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electrostatic boundary conditions

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If there is a field described by the vectors A and B. These are related by =k where k is a constant. The two vectors are found to satisfy the differential equations
1) =0
2) =0

1. Using these relations derive the boundary conditions that relate the values of the components of the vectors and on the two sides of a boundary between materials 1 and 2, where the constant k has two different values k and k in materials 1 and 2 respectively. So please give your answer in terms of k and k .
2. Now suppose that the boundary occupies the x-y plane (z=0). In medium 1 vector has components and =1. In medium 2, =2. Find the components of and in medium 2.

Please explain the steps as you go. I want to be able to understand how this problem was solved and what is happening.

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Solution Preview

Please see the attached file.

Note:- Comparison of A and B
From the description in the introductory part of the question we can assume the electric field instead of and the displacement vector instead of . It is not correct to consider them as the magnetic and electric fields or the magnetic vector potential and the magnetic fileds because the relations given in the question never holds for them.
Here and hold the relation or more comfortably
, where
In terms of A and B we will get =k or
from the conditions 1) =0 and 2) =0 it clear that the dielectric interface is charge free.

1. From =0
The boundary condition at a ...

Solution Summary

The electrostatic boundary conditions on the two sides of a boundary between materials 1 and 2, where the constant k has two different values k1 and k2 in materials 1 and 2 respectively is calculated in terms of k1 and k2 . and teh components of the vectors in both media were calculated. .