Chapter 9: Problem 20
Given Maxwell's equations in point form, assume that all fields vary as \(e^{s t}\) and write the equations without explicitly involving time.
Chapter 9: Problem 20
Given Maxwell's equations in point form, assume that all fields vary as \(e^{s t}\) and write the equations without explicitly involving time.
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Get started for freeA perfectly conducting filament is formed into a circular ring of radius \(a\). At one point, a resistance \(R\) is inserted into the circuit, and at another a battery of voltage \(V_{0}\) is inserted. Assume that the loop current itself produces negligible magnetic field. ( \(a\) ) Apply Faraday's law, Eq. (4), evaluating each side of the equation carefully and independently to show the equality; \((b)\) repeat part \(a\), assuming the battery is removed, the ring is closed again, and a linearly increasing \(\mathbf{B}\) field is applied in a direction normal to the loop surface.
In a sourceless medium in which \(\mathbf{J}=0\) and \(\rho_{v}=0\), assume a rectangular coordinate system in which \(\mathbf{E}\) and \(\mathbf{H}\) are functions only of \(z\) and \(t .\) The medium has permittivity \(\epsilon\) and permeability \(\mu .(a)\) If \(\mathbf{E}=E_{x} \mathbf{a}_{x}\) and \(\mathbf{H}=H_{y} \mathbf{a}_{y}\), begin with Maxwell's equations and determine the second-order partial differential equation that \(E_{x}\) must satisfy. \((b)\) Show that \(E_{x}=E_{0} \cos (\omega t-\beta z)\) is a solution of that equation for a particular value of \(\beta .(c)\) Find \(\beta\) as a function of given parameters.
A voltage source \(V_{0} \sin \omega t\) is connected between two concentric conducting spheres, \(r=a\) and \(r=b, b>a\), where the region between them is a material for which \(\epsilon=\epsilon_{r} \epsilon_{0}, \mu=\mu_{0}\), and \(\sigma=0 .\) Find the total displacement current through the dielectric and compare it with the source current as determined from the capacitance (Section \(6.3\) ) and circuit-analysis methods.
A vector potential is given as \(\mathbf{A}=A_{0} \cos (\omega t-k z) \mathbf{a}_{y} .(a)\) Assuming as many components as possible are zero, find \(\mathbf{H}, \mathbf{E}\), and \(V .(b)\) Specify \(k\) in terms of \(A_{0}, \omega\), and the constants of the lossless medium, \(\epsilon\) and \(\mu\).
Find the displacement current density associated with the magnetic field \(\mathbf{H}=A_{1} \sin (4 x) \cos (\omega t-\beta z) \mathbf{a}_{x}+A_{2} \cos (4 x) \sin (\omega t-\beta z) \mathbf{a}_{z}\)
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