# Need help with Poisson's equation for a charged cylinder

• snatchingthepi
In summary, the problem asks for the potential difference inside and outside of an infinite cylinder with uniform charge to length density. Using Gauss' law, the potential can be determined easily. However, the potential must be found for both inside and outside of the cylinder using Poisson's equations. Once the two constants for the potential are found, the potential can be calculated using the equation.
snatchingthepi
Homework Statement
Compute the scalar potential and the electric field for an infinite charged cylinder. Assume the length density q/L is constant. Solve using Gauss' law and Poisson's equation
Relevant Equations
##\int \vec E \cdot d\vec a \ = \left(\frac Q \epsilon \right)##

##\nabla^2 V = \left(\frac \rho \epsilon \right)##
So I'm trying to solve for the field and potential inside and outside of an infinite cylinder with uniform charge to length density.

Using Gauss' law I am able to do this very easily and get the answers.

## V = \left(\frac {-\lambda} {2\pi\epsilon} \right) \ln\left(\frac b a \right)## for outside
## V = (a^2 - b^2) \left(\frac \lambda {4\pi\epsilon R^2} \right) ## for inside

where 'a' and 'b' are just points used to perform the integration.

And now I got to do it with Poisson's equations, and working back through the equation, and realizing that I have symmetry to help me out here, I end up with the results that

## V = - \left(\frac {\lambda r^2} {4\epsilon}\right) + C_0 \ln r + C_1 ##

where 'r' is the distance from the axis of the cylinder. I don't really know where to go next with this. I realize I have to find these two constants, but I don't really know how. I searched the forum and found this post (https://www.physicsforums.com/threa...of-a-cylinder-using-poissons-equation.598097/) but I haven't gotten much out of it.

I have been ill for the past week and haven't been able to do any work, but this assignment is due tomorrow afternoon and I feel very time-pressured. I'd appreciate any help with this.

Last edited:
snatchingthepi said:
where 'a' and 'b' are just points used to perform the integration.
?
Is this a solid cylinder, or a shell? a and b must be geometries, not just "..used to perform the integration..."?

Restate the problem accurately & we may be able to provide assistance.

rude man said:
?
Is this a solid cylinder, or a shell? a and b must be geometries, not just "..used to perform the integration..."?

Restate the problem accurately & we may be able to provide assistance.

Rudeman all the information is present in the problem statement, and the use of arbitrary points to determine a potential difference is acceptable with infinite cylinders since the potential does not behave at zero or infinity.

I understand about a and b, yes, potential must be between two finite points.

But - is b "outside" and a can be either "outside" or "inside"? Or vice-versa? Or, are both a and b either "outsie" or "inside"? I'm guessing the last.

And I a ssume the cylinder is solid.

The problem is ill-stated. You should be able to answer these questions or ask the poser to.

rude man said:
I understand about a and b, yes, potential must be between two finite points.

But - is b "outside" and a can be either "outside" or "inside"? Or vice-versa? Or, are both a and b either "outsie" or "inside"? I'm guessing the last.

And I a ssume the cylinder is solid.

The problem is ill-stated. You should be able to answer these questions or ask the poser to.

I feel as though you are completely neglecting the actual purpose of this post. Regardless of how you feel my problem is stated, you are pedantically distracting attention away from the actual question.

But yes, you are right in your assumptions.

## 1. What is Poisson's equation?

Poisson's equation is a mathematical expression that describes the relationship between the distribution of charges and the electric potential in a given space. It is a fundamental equation in electrostatics and is used to calculate the electric field and potential in various situations.

## 2. How is Poisson's equation used for a charged cylinder?

For a charged cylinder, Poisson's equation is used to calculate the electric potential at any point outside or inside the cylinder. The equation takes into account the charge density of the cylinder and the distance from the center of the cylinder to the point in question.

## 3. Can Poisson's equation be solved analytically for a charged cylinder?

Yes, Poisson's equation can be solved analytically for a charged cylinder. The solution involves integrating the equation with appropriate boundary conditions to find the electric potential at any point inside or outside the cylinder.

## 4. What are the units of the constants in Poisson's equation?

The units of the constants in Poisson's equation depend on the system of units used. In SI units, the constant ε0 (permittivity of free space) has units of Farads per meter (F/m), while the charge density ρ has units of coulombs per cubic meter (C/m3).

## 5. What are some real-world applications of Poisson's equation for a charged cylinder?

Poisson's equation for a charged cylinder has many practical applications, such as modeling the electric field and potential in a coaxial cable, calculating the capacitance of a parallel plate capacitor, and determining the electric potential inside a charged particle accelerator. It is also used in various engineering and scientific fields, including electromagnetics, plasma physics, and semiconductor device design.

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