Given Maxwell's equations in point form, assume that all fields vary as \(e^{s t}\) and write the equations without explicitly involving time.

Short Answer

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Answer: The Maxwell's equations in point form without involving time explicitly are: 1. Gauss's law for electric fields: \(\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}\) 2. Gauss's law for magnetic fields: \(\nabla \cdot \mathbf{B} = 0\) 3. Faraday's law of electromagnetic induction with the given assumption: \(\nabla \times \mathbf{E} = -s\mathbf{B}(\mathbf{r},t)\) 4. Ampere's circuital law with Maxwell's addition (displacement current) with the given assumption: \(\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\epsilon_0s\mathbf{E}(\mathbf{r},t)\)

Step by step solution

01

Write down Maxwell's equations in point form

Maxwell's equations in point form are the following four equations: 1. Gauss's law for electric fields: \(\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}\) 2. Gauss's law for magnetic fields: \(\nabla \cdot \mathbf{B} = 0\) 3. Faraday's law of electromagnetic induction: \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\) 4. Ampere's circuital law with Maxwell's addition (displacement current): \(\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}\)
02

Apply the assumption of fields varying as \(e^{st}\)

We are given that all fields vary as \(e^{st}\), so we can represent the electric and magnetic fields as: \(\mathbf{E}(\mathbf{r},t) = \mathbf{E}(\mathbf{r})e^{st}\) \(\mathbf{B}(\mathbf{r},t) = \mathbf{B}(\mathbf{r})e^{st}\) Now, we will differentiate these fields with respect to time and apply the chain rule.
03

Differentiate the fields with respect to time

For electric field: \(\frac{\partial \mathbf{E}}{\partial t} = \frac{\partial}{\partial t}(\mathbf{E}(\mathbf{r})e^{st}) = s\mathbf{E}(\mathbf{r})e^{st} = s\mathbf{E}(\mathbf{r},t)\) For magnetic field: \(\frac{\partial \mathbf{B}}{\partial t} = \frac{\partial}{\partial t}(\mathbf{B}(\mathbf{r})e^{st}) = s\mathbf{B}(\mathbf{r})e^{st} = s\mathbf{B}(\mathbf{r},t)\)
04

Substitute the derivatives into Maxwell's equations

Now, we will substitute these derivatives into the Faraday's law and Ampere's circuital law. For Faraday's law: \(\nabla \times \mathbf{E} = -s\mathbf{B}(\mathbf{r},t)\) For Ampere's circuital law: \(\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\epsilon_0s\mathbf{E}(\mathbf{r},t)\) We have obtained the Maxwell's equations without involving time explicitly.

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Most popular questions from this chapter

A 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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