Solving Electric Field Boundary Conditions Across a Dipole Layer

[Norman]

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

 

[Aaron Barnard]

The first step is to find the electric potential $\phi$ due to the dipole layer. To do this, you need to use the method of images to solve Laplace's equation in the plane x=0. Once you have the potential, then you can calculate the electric field components from the gradient of the potential.

The boundary condition that states that the tangential component of E is continuous across a boundary is based on Gauss' law. To review the derivation of this boundary condition, look at the proof for Gauss' law which states that the total flux of a closed surface is zero. This implies that the tangential component of the electric field must be continuous across a boundary as the electric field points in the same direction on both sides of the boundary. If the electric field points in different directions on either side of the boundary then the total flux across the boundary would not be zero.

 

[M98Ranger]

To solve this problem, you need to use the boundary conditions for electric fields. These conditions state that the tangential component of the electric field must be continuous across a boundary, while the normal component may have a discontinuity if there is a surface charge present.

In this case, the dipole layer introduces a surface charge density on the plane x=0. This means that there will be a discontinuity in the normal component of the electric field at this boundary. To find the electric field components, you can use the electric field equations:

E_x = -d[phi]/dx
E_y = -d[phi]/dy
E_z = -d[phi]/dz

Using these equations, you can determine the electric field components on either side of the boundary x=0. The discontinuity in the normal component of the electric field can be calculated using the surface charge density D(y,z) and the permittivity of the medium, ε0.

Regarding the belief in the boundary condition that the tangential component of the electric field is continuous, you can review the derivation of this boundary condition to see if and where it breaks down. One way to do this is to consider a simple case where there is no surface charge present (i.e. D(y,z)=0). In this case, the electric field should be continuous across the boundary x=0. From this, you can see if and where the derivation of the boundary condition breaks down.

Overall, to solve this problem, you need to use the boundary conditions for electric fields and the electric field equations. You can also review the derivation of the boundary condition to better understand its validity. I hope this helps point you in the right direction.