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

• snatchingthepi

#### 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.

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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.

?
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.

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.