(Solved): Divergence Theorem, determine the differential form of the magnetic flux conservation law. Problem ...

Divergence Theorem, determine the differential form of the magnetic flux conservation law. Problem 5 Maxwell's Equation (a) Compare the differential forms of Faraday's law, Ampere's law, Gauss' law and magnetic flux conservation law that you obtained in Problem 1(a), 2(a) 3(a) and 4 respectively, with the time varying Maxwell's equation (Lecture B2 or (8.23) Nortaros pp. 358), are they the same? What is/are missing in your earlier results? Why are they different? (b) A parallel-plate capacitor of plate area \( S \) is filled with a perfect, homogenous dielectric of permittivity of \( \varepsilon \). The distance between the plate, \( d \), is much lesser than the dimension of the plates, so the fringing effects can be neglected. The capacitor is connected to a slowly timevarying voltage \( v(t)=\mathrm{V}_{0} \cos \omega t \). Find the displacement current density in the dielectric. Problem 6 Maxwell's Equation Derive 3D Helmholtz equations for a lossless medium, i.e. \[ \nabla^{2} \underline{\mathbf{E}}+\beta^{2} \underline{\mathbf{E}}=0 \quad \text { and } \quad \nabla^{2} \underline{\mathbf{H}}+\beta^{2} \underline{\mathbf{H}}=0 \]