Solving E&M Problem on Infinitely Long Cylinder

[ayoshi]

Homework Statement

infinitely long conducting cylinder (hollow, so outter radius = b and inner radius = a) which is grounded is in a constant E-field (E=Eo in direction of cylinders' axis) has a dielectric substand "painted" on it, with dielectric constant c and thickness D. Find the electric field everywhere!

Homework Equations

Well, for dielectrics the usual BC's are:
E[parallel](inside) = E[parallel](outside)
D[perpendicular](inside) - D[perpendicular](outside) = sigma <-- charge density

Also, the general solution for potential in cylindrical coords is:
V = Ao + Bo*ln(r) + $$\sum(An*r^n +Bn*r^(^-^n^))*(Cn*cos(n*phi)+Dn*sin(n*phi)$$

BC for infinite cylinder:
dV/dz = 0
V = -Eo*r*cos(phi) + constant, as r-->infinity (due to constant E-field.)

The Attempt at a Solution

I'm having some serious trouble with this problem. One thing I do know, however, is that all of the Dn*sin(n*phi) terms contribute nothing by symmetry and so we can kiss the Dn's goodbye. I think the BC's I stated above are correct, but I'm having problems using them. Other questions I have are: i) how does the dielectric effect V, and, ii)despite being grounded, wouldn't the external field effect the cylinder hence making it act like a dipole?

Finally, can we assume the E-field inside is zero since it is a conductor? Or does the dielectric and external E-field change this? The E-field in the region a<r<b should be zero too, shouldn't it?

Any help would be great!

 

[Jhero]

Hello, thank you for your post. I can see that you have a good understanding of the problem and the relevant equations. However, I would like to clarify a few points and provide some guidance on how to approach this problem.

Firstly, you are correct in assuming that the Dn*sin(n*phi) terms contribute nothing by symmetry and can be ignored. This is because the dielectric coating is axisymmetric and does not have any angular dependence.

Next, let's address your questions. The dielectric coating will affect V by changing the permittivity (epsilon) in the region between the inner and outer radii of the cylinder. This will change the electric field inside the cylinder, as well as the potential. As for your second question, the external E-field will certainly have an effect on the cylinder and may cause it to act like a dipole. However, since the cylinder is grounded, any induced charges on the surface will be neutralized and will not affect the overall electric field.

Now, to solve for the electric field everywhere, we can use the boundary conditions you have stated. However, we need to take into account the fact that the dielectric coating will also affect the electric field inside the cylinder. This can be accounted for by using the relationship between D and E in dielectrics: D = epsilon*E. We can also use the fact that the electric field is conservative, which means that the potential V can be written as the negative gradient of V. This will allow us to use the general solution for potential in cylindrical coordinates that you have provided.

Overall, to solve for the electric field everywhere, we need to solve for the potential V(r, phi, z) using the boundary conditions and incorporating the dielectric coating. Then, we can use the relationship between D and E to solve for the electric field E(r, phi, z) everywhere.

I hope this helps and provides some guidance for solving this problem. Good luck!