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I've been going through the "Foundations for microwave engineering" textbook by Robert E. Collin in order to study coplanar waveguides. In doing so I was introduced to Conformal mapping techniques, a beautiful way of calculating the capacitance of such systems by exploiting the Schwarz-Christoffel transformations.

In doing the derivation it states that the "By symmetry the two boundaries are magnetic walls on which $\frac{d\phi}{dv}=0$". Here v is the imaginary part of the complex plane, yet it can be seen as the physical y direction. Without worrying about the symmetry, what does it mean by magnetic walls? there seems to be no answer on the internet.

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  • $\begingroup$ Can you share a diagram, showing which direction is called "y" relative to the dimensions of the waveguide? $\endgroup$ – The Photon Jul 22 '19 at 21:58
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Magnetic wall usually means "perfect magnetic conductor" (PMC), which is the dual (or magnetic analogue) of a perfect electric conductor. Inside the PMC, $\vec{H} = 0$ and $\rho_m = 0$, where $\rho_m$ is the magnetic charge density.

The boundary conditions for a PMC are $$\hat{n} \times \vec{E} = -\vec{K}_m$$ $$\hat{n} \times \vec{H} = 0 $$ $$\hat{n} · \vec{D} = 0 $$ $$\hat{n} · \vec{B} = \rho_m $$ where $\hat{n}$ is the outward unit normal vector and $\vec{K}_m$ is the magnetic surface charge density.

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The text you quote is defining the term. The "magnetic wall" is a surface on which $\frac{d\phi}{dv}=0$.

Although I'm not 100% clear on the definition of terms in the text you're reading, it's likely called a magnetic wall because you could obtain the same boundary condition by putting a wall formed from a perfect magnetic conductor at the same location (if there were any such material).

It's an analogy to the placing an electrically conductive wall at a plane of symmetry where the appropriate components of the E-field go to zero.

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