Bound surface charge on a linear dielectric half-cylinder

[Daniel Sellers]

Homework Statement

Problem statement in attached photo. This is an ungraded assigned problem which I am using to study for an exam, so I don't need the whole solution just help with a couple of points I am confused about.

One: Part d) is really important to how I will answer part b). If we can use Gauss's law to find D then part b) should be very easy. If we can't then I suppose I will have to use boundary values to solve for P somehow in order to find the bound charges.

I don't see any reason why we shouldn't be able to, but one (very flawed) online solution said we could not, and the fact that we are asked the question makes me nervous about proceeding this way, so I wanted to ask some of the critical experts that frequent this forum.

Two: Why is the fact X_e << 1 relevant? This also seems to point me towards using boundary value equations to solve the problem, but I'm simply not familiar enough with this material yet to confidently set up the problem.

Homework Equations

P = ε₀χ_eE

Gauss's law for electric displacement?

D = ε₀E + P = εE

The Attempt at a Solution

I am fairly confident that I have solved part a) correctly and that I can solve the rest if the above points are clarified

 

[TSny]

Homework Statement

One: Part d) is really important to how I will answer part b). If we can use Gauss's law to find D then part b) should be very easy. If we can't then I suppose I will have to use boundary values to solve for P somehow in order to find the bound charges.

I don't see any reason why we shouldn't be able to...

In order to use Gauss' law, there needs to be enough symmetry so that there will exist a particular shape of the Gaussian surface that will allow you to simplify $\oint \mathbf{D \cdot da}$. So, think about whether there is sufficient symmetry in this problem.

Two: Why is the fact X_e << 1 relevant?

This will allow you to use approximations in which you only need to be accurate to first order in $\chi_e$. For example, it should be easy to find an expression for the polarization $\mathbf{P}$ that is accurate to first order in $\chi_e$ if you know $\mathbf{E}$ due to the line charge.

 

[Daniel Sellers]

When considering the free charge only, we have cylindrical symmetry, but that symmetry does not apply to the entire problem. So for now I will abandon using Gauss's law.

As for χ_e, assuming that it is very small let's us say that ε ≈ ε₀. This would suggest that the electric field in the dielectric is approximately the field we would see from the charged wire alone.

Then P ≈ χ_eλ/(2πr) in the radial direction.

Is this close to what you meant?

 

[TSny]

When considering the free charge only, we have cylindrical symmetry, but that symmetry does not apply to the entire problem. So for now I will abandon using Gauss's law.

Right.

As for χ_e, assuming that it is very small let's us say that ε ≈ ε₀. This would suggest that the electric field in the dielectric is approximately the field we would see from the charged wire alone.

Then P ≈ χ_eλ/(2πr) in the radial direction.

Is this close to what you meant?

Yes, exactly what I meant.

 

[Daniel Sellers]

Terrific! Back to practicing electrostatics problems then.

Thanks for your help!