[NOTES/EM-11001]-Electromagnetic Potentials in Electrodynamics

1. Maxwell's equations

\begin{align*} \bar{\nabla}\cdot\bar{E} = & \frac{\rho}{\epsilon_0} &\bar{\nabla}\cdot\bar{B}=0&\\ \bar{\nabla}\times\bar{E}=&-\frac{\partial\vec{B}}{\partial t} &\vec{\nabla}\times\vec{B}=&\mu_0\left(\vec{j}+\epsilon_0 \frac{\partial\vec{E}}{\partial t}\right)\\ \end{align*} $\bar{\nabla}\cdot\bar{B}=0$ implies that there exists a vector field $\vec{A}$ such that $$ \vec{B} = \vec{\nabla}\times\vec{A} $$ $\vec{A}$ is called \textit{vector potential}. Substituting $\vec{B}$ in terms of $\vec{A}$ in ${\rm curl}~\vec{E}$ equation we get \begin{align*} \bar{\nabla}\times\bar{E} = & - \frac{\partial}{\partial t}(\bar{\nabla}\times\bar{A})\\ \bar{\nabla}\times\bar{E} = & -\nabla \times \frac{\partial\bar{A}} {\partial t}\\ {\rm or}~~~~~~~\bar{\nabla}\times\bar{E}&+\bar{\nabla} \times \frac{\partial\vec{A}}{\partial t} = 0\\ \bar{\nabla}\times&(\bar{E}+\frac{\partial\vec{A}}{\partial t})=0. \end{align*} Vector calculus tells us that there exists a function $\phi$ such that \begin{align*} \vec{E}+&\frac{\partial\vec{A}}{\partial A} = -\vec{\nabla}\phi\\ {\rm or}~~~~~~\vec{E}= & -\vec{\nabla}\phi - \frac{\partial\vec{A}}{\partial t} \end{align*} $\vec{A}=$ and $\phi$ are known as the vector potential and the scalar potential. \noindent Given $\vec{E}$ and $\vec{B}$ the potentials are not fixed uniquely. Thus $\vec{A},\phi$ and $\vec{A}^\prime,\phi^\prime$ related by \begin{align*} \vec{A}^\prime = \vec{A}+\vec{\nabla} \Lambda (\vec{r},t),\qquad \phi^\prime = \phi - \frac{\partial\Lambda(\vec{r},t)}{\partial t}, \end{align*} where $\Lambda(\vec{r},t)$ is an arbitrary potential, give rise to the same electric and magnetic field. The transformation \begin{align*} \vec{A}\rightarrow \vec{A}^\prime = & \bar{A}+\bar{\nabla}\Lambda\\ \phi \rightarrow \phi^\prime = & \phi - \frac{\partial\Lambda}{\partial t} \end{align*} is called \textit{gauge transformation}. Under a gauge transformation, $\vec{E}$ and $\vec{B}$, and hence all physical quantities, remain unchanged.

• This principle --- gauge invariance of physical quantities --- is an important principle
• Quantum Theory of light is formulated in terms of potentials $\vec{A},\phi$ rather than $\vec{E}$ and $\vec{B}$.\\
• Potentials are not unique and can be modified by a gauge transformations. To fix potentials uniquely extra conditions must be imposed on the potentials. For example $$\phi=0, \qquad \nabla\cdot\vec{A} =0 $$ This condition is known as the Coulomb gauge. This is an important gauge condition and provides one of the ways to quantize the electromagnetic field by eliminating the unphysical degrees of freedom.

2.Wave equation in Lorentz gauge

The Lorentz gauge condition is $$ \mu_0\epsilon_0~\frac{\partial\phi}{\partial t}\,+\,\vec{\nabla}\cdot\vec{A} = 0 $$ and is used very frequently. In the Lorentz gauge the potentials satisfy. \begin{align*} \nabla^2\phi - \mu_0\epsilon_0~\frac{\partial^2\phi}{\partial t^2} = & -\frac{\rho}{\epsilon_0}\\ \nabla^2\vec{A}-\mu_0\epsilon_0~\frac{\partial\vec{A}}{\partial t} = & -\mu_0\vec{j} \end{align*}

Proof

We begin with the Maxwell equation $\bar{\nabla}\cdot\bar{E}=+\rho/\epsilon_0$ and express the field in terms of potentials. \begin{align*} &\nabla\left(-\nabla\phi-\frac{\partial\vec{A}}{\partial t}\right)=\rho/\epsilon_0,\\ \text{or }~~& \nabla^2\phi+\frac{\partial}{\partial t}(\nabla\bar{A})=-\rho/\epsilon_0,\\ \text{or }~~ & \nabla^2\phi-\mu_0\epsilon_0\frac{\partial^2}{\partial t^2}\phi=-\rho/\epsilon_0. \end{align*} Next we consider the Maxwell's equation \[\bar{\nabla}\times\bar{B}=+\mu_0\vec{j}+\mu_0\epsilon_0\frac{\partial\vec{E}}{\partial t}\] and write the fields again in terms of the potentials. \begin{eqnarray} &&\nabla\times({\nabla}\vec{A}) = +\mu_0\vec{j}+\mu_0\epsilon_0~ \frac{\partial}{\partial t}\left(-\nabla\phi-\frac{\partial\bar{A}} {\partial t}\right),\\ \text{or }~~&&{\nabla}{\big(\nabla\cdot\vec{A}\big)}-\nabla^2\vec{A} = \mu_0\vec{j}- \mu_0\epsilon_0\dd[\nabla \phi]{t} + \mu_0\epsilon_0~\frac{\partial^2\vec{A}}{\partial t^2},\\ \text{or }~~&&~\nabla^2\bar{A}-\mu_0\epsilon_0~\frac{\partial^2}{\partial t}~\vec{A}=-\mu_0\vec{j}. \end{eqnarray} where in the last step the Lorentz gauge condition has been used. Define $\square$ d'Alembertian operator by \begin{align*} \square =&\left(\nabla^2-\mu_0\epsilon_0~\frac{\partial^2}{\partial t^2}\right) \end{align*} The equations for the potentials now take a simple form \begin{equation} \square~\vec{A}_0=-\mu_0\vec{j}, \qquad \square~\phi = -\frac{1}{\epsilon_0}~\rho. \end{equation} Note that when $\rho=0,\vec{j}=0$ we get wave equations for $\vec{A}$ and $\phi$ \begin{align*} \nabla^2\vec{A} - \frac{1}{c^2}~\frac{\partial^2}{\partial t^2}~\vec{A} = &0\\ \nabla^2\phi-\frac{1}{c^2}~\frac{\partial^2}{\partial t^2}~\phi = & 0 \end{align*} where \(c=\frac{1}{\sqrt{\mu_0\epsilon_0}}\) can now be identified with the velocity of electromagnetic waves.

References : 

1. Sec 18-6 Solving Maxwell's equations; the potentials and the wave equation R. P. Feynman, Robert B. Leighton and Mathew Sands{\it Lectures on Physics}, vol-II, B.I. Publications (1964)

2. Sec 10.1 The potential formulation Introduction to Electrodynamics, 3rd EEE edn, Prentice Hall of India Pvt Ltd New Delhi, (2002).