Plasma Physics Debye Shielding

Q = q*n, where n is the number density of ions in the Debye shielding cloud.Substituting the equation for f(u), we get:Q = q*A*exp[-(q*phi)/k_b*T_e]*int(u1,u2)[exp(-(1/2*m*u^2)/k_b*T_e)*du]Since e*phi <<< kTe, we can approximate the integral as:int(u1,u2)[exp(-(1/2*m*u^2)/k_b*T_e)*du] = sqrt(2*pi*k_b*T_e/m) * [exp(-(1/2*m*u1^2)/k_b*T_e) - exp(-(1/2*m*u
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Homework Statement



Consider a positive point charge +q, immersed in plasma. Show that the net charge in the Debye shielding cloud exactly cancels the test charge. Assume the ions are fixed and that e*phi <<< kTe.

Homework Equations



f(u) = A exp [-(1/2*m*u^2 + q*phi)/k_b*T_e]
u is the x component of velocity

The Attempt at a Solution



Really lost on this one. I think I could make a hand waivey argument with Gauss's Law but I don't see where to go on this one.
 
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  • #2


Hello,

Thank you for bringing up this interesting question. Let's break down the problem and see if we can come up with a solution.

First, let's define some variables and assumptions:
- q is the positive point charge
- plasma is a collection of ions and electrons
- A is a constant
- m is the mass of the ions
- u is the x-component of velocity of the ions
- phi is the electric potential
- k_b is the Boltzmann constant
- T_e is the electron temperature
- We assume that the ions are fixed, meaning they do not move
- We also assume that e*phi <<< kTe, which means that the potential energy of the ions is much smaller than their thermal energy

Now, let's take a look at the equation given:
f(u) = A exp [-(1/2*m*u^2 + q*phi)/k_b*T_e]

This equation represents the velocity distribution of the ions in the plasma. It tells us the probability of finding an ion with a certain velocity u. The term inside the exponential is the energy of the ion, which consists of two parts: the kinetic energy (1/2*m*u^2) and the potential energy (q*phi).

We can see that as the potential energy (q*phi) becomes larger, the probability of finding an ion with a certain velocity decreases. This makes sense because if the potential energy is large, it means that the ions are attracted to the positive point charge and they will have a lower velocity.

Now, let's consider the Debye shielding cloud. This is a region around the positive point charge where the plasma is shielded from its electric field. In this region, the potential energy of the ions is significantly reduced because they are repelled by the positive point charge. This means that the ions in this region will have a higher velocity compared to the ions outside the Debye shielding cloud.

To show that the net charge in the Debye shielding cloud exactly cancels the test charge, we need to consider the total charge in this region. Since the ions are fixed, the only charges present in this region are the positive ions. The number of ions in this region can be calculated by integrating the velocity distribution function f(u) over the velocity range that corresponds to the Debye shielding cloud.

If we assume that the Debye shielding cloud extends from u1 to u2, we can calculate the total charge in this
 

1. What is Debye shielding in plasma physics?

Debye shielding is a phenomenon in plasma physics where a charged particle, such as an electron, is surrounded by a cloud of other charged particles that effectively shields the particle from the influence of external electric fields.

2. How does Debye shielding affect plasma behavior?

Debye shielding significantly affects the behavior of plasmas by reducing the strength of electric fields within the plasma. This allows for the stabilization of plasma instabilities and the formation of electric double layers.

3. What is the Debye length and how is it related to Debye shielding?

The Debye length is a measure of the distance over which the electric potential of a charged particle is significantly altered due to the presence of other charged particles. It is directly related to Debye shielding, as it is the characteristic length scale over which the electric field is screened.

4. How is Debye shielding relevant to plasma confinement?

Debye shielding is crucial for plasma confinement, as it helps to prevent the plasma from expanding and escaping its confinement. This is important for plasma-based technologies such as fusion reactors, where maintaining the plasma at high temperatures and densities is necessary for sustained fusion reactions.

5. Can Debye shielding be observed in laboratory experiments?

Yes, Debye shielding has been observed in laboratory experiments using various plasma devices such as tokamaks and plasma guns. It is a fundamental concept in plasma physics and is routinely studied and measured in plasma experiments.

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