Well, in what another way can I approach this problem?

What would be the potential of a general charge distribution $\rho(\vec x)$?

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.

 

[sid0123]

What would be the potential of a general charge distribution $\rho(\vec x)$?

This one?
Φ = ∫ (ρ×dV)/r

 

[Orodruin]

With the introduction of some additional constants, yes.

 

[sid0123]

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

 

[Orodruin]

Addition of which additional constant?

https://en.wikipedia.org/wiki/Coulomb's_constant

 

[sid0123]

https://en.wikipedia.org/wiki/Coulomb's_constant

Ahh. We follow CGS in our university. Our textbook (L&L) is CGS based.

 

[haruspex]

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?

 

[sid0123]

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.