[LECS/EM-10003] -- Energy Conservation ---- Poynting Theorem

In section we will discuss conservation of energy for charges in interaction with electromagnetic fields. An expression for rate of doing work by electromagnetic forces will be derived. This will then lead us to the rate of change of mechanical energy. The resulting conservation law is a local conservation law. For more details see end of this section. \begin{equation} U_{em}= \frac{\epsilon_0}{2}|\vec{E}|^2 +\frac{1}{2\mu_0} |\vec{B}|^2 \end{equation} It may be noted that the above form of energy per unit volume coincides with with sum of expression obtained for charge and current distributions for stationary charge and current distributions. The new insight that we gain here is the possibility of exchange of energy between charges, currents and electromagnetic fields. The resulting equation will be of the form of equation of continuity and is a local conservation law for energy for charges and electromagnetic fields. For mechanical systems, the When a point charged particle moves in electromagnetic field, it experiences a force \begin{equation}\label{EQ01} \vec{F} = q(\vec{E} + \vec{v}\times \vec{B}). \end{equation} For a continuous charge distribution, the force on a small volume element\(dV\) is given by \begin{equation}\label{EQ02} \Delta\vec{F} = dq(\vec{E} + \vec{v}\times \vec{B})=(\rho dV) (\vec{E} + \vec{v}\times \vec{B}). \end{equation} We now wish to compute work done on charges present in a small volume \(\Delta V\). The displacement of the charge element in time \(\Delta t\) is \(\vec{v}\Delta t\) and hence the work done by the electromagnetic forces in time \(\Delta t\) is \(\Delta \vec{F}\cdot\vec{v}\Delta t\). Therefore, work done per sec on all charges is given by \begin{equation} \dd[W]{t}= \iiint_V \rho (\vec{E} + \vec{v}\times \vec{B}).\vec{v}\,dV. \end{equation} By work energy theorem, this work done will be equal to the rate of change of mechanical energy (kinetic energy etc.) . \begin{eqnarray}\nonumber \dd[W]{t} &=& \iiint_V \rho (\vec{E} + \vec{v}\times \vec{B}).\vec{v}dV.\\\nonumber &=&\iiint_V \rho \vec{v}\cdot\vec{E} dV = \iiint_V \vec{j}\cdot\vec{E} dV\\ &=& \iiint_V \frac{1}{\mu_0}\vec{E}\cdot(\nabla\times\vec{B})dV - \epsilon_0 \iiint \vec{E}\cdot\pp[\vec{E}]{t}\, dV.\label{EQ03} \end{eqnarray} where the Maxwell's fourth equation \begin{equation}\label{EQ04} \nabla\times\vec{B}=\mu_0 \vec{j}+\epsilon_0\mu_0\pp[\vec{E}]{t} \end{equation} has been used to write the current density, \(\vec(j)\), in terms of \(\vec{E}, \vec{B}\). Next we use the vector calculus identity \begin{equation}\label{EQ05} \nabla\cdot(\vec{E}\times \vec{B})= \vec{B}\cdot(\nabla\times\vec{E}) - \vec{E}\cdot(\nabla\times\vec{B}). \end{equation} On use of the Maxwell's equation \(\displaystyle\nabla\times \vec{E} = -\pp[\vec{B}]{t}\), \EqRef{EQ04} becomes \begin{eqnarray}\nonumber \vec{E}\cdot(\nabla\times\vec{B}) &=&-\nabla\cdot(\vec{E}\times \vec{B})-\vec{B}\cdot(\nabla\times\vec{E})\\ &=& -\nabla\cdot(\vec{E}\times \vec{B})-\vec{B}\cdot\Big(\dfrac{\partial \vec{B}}{\partial t}\Big). \label{EQ06} \end{eqnarray} \EqRef{EQ03} and \eqRef{EQ06}lead to the following equation \begin{eqnarray} \dd[W]{t}\nonumber &=& \epsilon_0 \iiint_V \vec{E}\cdot\pp[\vec{E}]{t}\, dV - \frac{1}{\mu_0}\iiint_V \vec{B}\cdot\Big(\dfrac{\partial \vec{B}}{\partial t}\Big) -\frac{1}{\mu_0} \iiint_V \nabla\cdot(\vec{E}\times \vec{B})\\ &=& \epsilon_0 \iiint_V \vec{E}\cdot\pp[\vec{E}]{t}\, dV - \frac{1}{\mu_0}\iiint_V \vec{B}\cdot\Big(\dfrac{\partial \vec{B}}{\partial t}\Big) -\frac{1}{\mu_0} \iiint_S (\vec{E}\times \vec{B}) \end{eqnarray} where \(S\) is the surface enclosing the volume \(V\). Introducing the Poynting vector \begin{equation}\label{EQ10} S = \frac{1}{\mu_0} (\vec{E}\times \vec{B}), \end{equation} and using Gauss divergence theorem, we rewrite \EqRef{EQ06} in the form \begin{eqnarray} \dd[W]{t}= -\iiint_V \pp{t}\left( \frac{\epsilon_0}{2}|\vec{E}|^2 +\frac{1}{2\mu_0} |\vec{B}|^2 \right) -\iint_S \hat{n}\cdot \vec{S} dS. \end{eqnarray} or \begin{eqnarray}\label{EQ12} \dd{t} \left[ W+ \iiint_V \left( \frac{\epsilon_0}{2}|\vec{E}|^2 +\frac{1}{2\mu_0} |\vec{B}|^2 \right)\right] = -\iint_S \hat{n}\cdot \vec{S} dS. \end{eqnarray} The expression \begin{equation} U_{em}= \frac{\epsilon_0}{2}|\vec{E}|^2 +\frac{1}{2\mu_0} |\vec{B}|^2 \end{equation}
is the energy density associated with the electromagnetic field. This expression already appears for energy density of static charge and current distributions.

•  EQ12 has the interpretation that  change in energy of charges per sec + change in energy of e.m. field in volume V = flow of energy through the surface \(S\)per sec, and the flow of energy per unit area per sec is given by the Poynting vector \(\vec{S}\)

Here the following points should not be missed.

• that the electromagnetic fields carry energy density
• electromagnetic fields can exchange energy energy with a mechanical systems
• energy conservation, just like charge conservation is a local conservation law
•  You should think about it carefully, and ask yourself if these are new results not present in case of static fields.

References

1. Sec 27-1  Energy conservation and electromagentism R. P. Feynman, Robert B. Leighton and Mathew Sands Lectures on Physics, vol-II, B.I. Publications (1964)
2. Sec 8.1.2 Poynting's Theorem David Griffiths,  Introduction to Electrodynamics, 3rd EEE edn, Prentice Hall of India Pvt Ltd New Delhi, (2002).