Solve Dipole Layer Help: Electric Field Components Boundary Conditions

[Norman]

(Sorry for the duplicate post but I couldn't delete the old one in classical physics and didn't see this forum until I posted the original)

This is a homework question so please do not just tell me the answer, but please point me in the right direction.

A dipole layer, D(y,z), exists on the plane x=0. Find the boundary conditions (discontinuities, if any) for [phi](x,y,z), E_x(x,y,z),
E_y(x,y,z), and E_z(x,y,z) across the plane x=0. In view of this result do you believe in the boundary condition that the tangential component of E is contiuous across a boundary? Review the derivation of the boundary condition and see if and where the derivation breaks down.

When I read the first part of the problem I was content with how to solve it. The potential is discontinuous by D/[epsilon_0]. Then I would argue using typical boundary value knowledge that E_y and E_z are continuous and that E_x should be discontinuous. But after finishing reading the problem, it seems that my so called "notions" of the situation might be incorrect. Where do I start with finding the Electric Field components? I am very confused and any help would be very appreciated.
Cheers

 

[Norman]

Well, I guess no one knew how to solve this (so it goes with internet help I guess).

Anyways here was my solution method:

You know that the potential has a discontinuity of D/[epsilon_0]. Since you are looking at an infinite plate of charge (think of dipole layer as a sheet of positive sigma, parallel and infinitely close to a sheet of of negative sigma.) By symmetry, if you were to rotate the sheet still in the x=0 plane, the Electric Field should not vary, since we are talking about a dipole layer that doesn't vary with position. Therefore the Electric Field should only be in the x direction. Therefore, E_y and E_z are both zero. The x component of the Electric Field (E_x) has a discontinuity of 2*[sigma]/[epsilon_0] at x=0 boundary. This is seen by carefully examining the field inside and outside our boundary.
Sorry this is abreviated, have to teach in a couple of minute.

Anyone know if this is correct?
Cheers

 

[M98Ranger]

First, let's review the definition of a dipole layer. A dipole layer is a distribution of dipoles (two equal and opposite charges separated by a small distance) on a surface or plane. In this case, the dipole layer is on the plane x=0, which means that at every point on this plane, there is a dipole with its positive charge located at (0,y,z) and its negative charge located at (0,y,z+Δz). The strength of the dipole layer is denoted by D(y,z).

Now, let's consider the boundary conditions for the electric field components across the plane x=0. The first thing to note is that there will be a discontinuity in the potential, as you correctly stated, due to the presence of the dipole layer. This discontinuity can be written as Δ[phi] = D(y,z)/[epsilon_0].

Next, let's look at the tangential components of the electric field, E_y and E_z. These components are continuous across the boundary because there are no charges or dipoles present in the y or z directions. The only discontinuity that exists is in the x direction due to the dipole layer.

However, the x component of the electric field, E_x, will have a discontinuity across the boundary. This can be seen by considering the definition of the electric field, E = -∇[phi]. As we move across the boundary, the potential changes by Δ[phi], which means that the electric field will also change by ΔE = -∇(Δ[phi]). This change in the electric field results in a discontinuity in the x component.

Now, to answer the question about the tangential component of the electric field being continuous across a boundary, we can see that this is not always the case. In this specific problem, the tangential components are continuous, but this is not always true. In general, the tangential component of the electric field will be continuous across a boundary if there are no charges or dipoles present in the tangential direction.

In conclusion, to find the electric field components across the plane x=0, you will need to use the definition of the electric field and consider the discontinuity in the potential due to the dipole layer. It is important to keep in mind that the tangential component of the electric field may or may not be continuous