Help Needed: Theoretical Plasma Problem

In summary: This relationship is important for understanding the behavior of the plasma and finding solutions to the diffusion equation. In summary, the equation sqrt(vi/D)=pi/L can be used to calculate the electron temperature in a one-dimensional plasma with boundary conditions set at x=+/- L/2 and x=0.
  • #1
maverik
9
0
Any help on the following theoretical plasma problem would be greatly appreciated!

The diffusion eqn for a one-dimensional plasma, varying in the x direction only is

D d2n/dx2 =-Q

Letting Q=nvi we can show the diffusion eqn has a solution of the form

n=Asin[sqrt(vi/D)x] + Bcos[sqrt(vi/D)x]

Supposing boundary conditions of plasma are set at x= +/- L/2 where n=0 and x=0 where n=n0 one can show

n=n0cos(x pi/L)

The bit I am finding trouble with is what the implications of sqrt(vi/D)=pi/L are with respect to the electron temperature Te, I have recalled that vi has an exponential dependence on Te but cannot reason what this means.

Many thanks in advance for any help.
 
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  • #2
The equation sqrt(vi/D)=pi/L implies that the ratio of the electron thermal velocity to the diffusion coefficient is equal to the reciprocal of the length of the plasma. This can be used to calculate the electron temperature, as the thermal velocity is related to the temperature through the Maxwell-Boltzmann distribution. Specifically, it can be shown that the thermal velocity is proportional to the square root of the temperature, so the temperature can be calculated as follows: Te = (pi/L)^2 * D/vi
 

1. What is a theoretical plasma problem?

A theoretical plasma problem is a scientific question or issue related to the behavior, properties, or applications of plasma, which is a state of matter consisting of ionized gas particles.

2. What are some common applications of plasma?

Plasma has a wide range of applications, including in electronics, energy production, materials processing, and medical treatments.

3. What are the main challenges in solving theoretical plasma problems?

The main challenges in solving theoretical plasma problems include the complex and non-linear nature of plasma, the lack of experimental data, and the need for advanced mathematical and computational techniques.

4. How do scientists approach solving theoretical plasma problems?

Scientists typically use a combination of theoretical models, computer simulations, and experimental data to study and solve theoretical plasma problems. This interdisciplinary approach allows for a better understanding of plasma behavior and properties.

5. What are some current areas of research in theoretical plasma problems?

Some current areas of research in theoretical plasma problems include fusion energy, plasma propulsion, and plasma-based materials processing. Other topics of interest include plasma instabilities, turbulence, and self-organization.

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