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\centerline{\bf Electrodynamics B (PHY 5347) Winter/Spring 2017 
Solutions} \bigskip

{\bf Set 7:} \bigskip

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\noindent
{\bf 17.\ Lorentz force and energy-momentum tensor (E.77).}
\smallskip

\noindent Given is
$$ \Theta^{\alpha\beta} = {1\over 4\pi}\, \left[
   g^{\alpha\mu}\, F_{\mu\lambda}\, F^{\lambda\beta} +
   {1\over 4}\, g^{\alpha\beta}\, F_{\mu\lambda}\, F^{\mu\lambda}
   \right]\,, $$
we have
\begin{eqnarray} \nonumber
  \partial_{\alpha} \Theta^{\alpha\beta} &=& {1\over 4\pi}\, \left[
  \partial^{\mu}\, \left(F_{\mu\lambda}\,F^{\lambda\beta}\right)
  +{1\over 4}\, \partial^{\beta}\,\left(F_{\mu\lambda}\,
  F^{\mu\lambda}\right) \right]\\ \nonumber &=& {1\over 4\pi}\, 
  \left[ F^{\lambda\beta}\, \partial^{\mu}\, F_{\mu\lambda} +
  F_{\mu\lambda}\,\partial^{\mu}\,F^{\lambda\beta} +
  {1\over 2}\,F_{\mu\lambda}\,\partial^{\beta}\,
  F^{\mu\lambda} \right]\,.
\end{eqnarray}
The last term is in this form due to $\left( \partial^{\beta}\,
F_{\mu\lambda}\right)\,F^{\mu\lambda} = F_{\mu\lambda}\,
\partial^{\beta}\,F^{\mu\lambda}$. Using the inhomogeneous Maxwell 
equation, $\partial^{\mu}\, F_{\mu\lambda} = (4\pi/c)\,J_{\lambda}$,
$$ \partial_{\alpha} \Theta^{\alpha\beta} = 
   F^{\lambda\beta}\,\frac{1}{c}\, J_{\lambda} + {1\over 4\pi}\, 
   F_{\mu\lambda}\,\left(\partial^{\mu}\,F^{\lambda\beta} +
   {1\over 2}\, \partial^{\beta}\, F^{\mu\lambda} \right)\,.  $$
Remembering the homogeneous Maxwell equation in the form
$$ \partial^{\beta}\, F^{\mu\lambda} = - \partial^{\mu}\,
   F^{\lambda\beta} - \partial^{\lambda}\,F^{\beta\mu}\,, $$
our expression becomes
$$ \partial_{\alpha} \Theta^{\alpha\beta} = 
   F^{\lambda\beta}\,\frac{1}{c}\, J_{\lambda} + {1\over 8\pi}\, 
   F_{\mu\lambda}\,\left( \partial^{\mu}\,F^{\lambda\beta} -
   \partial^{\lambda}\,F^{\beta\mu} \right) =
   F^{\lambda\beta}\,\frac{1}{c}\, J_{\lambda} = - f^{\beta} $$
as (first re-naming summation indices, then using anti-symmetry)
$$ F_{\mu\lambda}\,\partial^{\mu}\,F^{\lambda\beta} -
   F_{\mu\lambda}\,\partial^{\lambda}\,F^{\beta\mu} = 
   F_{\lambda\mu}\,\partial^{\lambda}\,F^{\mu\beta} -
   F_{\mu\lambda}\,\partial^{\lambda}\,F^{\beta\mu} = $$
$$ F_{\mu\lambda}\,\partial^{\lambda}\,F^{\beta\mu} -
   F_{\mu\lambda}\,\partial^{\lambda}\,F^{\beta\mu} = 0\ . $$
The same result is obtained starting with
$$ \Theta^{\alpha\beta} = {1\over 4\pi}\, \left[F^{\alpha}_{\lambda}\, 
   F^{\lambda\beta} + {1\over 4}\, g^{\alpha\beta}\, F_{\mu\lambda}\, 
   F^{\mu\lambda} \right]\,, $$
\bigskip
\eject


\noindent
{\bf 18.\ Field energy of a sphere (E.78).}
\smallskip

\noindent (1) For $\vec{B}=0$ the energy-momentum tensor is given by
$$ \Theta^{00} = \frac{1}{8\pi}\,\vec{E}^{\,2}\,,\qquad
   \Theta^{i0} = \Theta^{0i} = 0\,,\qquad
   \Theta^{ij} = -\frac{1}{4\pi} E^i\,E^j - g^{ij}\,\vec{E}^{\,2}\,.$$ 
Therefore,
$$ P^0=\int d^3x\,\Theta^{00}=\frac{1}{8\pi}\int d^3x\,\vec{E}^{\,2}
   ~~~{\rm and}~~~P^i = \int d^3x\, \Theta^{0i} = 0\ .$$
Let $q$ be the total charge on the sphere and we take the center of the 
sphere at $\vec{r}=0$. By Gauss' law the field is zero inside the sphere 
and
$$ \vec{E} = E(r)\,\hat{r} = \frac{q\,\hat{r}}{r^2}~~{\rm outside}\,.$$
So $P^0(R)$ becomes
$$ P^0(R) = \frac{4\pi}{8\pi} \int_R^{\infty} r^2\,dr\,\frac{q^2}{r^4}
   = \left. - \frac{q^2}{2}\,\frac{1}{r} \right|^{\infty}_R 
   = \frac{q^2}{2R}\ . $$

\noindent (2) Now
$$ P^0 = m_e\ \Rightarrow\ R = \frac{q_e^2}{2\,m_e\,c^2} $$
where $q_e=4.803\times 10^{-10}\,$[esu], 
$m_e= 0.911\times 10^{-27}\,$[g],
and $c=2.998\times 10^{10}\,$[cm/s].  From this we find
$$ R = \frac{(4.803)^2\times 10^{-20}}{1.822\times 10^{-27}
   \times (2.998)^2\times 10^{20}}\,{\rm cm} = 
   1.41\times 10^{-13}\,{\rm cm} = 1.41\,{\rm fermi}\ . $$
\bigskip



\noindent
{\bf 19.\ Covariant derivation of the wave equation for fields (E.82).}
\smallskip

\begin{enumerate}
\item
\begin{eqnarray} \nonumber
  &~& 0\ =\ \epsilon^{\alpha\beta\gamma\delta}\partial_{\beta}
   \partial^{\hat{\beta}}\ ^*\!F_{\alpha\hat{\beta}} =
   \frac{1}{2}\, \epsilon^{\alpha\beta\gamma\delta}
   \epsilon_{\alpha\hat{\beta}\hat{\gamma}\hat{\delta}}
   \partial_{\beta}\partial^{\hat{\beta}}\, 
   F^{\hat{\gamma}\hat{\delta}} = 
  \\ \nonumber &~& \frac{1}{2}\, \left(
  - \delta^{\beta}_{~\hat{\beta}}\delta^{\gamma}_{~\hat{\gamma}}
    \delta^{\delta}_{~\hat{\delta}}
  - \delta^{\beta}_{~\hat{\gamma}}\delta^{\gamma}_{~\hat{\delta}}
    \delta^{\delta}_{~\hat{\beta}}
  - \delta^{\beta}_{~\hat{\delta}}\delta^{\gamma}_{~\hat{\beta}}
     \delta^{\delta}_{~\hat{\gamma}}
    \right. \\ \nonumber &~&~~~~\left. 
   + \delta^{\beta}_{~\hat{\beta}}\delta^{\gamma}_{~\hat{\delta}}
     \delta^{\delta}_{~\hat{\gamma}}
   + \delta^{\beta}_{~\hat{\delta}}\delta^{\gamma}_{~\hat{\gamma}}
     \delta^{\delta}_{~\hat{\beta}}
   + \delta^{\beta}_{~\hat{\gamma}}\delta^{\gamma}_{~\hat{\beta}}
     \delta^{\delta}_{~\hat{\delta}} \right) \partial_{\beta}
     \partial^{\hat{\beta}}\,F^{\hat{\gamma}\hat{\delta}} =
  \\ \nonumber &~& \frac{1}{2}\, \left(
  - \partial_{\beta}\partial^{\beta}\,F^{\gamma\delta} 
  - \partial_{\beta}\partial^{\delta}\,F^{\beta\gamma} 
  - \partial_{\beta}\partial^{\gamma}\,F^{\delta\beta} 
    \right. \\ \nonumber &~&~~~~\left. \ 
  + \partial_{\beta}\partial^{\beta}\,F^{\delta\gamma} 
  + \partial_{\beta}\partial^{\delta}\,F^{\gamma\beta} 
  + \partial_{\beta}\partial^{\gamma}\,F^{\beta\delta} 
     \right)\ .
\end{eqnarray}
Using $\partial_{\beta}F^{\beta\alpha}=0$ and the antisymmetry of the
field tensor, this becomes
$$  \partial_{\beta}\partial^{\beta}\,F^{\delta\gamma} 
    = \square\,F^{\delta\gamma} = 0\,, $$
which is the wave equation for the fields.
\item We find the wave equations
$$  \partial_{\beta}\partial^{\beta}\,F^{\delta\gamma} 
    = \square\,F^{\gamma\delta} = \frac{4\pi}{c}\,\left(
      \partial^{\gamma} J^{\delta} 
    - \partial^{\delta} J^{\gamma}\right)\,. $$
with antisymmetric sources on the right-hand side.
\end{enumerate}
\vfil

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