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 \(b\).

Key concepts

Displacement Field

The displacement field, denoted as \( \mathbf{D} \), is an essential concept in the study of electromagnetics, especially in materials with varying permittivity. This field accounts for the electric field \( \mathbf{E} \) within a medium and the effect of the medium's permittivity. The relationship between the displacement field and the electric field is given by the equation:

• \( \mathbf{D} = \epsilon \mathbf{E} \)

Here, \( \epsilon \) is the permittivity of the material, which can be calculated using the relative permittivity \( \epsilon_r \) and the vacuum permittivity \( \epsilon_0 \).
This concept helps us understand how materials respond to electric fields and how energy is stored within them. The displacement field is particularly pivotal in non-conducting materials, where it interacts solely with bound charges, while in conducting materials, free charges also come into play. Understanding \( \mathbf{D} \) helps in solving complex problems involving wave propagation and fields in materials.

Magnetic Fields

Magnetic fields are fundamental components of electromagnetics, represented by \( \mathbf{B} \) and \( \mathbf{H} \). The magnetic flux density \( \mathbf{B} \) and the magnetic field intensity \( \mathbf{H} \) are related through the permeability of the material, \( \mu \):

• \( \mathbf{B} = \mu \mathbf{H} \)

The magnetic field describes how magnetic forces are circulated in space and can rate the strength of these forces.
In the presence of currents and changing electric fields, magnetic fields are induced (as explained by Faraday's Law). These fields are crucial in various applications, like inductors, transformers, and motors, as they can influence and be influenced by electrical current's paths. Magnetic fields are the essence behind technologies such as MRI in medical diagnostics and magnetic levitation for transport.

Faraday's Law

Faraday's Law of electromagnetic induction is a core principle that connects time-varying magnetic fields with induced electric currents. This law states that a change in magnetic field within a closed loop induces an electromotive force (EMF) in the wire:

• \( abla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \)

The key concept here is that of transformation—magnetic fields changing over time to produce electric currents. Faraday's Law is foundational to the working mechanics of electric generators and transformers, where mechanical energy is converted into electrical energy and vice versa.
Understanding this law is crucial for grasping how energy conversion and transmission work in electrical systems.

Permittivity and Permeability

Permittivity (\( \epsilon \)) and permeability (\( \mu \)) are properties of materials that describe their response to electric and magnetic fields, respectively. Permittivity measures how much electric field is reduced inside a medium, while permeability describes the ability of a material to support the formation of magnetic fields.

• Effective permittivity: \( \epsilon = \epsilon_r \epsilon_0 \)
• Effective permeability: \( \mu = \mu_r \mu_0 \)

Both of these constants are critical in defining the electromagnetic behavior of materials in terms of propagation speed and how they interact with electromagnetic waves.
For instance, in designing capacitors, high permittivity materials are favored to enhance capacitance, while inductors rely on materials with high permeability for efficient operation. Engineers and scientists utilize these properties extensively to tailor materials for specific technological applications, ensuring optimal performance.