Title: ETH_Sem5 / AdvQC / 02_Electrodynamics

Q: Write down Maxwell's equations in Gaussian units (\(4\pi\epsilon_0=1\))!
A: Inhomogeneous Maxwell equations (\(\varrho\), \(\bm{j}\): sources of \(\Efield\), \(\Bfield\))
1. Gauss' law | 2. Ampere's law
\(\displaystyle\dop\Efield(\pos,t) = 4\pi\varrho(\pos,t)\) | \(\displaystyle\cop\Bfield(\pos,t)-\frac{1}{c}\pdv{\Efield(\pos,t)}{t} = \frac{4\pi}{c}\bm{j}(\pos,t)\)
Homogeneous Maxwell equations
3. Faraday's law | 4. No magn. monopoles
\(\displaystyle\cop\Efield(\pos,t)+\frac{1}{c}\pdv{\Bfield(\pos,t)}{t} = 0\) | \(\displaystyle\dop\Bfield(\pos,t) = 0\)
a time-varying magnetic field causes an electric vortex field & guarantees the absence of magnetic charges or monopoles.

Q: Write down the continuity equation! What is \(\varrho(\pos,t)\) and \(\bm{j}(\pos,t)\)?
A: This equation is a direct consequence of the inhomogeneous Maxwell eqs. (see ex. 1):
\[\colorboxed{BrickRed}{\pdv{\varrho}{t}+\dop\bm{j}=0}\]
It is the differential from of charge conservation.
\(\varrho(\pos,t)\): charge density | \(\bm{j}(\pos,t)\): current density
For  \(N\) point charges with charge  \(q_i\)  at | For  \(N\) (moving) point charges with
position  \(\pos_i\)  and total charge  \(Q_V\) | velocity  \(\vel_i\)
\(\displaystyle\varrho(\pos,t)=\sumitoN q_i\delta^{(3)}\left(\pos-\pos_i(t)\right)\) &
\(\displaystyle\bm{j}(\pos,t)=\sumitoN q_i\vel_i(t)\delta^{(3)}\left(\pos-\pos_i(t)\right)\)
\(\displaystyle Q_V(t)= \int_V\mathrm{d}^3\pos\,\varrho(\pos,t) = \sumitoN q_i\)
\(\displaystyle\bm{j}(\pos,t)=\sumitoN q_i\dpos_i(t)\delta^{(3)}\left(\pos-\pos_i(t)\right)\)

Q: What is the formula for the Lorentz force?
A: The electromagnetic force (Lorentz force) on particle \(i\) with charge \(q_i\):
\[\bm{F}_{L,i}=\bm{F}_L(\pos_i,t)=q_i\left(\Efield(\pos_i,t)+\frac{1}{c}\dpos_i\times\Bfield(\pos_i,t)\right)\]
The force resulting from the magnetic field acts only on moving particles and is perpendicular to both their velocity and the magnetic field.

Q: What are the scalar and vector potential \(\phi\) and \(\vecfield\)?
A: The electric and magnetic fields are physical observables. Consequence of homogeneous Maxwell eqs.: electric and magnetic fields are completely determined by a set of only four fields - the scalar
potential \(\phi=\phi(\pos,t)\) and the vector potential \(\vecfield=\vecfield(\pos,t)\):
\[\Efield=-\gop\phi-\frac{1}{c}\pdv{\vecfield}{t}\quad\text{and}\quad\Bfield=\cop\vecfield\]
instead of the six fields \(\Efield\) and \(\Bfield\).
However, they are NOT gauge invariant, and hence no physical observables!
\[\begin{aligned} \phi \quad & \longrightarrow\quad \phi\strich = \phi -\frac{1}{c}\pdv{\chi(\pos,t)}{t} \vecfield \\ & \longrightarrow \quad \vecfield\strich = \vecfield +\gop \chi(\pos,t) \end{aligned}\]

Q: What are Lorenz and Coulomb gauge?
A: Lorenz gauge
\[\partial_\mu A^\mu = \frac{1}{c}\pdv{\phi}{t} + \dop\bm{A} = 0\]
In Lorenz gauge the inhomog. Maxwell eqs. are decoupled:
\[\dAlembert\phi=4\pi\varrho\quad\text{and}\quad\dAlembert\vecfield=\frac{4\pi}{c}\bm{j}\]
with the d'Alembert operator
\[\dAlembert = \frac{1}{c^2}\pdv{^2}{t^2}-\Laplace\]
Coulomb gauge
\[\dop\bm{A} = 0\]
In Coulomb gauge, one of the inhomog. Maxwell eqs. is reduced to Poisson's equation:
\[\Laplace\phi=-4\pi\varrho\]
Especially useful in static situations \(\bm{j}=0\), as then all magnetic fields vanish and \(\vecfield=0\). Solving Poisson's equation leads to
\[\Efield(\pos)=\int\mathrm{d}^3\pos^\prime\,\varrho(\pos\strich)\frac{\pos-\pos\strich}{\abs{\pos-\pos\strich}^3}\]
and with \(\bm{F}=q\Efield\) to the familiar electrostatic Coulomb force:
\[\bm{F}_{12}=-q_1q_2\frac{\pos_1-\pos_2}{\abs{\pos_1-\pos_2}^3} = -q_1q_2\frac{\hat{\pos}_{12}}{{r_{12}^2}}\]

Q: What is the formula for the Lagrangian that describes one classical particle in electromagnetic fields? And what is the Hamiltonian for a charged particle moving in electromagnetic fields?
A: Lagrangian:
\[\colorboxed{BrickRed}{L(\pos,\dpos,t)=\underbrace{\frac{1}{2}m\dpos^2}_{=T(\dpos)} \underbrace{-q\phi(\pos,t)+\frac{q}{c}\dpos\cdot\bm{A(\pos,t)}}_{=U(\pos,\dpos,t)}}\]
Hamiltonian:
\[\colorboxed{BrickRed}{H(\pos,\mom,t) = \frac{1}{2m}\left[\mom-\frac{q}{c}\vecfield(\pos,t)\right]^2+q\phi(\pos,t)}\]