Changes between Version 15 and Version 16 of u/erica/d


Ignore:
Timestamp:
12/14/15 14:53:14 (9 years ago)
Author:
Erica Kaminski
Comment:

—

Legend:

Unmodified
Added
Removed
Modified

  • u/erica/d

    v15 v16  
    100100where
    101101
     102[[latex($\theta_1 = \int_{r_i}^{r_o} \frac{B_0 r_0}{r} 2 \pi r dr = 2 \pi B_0 r_o (r_o - r_i)$)]]
    102103
     104[[latex($\theta_2 = \int_{R}^{r_o} B_0 2 \pi r dr = \pi B_0(r_o^2 - R^2)$)]]
     105
     106this second integral is the flux as measured before the field has been distorted (i.e. when R=r_i).
     107
     108Setting these 2 fluxes equal and solving for r_i gives
     109
     110[[latex($r_i = \frac{r_o^2 + R^2}{2r_o}$)]]
     111
     112Plugging in for r_o found above gives
     113
     114[[latex($r_i = R \frac{(\beta_{ram} + 1)}{\sqrt{4 \beta_{ram}}}$)]]
     115
     116which is approximately
     117
     118[[latex($r_i \approx R \sqrt{\frac{\beta_{ram}}{4 }}$)]]
     119
     120or
     121
     122[[latex($\boxed{r_i \approx \frac{1}{2}r_o}$)]]
     123
     124or
     125
     126[[latex($\boxed{r_i \approx 3R}$)]]