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Consider the region defined by \(|x|,|y|\), and \(|z|<1\). Let \(\epsilon_{r}=5, \mu_{r}=4\), and \(\sigma=0 .\) If \(J_{d}=20 \cos \left(1.5 \times 10^{8} t-b x\right) \mathbf{a}_{y} \mu \mathrm{A} / \mathrm{m}^{2}(a)\) find \(\mathbf{D}\) and \(\mathbf{E} ;(b)\) use the point form of Faraday's law and an integration with respect to time to find \(\mathbf{B}\) and \(\mathbf{H} ;(c)\) use \(\nabla \times \mathbf{H}=\mathbf{J}_{d}+\mathbf{J}\) to find \(\mathbf{J}_{d} \cdot(d)\) What is the numerical value of \(b\) ?

Short Answer

Expert verified
In summary, use the given values for \(\epsilon_r\), \(\mu_r\), and \(\sigma\) to calculate the absolute permittivity \(\epsilon\) and permeability \(\mu\). Then, find the electric and displacement fields using the given conduction current density and the relationships between the fields. Calculate the magnetic fields using Faraday's law and the relationship between \(\mathbf{B}\) and \(\mathbf{H}\). Find the displacement current density using Ampère's circuital law and the computed values of \(\mathbf{H}\) and \(\mathbf{J}_d\). Finally, compare the \(x\) and \(y\) components of \(\mathbf{J}_d\) and \(\mathbf{H}\) to determine the numerical value of \(b\).

Step by step solution

01

Calculate \(\epsilon\) and \(\mu\)

Since we are given the relative permittivity \(\epsilon_r\) and the relative permeability \(\mu_r\), we can calculate the absolute permittivity \(\epsilon\) and permeability \(\mu\) using the following formulas: $$ \epsilon = \epsilon_{r} \epsilon_{0} $$ $$ \mu = \mu_{r} \mu_{0} $$ Where \(\epsilon_{0}\) and \(\mu_{0}\) are the vacuum permittivity and permeability, respectively. \(\epsilon_{0} = 8.854 \times 10^{-12} \, \mathrm{F/m}\) and \(\mu_{0} = 4 \pi \times 10^{-7} \, \mathrm{T \cdot m/A}\). Plug in the given values to find \(\epsilon\) and \(\mu\).
02

Calculate the displacement and electric fields

Using the given conduction current density, \(J_d = 20 \cos(1.5 \times 10^8 t - b x) \mathbf{a}_y \mu A/m^2\), we can derive the displacement field \(\mathbf{D}\) and electric field \(\mathbf{E}\) using the following relationships: $$ \mathbf{D} = \epsilon \mathbf{E} $$ $$ \mathbf{E} = \frac{1}{\sigma} \mathbf{J}_d $$ Substitute the given values for \(J_d\) and \(\sigma\), and then calculate \(\mathbf{E}\) and \(\mathbf{D}\).
03

Find the magnetic fields

Use the point form of Faraday's law and integrate with respect to time to find the magnetic fields \(\mathbf{B}\) and \(\mathbf{H}\): $$ \nabla \times \mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t} $$ $$ \mathbf{B} = \int[\nabla \times \mathbf{E}] \, dt $$ Compute \(\nabla \times \mathbf{E}\), then integrate with respect to time to find \(\mathbf{B}\). Next, use the relationship between \(\mathbf{B}\) and \(\mathbf{H}\) to find \(\mathbf{H}\): $$ \mathbf{B} = \mu \mathbf{H} $$
04

Determine the displacement current density

Use Ampère's circuital law to find the displacement current density \(\mathbf{J}\) $$ \nabla \times \mathbf{H} = \mathbf{J}_d + \mathbf{J} $$ Substitute the given values and computed \(\mathbf{H}\) to find \(\mathbf{J}\).
05

Find the numerical value of \(b\)

Now that we have all of the fields, we can find the numerical value of \(b\) in the expression for \(J_d\). To do this, observe the relationship between the fields and use the given values of the coefficients. Comparing the \(x\) and \(y\) components of \(\mathbf{J}_d\) and \(\mathbf{H}\) will reveal the value of \(b\).

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Most popular questions from this chapter

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 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 \(\mu\).

A square filamentary loop of wire is \(25 \mathrm{~cm}\) on a side and has a resistance of \(125 \Omega\) per meter length. The loop lies in the \(z=0\) plane with its corners at \((0,0,0),(0.25,0,0),(0.25,0.25,0)\), and \((0,0.25,0)\) at \(t=0\). The loop is moving with a velocity \(v_{y}=50 \mathrm{~m} / \mathrm{s}\) in the field \(B_{z}=8 \cos (1.5 \times\) \(\left.10^{8} t-0.5 x\right) \mu \mathrm{T}\). Develop a function of time that expresses the ohmic power being delivered to the loop.

In a region where \(\mu_{r}=\epsilon_{r}=1\) and \(\sigma=0\), the retarded potentials are given by \(V=x(z-c t) \mathrm{V}\) and \(\mathbf{A}=x\left(\frac{z}{c}-t\right) \mathbf{a}_{z} \mathrm{~Wb} / \mathrm{m}\), where \(c=1 \sqrt{\mu_{0} \epsilon_{0}}\) (a) Show that \(\nabla \cdot \mathbf{A}=-\mu \epsilon \frac{\partial V}{\partial t} .(b)\) Find \(\mathbf{B}, \mathbf{H}, \mathbf{E}\), and \(\mathbf{D} .(c)\) Show that these results satisfy Maxwell's equations if \(\mathbf{J}\) and \(\rho_{v}\) are zero.

In region \(1, z<0, \epsilon_{1}=2 \times 10^{-11} \mathrm{~F} / \mathrm{m}, \mu_{1}=2 \times 10^{-6} \mathrm{H} / \mathrm{m}\), and \(\sigma_{1}=\) \(4 \times 10^{-3} \mathrm{~S} / \mathrm{m} ;\) in region \(2, z>0, \epsilon_{2}=\epsilon_{1} / 2, \mu_{2}=2 \mu_{1}\), and \(\sigma_{2}=\sigma_{1} / 4\). It is known that \(\mathbf{E}_{1}=\left(30 \mathbf{a}_{x}+20 \mathbf{a}_{y}+10 \mathbf{a}_{z}\right) \cos 10^{9} t \mathrm{~V} / \mathrm{m}\) at \(P\left(0,0,0^{-}\right) \cdot(a)\) Find \(\mathbf{E}_{N 1}, \mathbf{E}_{t 1}, \mathbf{D}_{N 1}\), and \(\mathbf{D}_{t 1}\) at \(P_{1} \cdot(b)\) Find \(\mathbf{J}_{N 1}\) and \(\mathbf{J}_{t 1}\) at \(P_{1} \cdot(c)\) Find \(\mathbf{E}_{t 2}\), \(\mathbf{D}_{t 2}\), and \(\mathbf{J}_{t 2}\) at \(P_{2}\left(0,0,0^{+}\right) .(d)\) (Harder) Use the continuity equation to help show that \(J_{N 1}-J_{N 2}=\partial D_{N 2} / \partial t-\partial D_{N 1} / \partial t\), and then determine \(\mathbf{D}_{N 2}\) \(\mathbf{J}_{N 2}\), and \(\mathbf{E}_{N 2}\).

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