Changes between Version 4 and Version 5 of u/johannjc/scratchpad28


Ignore:
Timestamp:
12/30/20 18:23:58 (4 years ago)
Author:
Jonathan
Comment:

Legend:

Unmodified
Added
Removed
Modified
  • u/johannjc/scratchpad28

    v4 v5  
     1== Units in Astrobear ==
     2
     3Astrobear uses something like rationalized electromagnetic units (extra factor of $\sqrt{4 \pi}$ in the electric and magnetic fields) - or Lorentz-Heaviside but scaling the $E$ field by $c$ and the charge density $\rho$ (and current $J$ ) by $\frac{1}{c}$
     4
     5|| $E = c E^{LH} = \frac{c }{\sqrt{4\pi}}E^{G}$ ||
     6|| $\rho = \frac{1}{c}\rho^{LH} = \frac{\sqrt{4 \pi} }{c}\rho^{G}$||
     7|| $J = \frac{1}{c} J^{LH} = \frac{\sqrt{4 \pi}}{c} J^{G}$||
     8|| $B = B^{LH} = \frac{1}{\sqrt{4\pi}} B^{G}$ ||
     9
     10Using the approach in the appendix of Jackson, we have
     11|| $k_1 = \frac{c^2}{4 \pi} $ ||
     12|| $k_2 = \frac{1}{4\pi}$ ||
     13|| $k_3 = 1$ ||
     14|| $\alpha = 1$ ||
     15|| $\mu_0 = 1$ ||
     16|| $\epsilon_0 = \frac{1}{c^2}$ ||
     17This allows us to write Maxwell's equations as
     18
     19|| $\nabla \cdot \mathbf{E} = c^2 \rho$ ||
     20|| $\nabla \times \mathbf{B} = \mathbf{J} + \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t}$ ||
     21|| $\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$ ||
     22|| $\nabla \cdot \mathbf{B} = 0$ ||
     23
     24as well as
     25|| Lorentz Force Law || $\mathbf{F} = q \left ( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right)$ ||
     26|| Coulomb's Law || $\mathbf{F} = -\frac{c^2}{4 \pi} \frac{q_1 q_2}{r^2}\hat{\mathbf{r}}$ ||
     27
     28
     29Most of the time we don't care about $E$, $\rho$, or $J$, however for the Hall MHD terms, we need to determine the electron charge in our system of units.
     30
     31For our system of equations, the elementary charge is
     32
     33|| $e = 4.80320425 \times 10^{-10} \mbox{statC} = 1.70269157 \times 10^{-9} \mbox{hsu} = 5.67956774 \times 10^{-20} \left [\mbox{g}^{1/2} \mbox{cm}^{1/2} \right]$
     34
     35
     36
     37
     38
     39
    140=== MHD equations ===
    241
     42Using the system of units described above, we can write the following simplified equations
    343
    4 || Poynting's theorem (electromagnetic energy) || $\frac{\partial e}{\partial t} = - \nabla \cdot \mathbf{S} - \mathbf{J}\cdot \mathbf{E}  $ ||
    5 || Poynting vector || $\mathbf{S} = \mathbf{E} \times \mathbf{B} $ ||
     44
    645|| Induction equation || $\frac{\partial \mathbf{B}}{\partial t} = -\nabla \times \mathbf{E}$ ||
    746|| Ampere's equation (without Maxwell's correction) || $\mathbf{J} = \nabla \times \mathbf{B}$ ||
    847|| Lorentz Force Law || $\frac{\partial \rho \mathbf{v}}{\partial t}=\mathbf{J} \times \mathbf{B}$ ||
     48|| Poynting's theorem (electromagnetic energy) || $\frac{\partial e}{\partial t} = - \nabla \cdot \mathbf{S} - \mathbf{J}\cdot \mathbf{E}  $ ||
     49|| Poynting vector || $\mathbf{S} = \mathbf{E} \times \mathbf{B} $ ||
    950
    1051=== Equations for momentum, magnetic energy, and magnetic field ===