Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

The guide of Figure 7.4.1 is a perfectly conducting tube whose inner wall is the cylinder $ρ = a$ . It is filled with a homogeneous lossless medium $ε$ , $μ$ (air in most guides), and the wave travels along $+ z$ . The natural coordinates are the cylindrical ones of Section 2.3, $(ρ, ϕ, z)$ , because the wall is then the surface on which one coordinate is constant, and every boundary condition involves one coordinate only.

The circular waveguide. Left: a hollow metal tube of inner radius a, filled with a lossless medium, guides a wave that varies as \mathrm{e}^{-\mathrm{j}\tau z} along its axis. Right: the cross-section with the cylindrical coordinates (\rho,\phi) and the unit vectors \vec a_\rho, \vec a_\phi at a point. The wall is the coordinate surface \rho=a. — **Figure 7.4.1.** The circular waveguide. Left: a hollow metal tube of inner radius $a$ , filled with a lossless medium, guides a wave that varies as $e^{- j τ z}$ along its axis. Right: the cross-section with the cylindrical coordinates $(ρ, ϕ)$ and the unit vectors $a_{ρ}$ , $a_{ϕ}$ at a point. The wall is the coordinate surface $ρ = a$ .

On a perfect conductor the tangential electric field vanishes. The tangential directions on the wall $ρ = a$ are $a_{ϕ}$ and $a_{z}$ , so

E_{z} (a, ϕ) = 0 and E_{ϕ} (a, ϕ) = 0 for all ϕ . (7.4.4)

The radial component $E_{ρ}$ is normal to the wall and may stay nonzero there: it ends on a surface charge $ρ_{S} = - ε E_{ρ} (a)$ (the normal of the wall points into the guide, along $- a_{ρ}$ ). Since $B$ is zero in the metal and $B_{n}$ is continuous, the normal magnetic field $H_{ρ}$ vanishes at the wall as well. We shall see that this second condition is satisfied automatically once Eq. (7.4.4) is.

Why cylindrical coordinates

In Cartesian coordinates the circle $x^{2} + y^{2} = a^{2}$ mixes both coordinates, and a condition such as “ $E_{ϕ} = 0$ on the wall” becomes a relation between $E_{x}$ and $E_{y}$ that changes from point to point. In cylindrical coordinates it reads $E_{ϕ} = 0$ at $ρ = a$ , for every $ϕ$ at once. Choosing coordinates in which the boundary is a coordinate surface is the key step of every boundary-value problem, exactly as the planes $x = 0, a$ and $y = 0, b$ were for the rectangular guide.

7.4.1 The circular pipe and its boundary conditions