Question

a) (10 pts.) Consider first a generic non-conducting medium with a permeability \(\mu\) and dielectric constant \(\epsilon\). Write out the Maxwell's equations for such medium and derive the wave equation governing the propagation of electromagnetic waves in this medium. Provide the plane wave solutions for both \(\vec{E}\) and \(\vec{B}\). b) (15 pts.) Consider a square wave guide of side L filled with this medium (see figure below). The electric field is zero outside the wave guide. Using the boundary conditions for the tangential components of \(\vec{E}\) across the interface and \(\nabla \cdot \vec{E} = 0\), determine the electric \(\vec{E}\) and magnetic fields in matter \(\vec{H}\) for the lowest TE mode. We recall the magnetic field in vacuum \(\vec{B}_0\) is related to the electric field by \(\vec{B}_0 = \frac{1}{\omega} \vec{k} \times \vec{E}_0\), where \(\omega\) is the angular frequency of the plane wave. c) (10 pts.) Calculate the dispersion relation for that wave guide, and provide the dominant mode frequency. For what range of frequencies \(\omega\) is the mode found in (b) the only TE mode which can be excited? What happens to the other modes when this is the case?

          a) (10 pts.) Consider first a generic non-conducting medium with a permeability \(\mu\) and dielectric constant \(\epsilon\). Write out the Maxwell's equations for such medium and derive the wave equation governing the propagation of electromagnetic waves in this medium. Provide the plane wave solutions for both \(\vec{E}\) and \(\vec{B}\).
b) (15 pts.) Consider a square wave guide of side L filled with this medium (see figure below). The electric field is zero outside the wave guide. Using the boundary conditions for the tangential components of \(\vec{E}\) across the interface and \(\nabla \cdot \vec{E} = 0\), determine the electric \(\vec{E}\) and magnetic fields in matter \(\vec{H}\) for the lowest TE mode. We recall the magnetic field in vacuum \(\vec{B}_0\) is related to the electric field by \(\vec{B}_0 = \frac{1}{\omega} \vec{k} \times \vec{E}_0\), where \(\omega\) is the angular frequency of the plane wave.
c) (10 pts.) Calculate the dispersion relation for that wave guide, and provide the dominant mode frequency. For what range of frequencies \(\omega\) is the mode found in (b) the only TE mode which can be excited? What happens to the other modes when this is the case?

a) (10 pts.) Consider first a generic non-conducting medium with a permeability μ and dielectric constant ϵ. Write out the Maxwell's equations for such medium and derive the wave equation governing the propagation of electromagnetic waves in this medium. Provide the plane wave solutions for both E⃗ and B⃗.
b) (15 pts.) Consider a square wave guide of side L filled with this medium (see figure below). The electric field is zero outside the wave guide. Using the boundary conditions for the tangential components of E⃗ across the interface and ∇·E⃗ = 0, determine the electric E⃗ and magnetic fields in matter H⃗ for the lowest TE mode. We recall the magnetic field in vacuum B⃗0 is related to the electric field by B⃗0 = (1)/(ω)k⃗×E⃗0, where ω is the angular frequency of the plane wave.
c) (10 pts.) Calculate the dispersion relation for that wave guide, and provide the dominant mode frequency. For what range of frequencies ω is the mode found in (b) the only TE mode which can be excited? What happens to the other modes when this is the case?

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