Question

A transverse electromagnetic wave (TEM) in a coaxial waveguide has an electric field \(\mathbf{E} = \mathbf{E}(\rho, \phi)e^{i(kz-\omega t)}\) and a magnetic induction field of \(\mathbf{B} = \mathbf{B}(\rho, \phi)e^{i(kz-\omega t)}\). Since the wave is transverse, neither \(\mathbf{E}\) nor \(\mathbf{B}\) has a z component. The two fields satisfy the vector Laplacian equation \(\nabla^2 \mathbf{E}(\rho, \phi) = 0\) \(\nabla^2 \mathbf{B}(\rho, \phi) = 0.\) (a) Show that \(\mathbf{E} = \hat{\rho} E_0(a/\rho)e^{i(kz-\omega t)}\) and \(\mathbf{B} = \hat{\phi} B_0(a/\rho)e^{i(kz-\omega t)}\) are solutions. Here a is the radius of the inner conductor and \(E_0\) and \(B_0\) are constant amplitudes. (b) Assuming a vacuum inside the waveguide, verify that Maxwell's equations are satisfied with \(B_0/E_0 = k/\omega = \mu_0 \epsilon_0 (\omega/k) = 1/c.\)

          A transverse electromagnetic wave (TEM) in a coaxial waveguide has an electric field
\(\mathbf{E} = \mathbf{E}(\rho, \phi)e^{i(kz-\omega t)}\) and a magnetic induction field of \(\mathbf{B} = \mathbf{B}(\rho, \phi)e^{i(kz-\omega t)}\). Since the
wave is transverse, neither \(\mathbf{E}\) nor \(\mathbf{B}\) has a z component. The two fields satisfy the vector
Laplacian equation
\(\nabla^2 \mathbf{E}(\rho, \phi) = 0\)
\(\nabla^2 \mathbf{B}(\rho, \phi) = 0.\)
(a) Show that \(\mathbf{E} = \hat{\rho} E_0(a/\rho)e^{i(kz-\omega t)}\) and \(\mathbf{B} = \hat{\phi} B_0(a/\rho)e^{i(kz-\omega t)}\) are solutions. Here
a is the radius of the inner conductor and \(E_0\) and \(B_0\) are constant amplitudes.
(b) Assuming a vacuum inside the waveguide, verify that Maxwell's equations are
satisfied with
\(B_0/E_0 = k/\omega = \mu_0 \epsilon_0 (\omega/k) = 1/c.\)

A transverse electromagnetic wave (TEM) in a coaxial waveguide has an electric field
𝐄 = 𝐄(ρ, ϕ)e^i(kz-ω t) and a magnetic induction field of 𝐁 = 𝐁(ρ, ϕ)e^i(kz-ω t). Since the
wave is transverse, neither 𝐄 nor 𝐁 has a z component. The two fields satisfy the vector
Laplacian equation
∇^2 𝐄(ρ, ϕ) = 0
∇^2 𝐁(ρ, ϕ) = 0.
(a) Show that 𝐄 = ρ̂ E0(a/ρ)e^i(kz-ω t) and 𝐁 = ϕ̂ B0(a/ρ)e^i(kz-ω t) are solutions. Here
a is the radius of the inner conductor and E0 and B0 are constant amplitudes.
(b) Assuming a vacuum inside the waveguide, verify that Maxwell's equations are
satisfied with
B0/E0 = k/ω = μ0 ϵ0 (ω/k) = 1/c.

Added by Rebecca K.

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