Bound surface charge of a cylinder

[Taylor_1989]

Homework Statement

Consider a cyclinder made of linear dielectric material with uniform dipole distribution. An externally applied field ##E_{ext}## is applied in the direction parallel to the axis of the cyclinder. What are the values of the bound chrages at the surface.

Homework Equations

##\sigma_{bound} = P . \hat n##

The Attempt at a Solution

I can't seem to upload the drawing from my phone, so I will do my best to explain what I have done so far.

I drew a cylinder along the x-axis and labled each surface as follows

end of cylinder in negtive x region ##S_1##

body of cyclinder ##S_2##

end of cyclinder in postive x region ##S_3##

The electric field is going from ##S1## to ##S3##

So my values for the give surfaces are as follows

##S1=|P||1|cos(180)=-P=\sigma_b##

##S2=|P||1|cos(90)=0=\sigma_b##

##S3=|P||1|cos(0)=P=\sigma_b##

are these the value that the question is looking for or have I missed what the question is asking me for

 

[kuruman]

I believe your answers must be in terms of the given external electric field, ##E_{ext}##. You may need to consider as given a permittivity ##\epsilon## for the material.

 

[Taylor_1989]

I believe your answers must be in terms of the given external electric field, ##E_{ext}##. You may need to consider as given a permittivity ##\epsilon## for the material.

Im not quite sure what you mean? Could you please expand

 

[kuruman]

Your answer would be correct if P were a given quantity. It is not. The given quantity is ##E_{ext}## so your answer must have ##E_{ext}## in the final expression on the right side of the expression for ##\sigma_b##. However, this cannot be done unless you assume that ##\epsilon## is also a given quantity because it too must appear on the right side of the expression for ##\sigma_b##. That's my interpretation of what the questioner expects the answer to be.

 

[Taylor_1989]

So polaristaion can be wirtten in the form of

##P=X\epsilon E##

 

[kuruman]

It's probably more straightforward to use ##\vec{D}=\epsilon_0\vec{E}+\vec{P}##, ##\vec{D}=\epsilon\vec{E}## and the boundary condition at the flat surface of the cylinder.