# Infinitely Long Charged Cylinder in a Maxwellian Plasma

• karlhoffman_76
In summary: C_2## by rearranging the equation: ##C_1 = \phi_0 \exp\left(\frac{-n_0 e^2}{k_B T_e \epsilon_0} r_0 \right)  C_2 = \exp\left(\frac{-n_0 e^2}{k_B T_e \epsilon_0} r_0 \right) $$Finally, we can substitute these values into the equation for ##\phi(r)## to get: ##\phi(r) = \phi_0 \exp\left(\frac{n_0 e^2}{k_B T_e \epsilon_0} (r-r_0) \right)$$In summary
karlhoffman_76
Hi guys, having a hard time figuring out where to start on a plasma physics related problem.

## Homework Statement

An inﬁnitely long metallic cylinder with a radius of ##r_0## is immersed in a Maxwellian plasma. An electric potential, ##\phi_0##, is applied to the cylinder. Assuming that the electrons are mobile but the ions are stationary, derive an expression for the potential as a function of radial distance, ##r##, also including ##r_0##, ##\phi_0##, and the Debye length, ##\lambda_D##.

## Homework Equations

Poisson's equation in cylindrical coordinates for the radial component only,

##\nabla^2\phi=\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial\phi}{\partial r})=\frac{-e}{\epsilon_0}(n_p-n_e)##

Boltzmann relation for electrons,

##n_e=n_0exp(e\phi/k_B T_e)##

## The Attempt at a Solution

Inside the cylinder the electric fields will cancel, hence there is no potential there. For ##0\leq r\leq r_0##,

##\phi(r)=0##

For ##r>r_0## first assume that far away from the cylinder ##n_0=n_p=n_e##. Thus the expression for the divergence of the electric field becomes,

##\nabla^2\phi=\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial\phi}{\partial r})=\frac{-e}{\epsilon_0}n_0[1-exp(e\phi/k_B T_e)]##

For large distances away from the cylinder ##|e\phi|\ll k_B T_e## and so expanding the exponential into a Taylor series and keeping only the first order term,

##\nabla^2\phi=\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial\phi}{\partial r})=\frac{n_0 e^2}{k_B T_e \epsilon_0}\phi##

Expanding out the LHS,

##\nabla^2\phi=\frac{d^2\phi}{dr^2}+\frac{1}{r}\frac{d\phi}{dr}=\frac{n_0 e^2}{k_B T_e \epsilon_0}\phi##

A second order, non-linear ODE in ##r##. I have not been able to figure out how to solve this equation. Although I only JUST now had a thought; at large distances from the cylinder the first order term will disappear. This is because as ##r\to \infty##,

##\frac{1}{r}\frac{d\phi}{dr}\to 0##

The resulting ODE is easy to solve. Am I on the right track?

Yes, you are on the right track! This is a good approach to solving this problem. To solve the resulting equation, you can use the method of separation of variables. We can write the equation as: ##\frac{d^2 \phi}{dr^2} + \frac{n_0 e^2}{k_B T_e \epsilon_0}\phi = 0 ## Let ##\phi(r) = R(r) \cdot F(r)##. Then we have: ## \frac{dF}{dr} = \frac{-n_0 e^2}{k_B T_e \epsilon_0} \cdot \frac{1}{R(r)}  \frac{dR}{dr} = \frac{n_0 e^2}{k_B T_e \epsilon_0} \cdot F(r) $$Solving these equations gives us: ##F(r) = C_1 exp\left(\frac{-n_0 e^2}{k_B T_e \epsilon_0} r \right)$$$$R(r) = C_2 \exp\left(\frac{n_0 e^2}{k_B T_e \epsilon_0} r \right)$$where ##C_1## and ##C_2## are constants. Now, we can find the potential by combining the two equations: ##\phi(r) = C_1 C_2 \exp\left(\frac{n_0 e^2}{k_B T_e \epsilon_0} r \right) $$To find the constants ##C_1## and ##C_2##, we need to apply the boundary conditions. At ##r = r_0##, the potential is equal to ##\phi_0##. Thus, we have: ## \phi_0 = C_1 C_2 \exp\left(\frac{n_0 e^2}{k_B T_e \epsilon_0} r_0 \right)$$We can solve for ##C_1## and

## 1. What is an infinitely long charged cylinder in a Maxwellian plasma?

An infinitely long charged cylinder in a Maxwellian plasma is a theoretical model used to study the behavior of charged particles in a plasma environment. It consists of an infinitely long cylinder with a uniform charge density surrounded by a plasma, which is a gas that has been ionized to create a collection of charged particles.

## 2. What is a Maxwellian plasma?

A Maxwellian plasma is a type of plasma in which the charged particles follow a Maxwellian distribution, meaning they have a range of speeds and energies that can be described by a specific probability function. This distribution is important for understanding the behavior of particles in a plasma environment.

## 3. How does the charge of the cylinder affect the plasma?

The charge of the cylinder has a significant effect on the plasma. It creates an electric field that can accelerate or decelerate the charged particles in the plasma, causing them to move in certain directions and at certain speeds. This can also cause the plasma to become unstable and lead to the formation of structures such as waves and instabilities.

## 4. What factors can influence the behavior of the charged particles in the plasma?

The behavior of charged particles in a plasma is influenced by several factors, including the strength of the electric field, the density and temperature of the plasma, the type of particles present, and any external forces acting on the system. These factors can all impact the motion and interactions of the charged particles, leading to complex and dynamic behavior.

## 5. How is the model of an infinitely long charged cylinder in a Maxwellian plasma used in scientific research?

The model of an infinitely long charged cylinder in a Maxwellian plasma is used in many areas of scientific research, including plasma physics, astrophysics, and engineering. It helps scientists to understand the behavior of charged particles in a plasma and how they interact with each other and their environment. This can be applied to many real-world scenarios, such as designing better plasma-based technologies or studying the behavior of plasma in space.

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