Induced surface charge on a dielectric

[camcors]

Homework Statement

A point charge Q is placed at a perpendicular distance a from an infinite planar LIH (linear, isotropic, homogeneous) dielectric of relative permittivity ε. By considering the normal component of the electric field at an arbitrary point just inside the dielectric to be the superposition of the normal components of the electric fields produced by the point charge Q and the induced surface charge
σ on the dielectric, show that

σ=- (ε-1)Qa /2∏(ε+1)r^3

Homework Equations

Normal component due the point charge will be:

E=Qa/(4∏ε0εr^3)

Induced surface charge due to the electric field from Q will be:

σ=P.n (P is the polarisation)
σ=(ε-1)E

The induced surface charge produced its own electric field which is obtained from gauss' law and is just the surface charge divided by ε0.

The Attempt at a Solution

So the difference between these two fields gives the resultant field at the point just inside the dielectric. But I'm uncertain as to why the questions then wants another value for surface charge density, will this change because of the resultant field. I thought this was induced anyway such that it already accounts for the field it produced and that of the point charge. I'm therefore confused as to how to proceed.

 

[camcors]

So yes just to clarify, I think the two fields I've calculated are correct. However, I don't really understand why the question is asking for the induced surface charge in this way when you already know it from the dotting the polarisation with the unit normal. Am I confusing some of the concepts here?

 

[TSny]

Hello camcors and welcome to PF!

σ=P.n (P is the polarisation)
σ=(ε-1)E

Shouldn't the second equation above be σ= ε_o(ε-1)E_n where E_n is the normal component of the total electric field just inside the surface of the dieletric?

I think the idea of the problem is to find σ by using this equation and writing E_n as the superposition of two electric fields: (1) the normal component of the field of the point charge and (2) the field due to the surface charge density σ.

Then you will have an equation with σ on both sides and you can solve for σ.

The induced surface charge produced its own electric field which is obtained from gauss' law and is just the surface charge divided by ε0.

Yes, but the field produced by the surface charge σ is not quite σ/ε_o. There should be a factor of 2 somewhere in there. (You might be thinking of the field at the surface of a conductor with charge density σ.) If you have a flat, infinite sheet of uniform charge density σ and use Gauss' law to find E on each side of the sheet, what do you get?

 

[camcors]

Yes that's what I meant slight typo! However, the way I defined E was already as the normal component. Thanks for your help I've solved this now, you're right it was a case of solving for sigma by including it in both terms.

 

[paranormal]

I can provide a response to the content and help clarify the confusion.

Firstly, it is important to understand that the induced surface charge is a result of the electric field produced by the point charge Q and the dielectric's own polarisation. The polarisation, P, is a measure of the dipole moment per unit volume of the dielectric material.

Now, let's consider the normal component of the electric field at an arbitrary point just inside the dielectric. This field is a superposition of the fields produced by the point charge Q and the induced surface charge σ. This can be expressed as:

E = E_Q + E_σ

Where E_Q is the electric field produced by the point charge Q and E_σ is the electric field produced by the induced surface charge σ.

Using Gauss' law, we can express E_σ in terms of the surface charge density σ as:

E_σ = σ/ε0

Now, we can substitute this value of E_σ in the equation for the normal component of the electric field and equate it to the superposition of the fields:

E = Qa/(4∏ε0εr^3) + σ/ε0

Substituting the value of σ from the equation given in the homework statement, we get:

E = Qa/(4∏ε0εr^3) - (ε-1)Qa/(4∏ε0ε^2r^3)

Simplifying the equation, we get:

E = Qa/(4∏ε0εr^3) * (1-ε+1)/(ε+1)

E = Qa/(4∏ε0εr^3) * (ε/ε+1)

Now, we can express the normal component of the electric field in terms of the polarisation P as:

E = P/ε0

Therefore, we can equate the two equations and solve for the induced surface charge density σ:

P/ε0 = Qa/(4∏ε0εr^3) * (ε/ε+1)

σ/ε0 = (ε-1)Qa/(4∏ε0εr^3) * (ε/ε+1)

σ = (ε-1)Qa/(4∏εr^3) * (ε/ε+1) * ε0