[NOTES/EM-10008]-Maxwell's Equations for Time Varying Fields and Ampere's Law

Several concepts and results valid for static fields need revision when one is considering the situation of time varying fields. Some of these are discussed here.

Results in static case need revision in time varying case

We shall now discuss changes that become necessary in time varying situations as a consequence of Maxwell's equations.

1. Most important consequences

1. Electric field forces are not conservative because $\nabla\times\vec{E}\ne0$.
2. Electrostatic potential cannot be defined. There is \underline{\underline{no}}. $\phi$ such that $-\nabla\phi=\vec{E}$.
3. Work done by $\vec{E}$ depends on path. $$ W = \int_a^b (\vec{E})\overline{dl} $$ depends on path joining points \(a,b\) and the work done is not zero in a loop. In electrostatic $I$ depends only on the end points and is defined as $W=V_a-V_b$.
4. The concept of potential, hence conductor being equipotential, {\it i.e.} a constant potential surface is FALSE.
5. Concept of capacitance is meaningless (not precise)??
6. $\vec{E}=0$ inside body/cavity of

What all is true in statics is not always true in dynamics. For a detailed discussion see Sec 15.6 of Feynman Lectures Vol-II.

2. Scalar and vector potentials

For time varying situations the vector potentials is defined as before in the static case. However the electric potential definition needs to be modified. % \begin{align*} &\nabla\cdot\vec{E} = \rho/\epsilon_0, & \qquad &\nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}.\\ &\nabla\cdot\vec{B} = 0, & \qquad &\nabla\times\vec{B}=\mu_0 j + \mu_0\epsilon_0 \frac{\partial\vec{E}}{\partial t}. \end{align*} $\nabla\cdot\vec{B}=0 \Rightarrow$ \text{There exists a vector field $\vec{A}$ such that} $$ \vec{B} = Curl~\vec{A} $$ substitute in $\text{Curl}~E$ equation. \begin{align*} \nabla\times\vec{E}+\frac{\partial}{\partial t} \nabla \times\vec{A} =& 0\\ \nabla \times \left(\vec{E} + \frac{\partial\vec{A}}{\partial t}\right) = & 0 \end{align*} Therefore there exists a function $\phi$ such that \begin{align*} \vec{E} +& \frac{\partial\vec{A}}{\partial t} = -\nabla \phi \Longrightarrow \vec{E} = -\nabla\phi - \frac{\partial\vec{A}}{\partial t} \end{align*} The function \(\phi\) is called the scalar potential.

Ref : R. P. Feynman, Robert B. Leighton and Mathew Sands  Lectures on Physics, vol-II, B.I. Publications (1964)