\[\bar{E}_{x} = E_{x} \qquad \bar{E}_y = \gamma\left(E_y-vB_z\right) \qquad \bar{E}_z = \gamma\left(E_z+vB_y\right)\]
\[\bar{B}_{x} = B_{x} \qquad \bar{B}_y = \gamma\left(B_y+\dfrac{v}{c^2}E_z\right) \qquad \bar{B}_z = \gamma\left(B_z-\dfrac{v}{c^2}E_y\right)\]
- Use these equations to show that both $\mathbf{E}\cdot\mathbf{B}$ and $(E^2 − c^2 B^2)$ are Lorentz invariants.
- We found earlier that $\mathbf{E}$ and $\mathbf{B}$ are mutually perpendicular for traveling EM waves. Given that this is true in some frame, can there be any other reference frame in which you would find $\mathbf{E}$ and $\mathbf{B}$ not perpendicular for traveling EM waves?
- Suppose $E > cB$ in some frame. Show that there is no possible frame in which $E=0$.
- If $E = 0$ in some frame, do these relations mean that $E$ is always equal to 0 in every other inertial frame?
- If $B = 0$ (but $E$ is nonzero) in some frame, can you always (ever?) find another frame in which $E = 0$ (but $B$ is nonzero)?
2. Maxwell’s equations written compactly
We found that we could write the field tensor like this
\[F^{\mu\nu}=\left(
\begin{array}{cccc}
0 & E_x/c & E_y/c & E_z/c\\
-E_x/c & 0 & B_z & -B_y\\
-E_y/c & -B_z & 0 & B_x\\
-E_z/c & B_y & -B_x & 0\\
\end{array}
\right)\]
and the dual tensor like this,
\[G^{\mu\nu}=\left(
\begin{array}{cccc}
0 & B_x & B_y & B_z\\
-B_x & 0 & -E_z/c & E_y/c\\
-B_y & E_z/c & 0 & -E_x/c\\
-B_z & -E_y/c & E_x/c & 0\\
\end{array}
\right)\]
With the current density 4-vector written as this: $J^{\mu} = (c\rho,J_x,J_y,J_z)$, we claim that Maxwell’s equations in vaccuum are given thusly,
\[\dfrac{\partial F^{\mu\nu}}{\partial x^{\nu}} = \mu_0 J^{\mu}, \qquad \dfrac{\partial G^{\mu\nu}}{\partial x^{\nu}} = 0\]
- Show by explicit calculation that you can recover all 4 Maxwell’s equations.
- What would have happened if $\dfrac{\partial G^{\mu\nu}}{\partial x^{\nu}} \neq 0$? Think about what happens physically?! It must be zero!
3. Self-Reflection (PHY 481 and 482)
We’ve talked about a lot of different things in E&M 1 and 2. In doing so, we’ve covered Chapters 1-9 and 12 of Griffiths along with a bunch of other special topics. You’ve studied systems on your own and learned a ton of E&M. Think about where your E&M knowledge was at the beginning of last fall and where it is now.
- Write up a half-page or so about how you feel your understanding of E&M has changed. Think about what you enjoyed learning most, what you felt you learned most deeply, and what you think that means for you in your future as a physicist.
- On the first day of class in PHY 481, I asked you to draw a concept map of E&M. How different concepts were connected and why they were connected that way. Draw another one now. Think about all the new knowledge you have gained and how it all fits together. Focus on the high level concepts and ideas and fill in as much detail as you feel you need to make your point.