Boundary Conditions in a Waveguide

[a.merchant]

Homework Statement

Derive the boundary conditions for the B field imposed by divB=0I'm lost with this question, I don't really understand how boundary conditions work. D.J. Griffiths only really mentions how to arrive at the conclusion that:
B$_{1}$$\bot$-B$_{2}$$\bot$=0
but doesn't outline the method enough so that I can understand it. Any help would greatly be appreciated, even if its just a link in the right direction to understanding the concept.

Thanks

 

[mark726]

for your post! Boundary conditions are an important concept in physics, and understanding them is crucial for solving many problems in electromagnetism. In the case of the B field, the boundary conditions are a set of rules that determine how the B field behaves at the interface between two different materials or regions.

To derive the boundary conditions for the B field from the equation divB=0, we need to consider two different regions, say region 1 and region 2, with different values of B. Let's assume that the regions are separated by a surface S, as shown in the diagram below.

We can apply the divergence theorem to the equation divB=0, which states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of that field over the enclosed volume. In mathematical terms, this can be written as:

∫∫B⋅dS = ∫∫∫(∇⋅B)dV

where ∫∫B⋅dS represents the flux of B through the surface S, and ∫∫∫(∇⋅B)dV represents the volume integral of the divergence of B over the enclosed volume.

Using this theorem, we can write the equation as:

∫∫B1⋅dS + ∫∫B2⋅dS = ∫∫∫(∇⋅B1)dV + ∫∫∫(∇⋅B2)dV

Now, since divB=0, the volume integrals on the right-hand side will be equal to zero. This means that:

∫∫B1⋅dS + ∫∫B2⋅dS = 0

This equation tells us that the total flux of the B field through the surface S is equal to zero. From this, we can derive the boundary conditions for the B field:

1. The tangential component of B must be continuous across the boundary, meaning that the tangential component of B1 must be equal to the tangential component of B2 at any point on the surface S.

2. The normal component of B may have a discontinuity across the boundary. In other words, the normal component of B1 may not be equal to the normal component of B2 at any point on the surface S.

To summarize