What does this boundary condition mean?

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SUMMARY

The boundary condition for a homogeneous uniform waveguide, represented as \(\frac{\partial H_z}{\partial n}=0\), indicates that the magnetic field component \(H_z\) does not change in the direction normal to the boundary. This condition arises from the equation \(n \cdot B = 0\) and is derived by applying the dot product to the relevant equations from Jackson's "Classical Electrodynamics." Specifically, it leads to the conclusion that \(H_z(x,y) \sim \cos(m\pi x/A) \cos(n\pi y/B)\) for an \(A \times B\) rectangular waveguide.

PREREQUISITES
  • Understanding of electromagnetic wave propagation
  • Familiarity with boundary conditions in waveguides
  • Knowledge of vector calculus, specifically derivatives and dot products
  • Access to Jackson's "Classical Electrodynamics" for reference
NEXT STEPS
  • Study the derivation of boundary conditions in waveguides from Jackson's Eq. (8.24)
  • Explore the implications of the boundary condition \(\partial_n(B_z)=0\) in cylindrical waveguides
  • Investigate the mathematical properties of the cosine functions in waveguide modes
  • Learn about different types of waveguides and their respective boundary conditions
USEFUL FOR

Students and professionals in electrical engineering, particularly those specializing in electromagnetic theory and waveguide design.

HasuChObe
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One of the boundary conditions for a homogeneous uniform waveguide is \frac{\partial H_z}{\partial n}=0. What does this mean physically?
 
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I tried to put it into words, but the equation is clearer.
d/dn means the derivative in the direction perpendicular to the wall.
 
Hello! I have a question related to this. This boundary condition yield from n.B = 0 , but i don't know how, from considering a cylindrical waveguide. I know that there is an equation (first of 8.24 from Jackson) but i don't realice how to use it. If u know, please let me know. Thankss
 
You must mean Eq. (8.30): \partial_n(B_z)=0 at the surface.
It follows from (8.24) by dotting it with n. The two terms on the LHS are zero, giving the BC on B_z. Then, it follows That H_z(x,y)\sim\cos(m\pi x/A)\cos(n\pi y/B)
for an AXB rectangular guide.
 
Last edited:
Thank you so much!
 

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