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- 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.

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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