Monday, January 11, 2016

Solutions to Space Plasma Physics Problems (1)


1)      Find the ion and electron gyrofrequencies for an ion and electron in solar plasma with a magnetic induction (field strength) of B = 0.0001 T.

Solution:   The ion gyrofrequency will be:

i  = qB/ m i  =   [(1.6 x 10 -19  C) (0.0001 T) ]/ 1.7 x 10 -27  kg
i   =  9.4 x 10 3 /s

 And the electron gyrofrequency is:

 e  = qB/ m e  =   [(1.6 x 10 -19  C) (0.0001 T) ]/ 9.1 x 10 -31  kg

e  =  1.7 x 10 7  /s


2)     If the perpendicular velocity component ( v) is 105  m/s for the electron, find its Larmor radius and its gyro-period.

 
Larmor radius:  r = m/ q [v / B] =  v/ (qB/ m e) = v/ e

 r = (10 5  m/s) / 1.7 x 10 7  /s   =     0.0056 m or:  0.56 cm

Gyro-period: T = 2 p  / e  =   2 p / 1.7 x 10 7  /s   =  3.5 x 10 -7  s

3)     Thence or otherwise obtain the gyration energy in eV.
(1.6 x 10 -19  J = 1 eV)

 Gyration energy E = m e  (v)2/ 2 =  
  (9.1 x 10 -31  kg) (10 5  m/s) 2  / 2

E =    4.5 x 10 -21  J  Or: in Electron volts:  
 
 (4.5 x 10 -21  J )/ (1.6 x 10 -19  J/eV) = 0.028 eV


4)     Find the guiding center positions for the electron referenced above (previous problems) if t = T/4.

 Guiding center positions:

 x – xo =   r sin (e t)   =    (0.0056 m)  sin (e  T/4) = 

=  (0.0056 m) sin [(1.7 x 10 7  /s) (3.5 x 10 -7  s/ 4)] = 0.0056 m

y – yo =   r cos (e t)   =    
(0.0056 m) sin [(1.7 x 10 7  /s) (3.5 x 10 -7  s/ 4)] = 0

 To see how these values can be, note that: (e T)   = 6.28 rad = 2p rad

 So:  (e T/ 4)   =1.57 rad =  p  / 2

 But:  sin (p /2) = 1 and (cos p /2) = 0

 So the value for x – xo is simply dictated by the value for  r

 5) What other critical information do you need to obtain the E-field at the guiding center?

From the information provided we see that:

v d     = (1 +    ¼ r 2  v2 ) o  X B/  B 2

 So the other information needed to obtain o  would be the drift velocity, v d  .

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