Question

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

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

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 \(\mu\).

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 \(\mathbf{D}\).

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 \(\mathbf{H}\): $$ \mathbf{B} = \mu \mathbf{H} $$

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 \(\mathbf{J}\).

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 \(b\).