Decoding Electromagnetic Fields From Scalar And Vector Potentials

(a) Determining {\vec{E}} and {\vec{B}} from Given Potentials

In this section, we embark on a journey to unravel the intricacies of electromagnetic fields, specifically focusing on how to derive the electric field (${\vec{E}}$) and magnetic field (${\vec{B}}$) from given scalar and vector potentials. This is a fundamental concept in electromagnetism, providing a powerful tool for analyzing and understanding the behavior of electromagnetic waves and fields. Understanding electromagnetic fields is crucial for various applications, including wireless communication, medical imaging, and particle physics. To begin, we are presented with a scenario where the scalar potential (${\phi(\vec{r},t)}$) is zero and the vector potential (${\vec{A}(\vec{r},t)}$) is given by ${\vec{A}(\vec{r},t) = \hat{j} A_0 \sin(kx - \omega t)}$, where ${A_0}$, ${k}$, and ${\omega}$ are constants. Our mission is to decipher the electric and magnetic fields that arise from these potentials. The relationship between electromagnetic fields and potentials is governed by the following fundamental equations:

• Electric Field: ${\vec{E} = -\nabla \phi - \frac{\partial \vec{A}}{\partial t}}$
• Magnetic Field: ${\vec{B} = \nabla \times \vec{A}}$

These equations are the cornerstone of our analysis. The first equation tells us that the electric field is the negative gradient of the scalar potential minus the time derivative of the vector potential. The second equation reveals that the magnetic field is the curl of the vector potential. Applying these equations to our given potentials, we can systematically determine the electric and magnetic fields. Since the scalar potential ${\phi(\vec{r},t)}$ is zero, the first term in the electric field equation vanishes, simplifying our calculations. The vector potential ${\vec{A}(\vec{r},t)}$ is oriented along the ${\hat{j}}$ direction, which means it only has a y-component. This will further simplify the curl operation when calculating the magnetic field. Let's dive into the step-by-step calculations to unveil the electric and magnetic fields.

Step-by-Step Calculation of Electric Field (${\vec{E}}$)

The electric field is given by:

${\vec{E} = -\nabla \phi - \frac{\partial \vec{A}}{\partial t}}$

Since ${\phi(\vec{r},t) = 0}$, the gradient term vanishes:

${\vec{E} = - \frac{\partial \vec{A}}{\partial t}}$

The vector potential is ${\vec{A}(\vec{r},t) = \hat{j} A_0 \sin(kx - \omega t)}$. Taking the partial derivative with respect to time, we get:

${\frac{\partial \vec{A}}{\partial t} = \hat{j} A_0 \frac{\partial}{\partial t} \sin(kx - \omega t) = \hat{j} A_0 (-\omega) \cos(kx - \omega t) = -\hat{j} A_0 \omega \cos(kx - \omega t)}$

Therefore, the electric field is:

${\vec{E} = -(-\hat{j} A_0 \omega \cos(kx - \omega t)) = \hat{j} A_0 \omega \cos(kx - \omega t)}$

Step-by-Step Calculation of Magnetic Field (${\vec{B}}$)

The magnetic field is given by:

${\vec{B} = \nabla \times \vec{A}}$

The vector potential is ${\vec{A}(\vec{r},t) = \hat{j} A_0 \sin(kx - \omega t)}$. The curl operation in Cartesian coordinates is:

(\nabla \times \vec{A} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ 0 & A_0 \sin(kx - \omega t) & 0 \end{vmatrix})

Expanding the determinant, we get:

${\vec{B} = \hat{i} \left(\frac{\partial}{\partial y}(0) - \frac{\partial}{\partial z}(A_0 \sin(kx - \omega t))\right) - \hat{j} \left(\frac{\partial}{\partial x}(0) - \frac{\partial}{\partial z}(0)\right) + \hat{k} \left(\frac{\partial}{\partial x}(A_0 \sin(kx - \omega t)) - \frac{\partial}{\partial y}(0)\right)}$

${\vec{B} = \hat{i}(0 - 0) - \hat{j}(0 - 0) + \hat{k} \left(A_0 k \cos(kx - \omega t) - 0\right)}$

Therefore, the magnetic field is:

${\vec{B} = \hat{k} A_0 k \cos(kx - \omega t)}$

Summary of Results

We have successfully determined the electric and magnetic fields from the given scalar and vector potentials:

• Electric Field: ${\vec{E} = \hat{j} A_0 \omega \cos(kx - \omega t)}$
• Magnetic Field: ${\vec{B} = \hat{k} A_0 k \cos(kx - \omega t)}$

These results reveal that the electric field is oriented along the ${\hat{j}}$ direction (y-axis) and the magnetic field is oriented along the ${\hat{k}}$ direction (z-axis). Both fields oscillate sinusoidally with the same phase and have amplitudes proportional to ${A_0 \omega}$ and ${A_0 k}$ respectively. Understanding the relationship between these fields and the original potentials is key to comprehending electromagnetic wave propagation and behavior. Furthermore, these findings align with the fundamental principles of electromagnetism, such as Faraday's law of induction and Ampere's law, which describe how changing magnetic fields induce electric fields and vice versa. This interconnectedness of electric and magnetic fields is what gives rise to electromagnetic waves, which are the foundation of many technologies we use every day, from radio communication to medical imaging. By mastering the techniques for deriving electromagnetic fields from potentials, we gain a deeper appreciation for the fundamental forces that govern our universe.

In conclusion, the process of deriving electric and magnetic fields from scalar and vector potentials is a cornerstone of electromagnetic theory. By applying the fundamental equations and carefully performing the necessary calculations, we can unveil the hidden electromagnetic fields that permeate our world. This knowledge is not only crucial for physicists and engineers but also provides a profound understanding of the nature of light and other electromagnetic phenomena. Mastering this concept opens doors to a wide range of applications and further exploration in the fascinating realm of electromagnetism.