Determining K For Electric And Magnetic Fields In A Given Medium

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In the realm of electromagnetics, understanding the relationship between electric and magnetic fields in different mediums is crucial for various engineering applications. This article delves into the process of determining the constant 'K' in the given electric field equation **${ \bar{E} = (20y - Kt) \hat{a}_x }∗∗V/m,consideringamediumwithaspecificpermittivity(** V/m, considering a medium with a specific permittivity ({ \epsilon = 4 \times 10^{-9} }$ F/m) and conductivity (σ=0{ \sigma = 0 }). The corresponding magnetic field is given by ${ \bar{H} = (y + 2 \times 10^{9}t) \hat{a}_z }$ A/m. Our analysis will heavily rely on Maxwell's equations, the fundamental laws governing electromagnetic phenomena, specifically focusing on Faraday's Law and Ampere's Law. These laws establish the relationship between electric and magnetic fields, and their temporal and spatial variations. This exploration is not only academically significant but also practically relevant in designing electromagnetic devices and systems.

Maxwell's equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields and their interactions. These equations are the cornerstone of classical electromagnetism, providing a comprehensive framework for understanding electromagnetic phenomena. The four equations are:

  1. Gauss's Law for Electricity: This law relates the electric field to the distribution of electric charges, stating that the electric flux through a closed surface is proportional to the enclosed electric charge.
  2. Gauss's Law for Magnetism: This law states that there are no magnetic monopoles, implying that magnetic field lines always form closed loops. Mathematically, it means the magnetic flux through any closed surface is zero.
  3. Faraday's Law of Induction: This law describes how a changing magnetic field creates an electric field. It is the principle behind electromagnetic induction, which is fundamental to the operation of transformers, generators, and many other electrical devices.
  4. Ampere-Maxwell's Law: This law relates magnetic fields to electric currents and changing electric fields. It is an extension of Ampere's circuit law, incorporating the concept of displacement current due to a changing electric field.

For our specific problem, we will primarily utilize Faraday's Law and Ampere-Maxwell's Law. These laws provide the necessary relationships between the electric and magnetic fields to determine the constant 'K'.

Faraday's Law

Faraday's Law of Induction states that the electromotive force (EMF) induced in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit. In its differential form, Faraday's Law is expressed as:

${ \nabla \times \bar{E} = -\frac{\partial \bar{B}}{\partial t} }$

where:

  • ${ \nabla \times \bar{E} }$ is the curl of the electric field, representing the spatial variation of the electric field.
  • ${ \frac{\partial \bar{B}}{\partial t} }∗∗isthetimerateofchangeofthemagneticfluxdensity(∗∗$Bˉ** is the time rate of change of the magnetic flux density (**\${ \bar{B} }), which is related to the magnetic field intensity (${ \bar{H} }∗∗)bythepermeabilityofthemedium(∗∗$μ**) by the permeability of the medium (**\${ \mu }), i.e., ${ \bar{B} = \mu \bar{H} }$.

Applying Faraday's Law to our problem, we need to compute the curl of the given electric field ${ \bar{E} = (20y - Kt) \hat{a}_x }∗∗V/mandthetimederivativeofthemagneticfluxdensity.Thecurlof∗∗$Eˉ** V/m and the time derivative of the magnetic flux density. The curl of **\${ \bar{E} } in Cartesian coordinates is given by:

${ \nabla \times \bar{E} = \left(\frac{\partial E_z}{\partial y} - \frac{\partial E_y}{\partial z}\right)\hat{a}_x + \left(\frac{\partial E_x}{\partial z} - \frac{\partial E_z}{\partial x}\right)\hat{a}_y + \left(\frac{\partial E_y}{\partial x} - \frac{\partial E_x}{\partial y}\right)\hat{a}_z }$

Since ${ \bar{E} = (20y - Kt) \hat{a}_x }∗∗,wehave∗∗$Ex=20y−Kt**, we have **\${ E_x = 20y - Kt }, ${ E_y = 0 }∗∗,and∗∗$Ez=0**, and **\${ E_z = 0 }. Thus, the curl simplifies to:

${ \nabla \times \bar{E} = (0 - 0)\hat{a}_x + (0 - 0)\hat{a}_y + (0 - 20)\hat{a}_z = -20 \hat{a}_z }$

Next, we find the time derivative of the magnetic flux density ${ \bar{B} }∗∗.Given∗∗$Hˉ=(y+2×109t)a^z**. Given **\${ \bar{H} = (y + 2 \times 10^{9}t) \hat{a}_z } A/m, we have ${ \bar{B} = \mu \bar{H} = \mu (y + 2 \times 10^{9}t) \hat{a}_z }∗∗.Thetimederivativeof∗∗$Bˉ**. The time derivative of **\${ \bar{B} } is:

${ \frac{\partial \bar{B}}{\partial t} = \mu \frac{\partial}{\partial t} (y + 2 \times 10^{9}t) \hat{a}_z = \mu (2 \times 10^{9}) \hat{a}_z }$

Substituting these results into Faraday's Law, we get:

${ -20 \hat{a}_z = -\mu (2 \times 10^{9}) \hat{a}_z }$

This equation provides one relationship that can help us determine 'K', but we need another equation to solve for 'K' definitively. This is where Ampere-Maxwell's Law comes into play.

Ampere-Maxwell's Law

Ampere-Maxwell's Law is an extension of Ampere's circuit law that accounts for the effects of a changing electric field. It states that magnetic fields can be generated not only by electric currents but also by time-varying electric fields. The differential form of Ampere-Maxwell's Law is:

${ \nabla \times \bar{H} = \bar{J} + \frac{\partial \bar{D}}{\partial t} }$

where:

  • ${ \nabla \times \bar{H} }$ is the curl of the magnetic field intensity.
  • ${ \bar{J} }$ is the current density.
  • ${ \frac{\partial \bar{D}}{\partial t} }∗∗isthedisplacementcurrentdensity,representingthetimerateofchangeoftheelectricfluxdensity(∗∗$Dˉ** is the displacement current density, representing the time rate of change of the electric flux density (**\${ \bar{D} }).

In our case, the conductivity ${ \sigma = 0 }∗∗,whichimpliesthatthecurrentdensity∗∗$Jˉ=σEˉ=0**, which implies that the current density **\${ \bar{J} = \sigma \bar{E} = 0 }. The electric flux density ${ \bar{D} }∗∗isrelatedtotheelectricfield∗∗$Eˉ** is related to the electric field **\${ \bar{E} } by the permittivity ${ \epsilon }∗∗,i.e.,∗∗$Dˉ=ϵEˉ**, i.e., **\${ \bar{D} = \epsilon \bar{E} }. Thus, Ampere-Maxwell's Law simplifies to:

${ \nabla \times \bar{H} = \frac{\partial \bar{D}}{\partial t} = \epsilon \frac{\partial \bar{E}}{\partial t} }$

We need to compute the curl of the given magnetic field ${ \bar{H} = (y + 2 \times 10^{9}t) \hat{a}_z }∗∗A/mandthetimederivativeoftheelectricfield.Thecurlof∗∗$Hˉ** A/m and the time derivative of the electric field. The curl of **\${ \bar{H} } in Cartesian coordinates is:

${ \nabla \times \bar{H} = \left(\frac{\partial H_z}{\partial y} - \frac{\partial H_y}{\partial z}\right)\hat{a}_x + \left(\frac{\partial H_x}{\partial z} - \frac{\partial H_z}{\partial x}\right)\hat{a}_y + \left(\frac{\partial H_y}{\partial x} - \frac{\partial H_x}{\partial y}\right)\hat{a}_z }$

Since ${ \bar{H} = (y + 2 \times 10^{9}t) \hat{a}_z }∗∗,wehave∗∗$Hx=0**, we have **\${ H_x = 0 }, ${ H_y = 0 }∗∗,and∗∗$Hz=y+2×109t**, and **\${ H_z = y + 2 \times 10^{9}t }. Thus, the curl simplifies to:

${ \nabla \times \bar{H} = \left(\frac{\partial}{\partial y}(y + 2 \times 10^{9}t) - 0\right)\hat{a}_x + (0 - 0)\hat{a}_y + (0 - 0)\hat{a}_z = \hat{a}_x }$

Next, we find the time derivative of the electric field ${ \bar{E} = (20y - Kt) \hat{a}_x }∗∗.Thetimederivativeof∗∗$Eˉ**. The time derivative of **\${ \bar{E} } is:

${ \frac{\partial \bar{E}}{\partial t} = \frac{\partial}{\partial t} (20y - Kt) \hat{a}_x = -K \hat{a}_x }$

Substituting these results into Ampere-Maxwell's Law, we get:

${ \hat{a}_x = \epsilon (-K) \hat{a}_x }$

Now we have two equations from Faraday's Law and Ampere-Maxwell's Law that we can use to solve for K.

From Faraday's Law:

${ -20 \hat{a}_z = -\mu (2 \times 10^{9}) \hat{a}_z }$

This gives us:

${ 20 = \mu (2 \times 10^{9}) }$

Thus,

${ \mu = \frac{20}{2 \times 10^{9}} = 10^{-8} \text{ H/m} }$

From Ampere-Maxwell's Law:

${ \hat{a}_x = \epsilon (-K) \hat{a}_x }$

Substituting the given permittivity ${ \epsilon = 4 \times 10^{-9} }$ F/m, we get:

${ 1 = (4 \times 10^{-9})(-K) }$

Solving for K:

${ K = -\frac{1}{4 \times 10^{-9}} = -2.5 \times 10^{8} }$

In conclusion, by applying Maxwell's equations, specifically Faraday's Law and Ampere-Maxwell's Law, we have successfully determined the value of 'K' for the given electric and magnetic fields in the specified medium. The value of K is found to be ${ -2.5 \times 10^{8} }$. This result is crucial for understanding the behavior of electromagnetic waves in the given medium and has significant implications for various engineering applications, including the design of antennas, waveguides, and other electromagnetic devices. The step-by-step application of Maxwell's equations highlights their fundamental role in solving electromagnetic problems and their practical relevance in engineering.

By meticulously calculating the curl of the magnetic field and the time derivative of the electric field, we applied Ampere-Maxwell's Law to derive a relationship involving K. This process underscored the importance of considering displacement current in time-varying electromagnetic fields. The final determination of K not only provides a numerical solution but also enhances our understanding of the interplay between electric and magnetic fields in dynamic systems. This understanding is paramount in advancing technology and solving real-world engineering challenges.