Question

Find the displacement current density associated with the magnetic field \(\mathbf{H}=A_{1} \sin (4 x) \cos (\omega t-\beta z) \mathbf{a}_{x}+A_{2} \cos (4 x) \sin (\omega t-\beta z) \mathbf{a}_{z}\)

Short answer

Answer: The displacement current density associated with the given magnetic field is: \(\mathbf{J_d} = \frac{4A_2\omega}{\mu_0} \cos(4x) \cos(\omega t-\beta z) \mathbf{a}_{y}\)

Step-by-step solution

Identify the magnetic field components

The given magnetic field H can be written as: \(\mathbf{H} = H_x \mathbf{a}_{x} + H_z \mathbf{a}_{z}\) Where \(H_x = A_1 \sin(4x) \cos(\omega t - \beta z) \) \(H_z = A_2 \cos(4x) \sin(\omega t - \beta z) \)

Calculate the time derivative of H

Now, we need to find the time derivative of the magnetic field, as follows: \(\frac{\partial \mathbf{H}}{\partial t} = \frac{\partial H_x}{\partial t}\mathbf{a}_{x} + \frac{\partial H_z}{\partial t}\mathbf{a}_{z}\) For \(H_x\): \(\frac{\partial H_x}{\partial t} = -A_1 \sin(4x) \sin(\omega t - \beta z)\cdot \omega\) For \(H_z\): \(\frac{\partial H_z}{\partial t} = A_2 \cos(4x) \cos(\omega t - \beta z)\cdot \omega\) So, \(\frac{\partial \mathbf{H}}{\partial t} = -A_1 \omega\sin(4x) \sin(\omega t - \beta z) \mathbf{a}_{x} + A_2 \omega \cos(4x) \cos(\omega t - \beta z) \mathbf{a}_{z}\)

Apply Maxwell's equation to find the displacement current density

The Maxwell's equation relating the displacement current density to the time derivative of the magnetic field is given by: \(\mathbf{J_d} = \frac{1}{\mu_0}(\nabla \times \frac{\partial \mathbf{H}}{\partial t})\) Now, let's calculate the curl of the time derivative of the magnetic field: $(\nabla \times \frac{\partial \mathbf{H}}{\partial t}) = \begin{vmatrix} \mathbf{a}_x & \mathbf{a}_y & \mathbf{a}_z \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ -A_1 \omega\sin(4x) \sin(\omega t - \beta z) & 0 & A_2 \omega \cos(4x) \cos(\omega t - \beta z) \\ \end{vmatrix} $ Solving this determinant, we get: \((\nabla \times \frac{\partial \mathbf{H}}{\partial t}) = 4A_2\omega \cos(4x) \cos(\omega t-\beta z) \mathbf{a}_{y}\)

Calculate the displacement current density

Finally, substitute the curl of the time derivative of the magnetic field into the Maxwell's equation to find the displacement current density: \(\mathbf{J_d} = \frac{1}{\mu_0}(4A_2\omega \cos(4x) \cos(\omega t-\beta z) \mathbf{a}_{y}) = \frac{4A_2\omega}{\mu_0} \cos(4x) \cos(\omega t-\beta z) \mathbf{a}_{y}\) Thus, the displacement current density associated with the given magnetic field is: \(\mathbf{J_d} = \frac{4A_2\omega}{\mu_0} \cos(4x) \cos(\omega t-\beta z) \mathbf{a}_{y}\)

Key concepts

Maxwell's Equations

Maxwell's equations are a set of four fundamental laws that describe the behavior of electric and magnetic fields and their interrelation with electric charges and currents. These equations are the foundation of classical electromagnetism, optics, and electric circuits and are critical in understanding the nature of electromagnetic waves. The equations can be summarized as follows:

• Gauss's Law for electricity, which states that the electric flux out of a closed surface is proportional to the charge enclosed.
• Gauss's Law for magnetism, which indicates that there are no magnetic monopoles; the magnetic flux out of a closed surface is zero.
• Faraday's Law of induction, explaining how a time-varying magnetic field creates an electromotive force.
• Ampère's Law with Maxwell's addition, which states that magnetic fields can be generated by electric currents and by time-varying electric fields (displacement current).

In the context of the textbook exercise, Maxwell's equation referencing the displacement current density is Ampère's Law with Maxwell's addition. This law is crucial in explaining that a changing electric field can produce a magnetic field, just as a current does. This insight bridged a significant gap in understanding how capacitors work in an electric circuit and led to the unified theory of electromagnetism.

Magnetic Field Time Derivative

When discussing electromagnetism, the concept of the magnetic field time derivative is vital. It refers to the rate of change of the magnetic field with respect to time. The time-varying magnetic fields are especially important because they can induce electric fields based on Faraday's Law, leading to the concept of electromagnetic induction.
In the given exercise, finding the time derivative of the magnetic field, \(\frac{\partial \mathbf{H}}{\partial t}\), is an essential step. This derivative provides the required value to calculate the displacement current density using Maxwell's equations. The time derivative of the magnetic field is the foundation for understanding how electric generators work, how transformers transfer energy, and even the operation of electric guitars, all of which rely on changing magnetic fields over time to function.

Curl of a Vector Field

In vector calculus, the curl of a vector field is a measure of the field's rotation at a point. In a physical sense, it describes how much and in what direction the field 'curls' or 'swirls' around a specific point. For a magnetic field, this concept is crucial as it helps to analyze the rotational characteristics of the field. It's calculated using partial derivatives and the cross product of vectors.
In electromagnetism, the curl of the magnetic field, represented by \(abla \times \mathbf{H}\), tells us about the presence and direction of electric current (including the displacement current) that generates the magnetic field. The exercise exemplifies this by determining the curl of the time derivative of the magnetic field, which is later used in calculating the displacement current density.

Electromagnetic Theory

Electromagnetic theory encompasses the study of electric and magnetic fields and their interactions with matter, and it constitutes the theoretical underpinning of electromagnetic fields in physics. Classical electromagnetic theory is based on the principles of Maxwell's equations, which not only explain electromagnetic phenomena but also predict the existence of electromagnetic waves that propagate through space at the speed of light.
In practice, electromagnetic theory is responsible for the principles behind various technologies such as radio, television, wireless communication, and radar. By solving the textbook exercise involving the displacement current density, we touch upon a small aspect of this broad theory, witnessing how dynamic electric and magnetic fields interact, a concept pioneered by James Clerk Maxwell that has profoundly impacted the modern world.