Classical Electrodynamics · Problem 9.21

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · tm-waveguide-angular-momentum

Problem

The fields representing a transverse magnetic wave propagating in a cylindrical waveguide of radius RR are: Ez=Jm(γr)eimϕeiβz−iωt,Hz=0Eϕ=−mβγ2Ezr,Hr=−kZ0βEϕEr=iβγ2∂Ez∂r,Hϕ=kZ0βEr\begin{aligned} E_z &= J_m(\gamma r)e^{im\phi}e^{i\beta z - i\omega t}, \quad H_z = 0 \\ E_\phi &= \frac{-m\beta}{\gamma^2}\frac{E_z}{r}, \quad H_r = -\frac{k}{Z_0 \beta} E_\phi \\ E_r &= \frac{i\beta}{\gamma^2}\frac{\partial E_z}{\partial r}, \quad H_\phi = \frac{k}{Z_0 \beta} E_r \end{aligned} where mm is the index specifying the angular dependence, β\beta is the propagation constant, γ2=k2−β2\gamma^2 = k^2 - \beta^2 (k=ω/ck = \omega/c), where γ\gamma is such that Jm(γR)=0J_m(\gamma R) = 0. Calculate the ratio of the zz component of the electromagnetic angular momentum to the energy in the field. It may be advantageous to perform some integrations by parts, and to use the differential equation satisfied by EzE_z, to simplify your calculations.
Intuitive explanation

This problem asks you to find how much angular momentum per unit energy is carried by a TM waveguide mode—essentially measuring the "twist" efficiency of the wave. The key insight is that angular momentum density goes as r⃗×(E⃗×B⃗)\vec{r} \times (\vec{E} \times \vec{B}), which in cylindrical coordinates picks out the zz-component through the ϕ\phi-dependent terms; you'll need to integrate both the numerator (angular momentum flux) and denominator (energy) over the waveguide cross-section, then use integration by parts and the radial differential equation that EzE_z satisfies (from ∇2Ez+k2Ez=0\nabla^2 E_z + k^2 E_z = 0) to cancel messy derivative terms and reveal a clean ratio.

Intuitive explanation

This problem asks you to find how much angular momentum per unit energy is carried by a TM waveguide mode—essentially measuring the "twist" efficiency of the wave. The key insight is that angular momentum density goes as r⃗×(E⃗×B⃗)\vec{r} \times (\vec{E} \times \vec{B}), which in cylindrical coordinates picks out the zz-component through the ϕ\phi-dependent terms; you'll need to integrate both the numerator (angular momentum flux) and denominator (energy) over the waveguide cross-section, then use integration by parts and the radial differential equation that EzE_z satisfies (from ∇2Ez+k2Ez=0\nabla^2 E_z + k^2 E_z = 0) to cancel messy derivative terms and reveal a clean ratio.

Related theory

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