Calculating Drifts in a Z-Pinch Plasma

[Firben]

Homework Statement

In a so called z-pinch the plasma is determined by:B =B₀r/a θ^{^}

where a is the pinch radius

Calculate the drift velocity due a inhomogenous magnetic field and a bended magnetic field in ions with charge q and mass M in the pinch. Assume that that the ions in the plasma has a isotropic speed distribution with the speed sqrt(KT/M) per degree of freedom

Homework Equations

V_∇B = μ/(qB²)*(B x ∇B)

V_R = (mv_||²)/(qBR_c²)*(R_c x B)

The Attempt at a Solution

From the solution manual i don't understand how they calculate the
perpendicular and
horizontal speed.

This is what i don't get:

Isotropic distribution of ions:

1/2M<v_{_}²> = 2* KT/M <----- Why multiply with 2 ?

and

1/2M<v_||²> = 1 * KT/M <----- Why multiply with 1 ?
[/B]

 

[mark726]

it is important to always question and understand the methods and calculations used in any solution. In this case, it seems that the solution manual is using the standard definition of an isotropic distribution, which means that the particles have equal probability of moving in any direction. This allows us to simplify some of the calculations.

For the first question, multiplying by 2 is used to account for the isotropic distribution. In an isotropic distribution, the average speed in any direction is the same, so we can use the average speed in one direction and multiply it by the number of directions (which is 2 in this case) to get the total average speed.

For the second question, multiplying by 1 is used because the ions in the plasma are assumed to be moving in one direction only (parallel to the magnetic field). In this case, there is no need to account for the isotropic distribution, so we simply use the average speed in that one direction.

It is always important to carefully analyze and understand the methods and assumptions used in any solution, as this helps us to improve our understanding and knowledge as scientists.