Using Gauss' law to find the induced surface charge density ##\sigma##

[Meow12]

Homework Statement

A thin insulating rod with charge density ##\lambda=\rm+5\ nC/m## is arranged inside a thin conducting cylindrical shell of radius ##R=\rm 3\ cm##. The rod and shell are on the same axis, and you can assume they are both infinite in length. What is the SURFACE charge density ##\sigma## induced on the OUTSIDE of the conducting shell in ##\rm C/m^2##?

Relevant Equations

Statement of Gauss's Law: ##\displaystyle\oint\limits\vec{E}\cdot d\vec{A} = \frac{Q}{\epsilon_0}##

My attempt:

The electric field in the interior of a conductor is ##0##.

By symmetry, the electric field is directed radially outward.

Take the Gaussian surface as the thin cylindrical shell of radius ##\rm 3\ cm## and length ##L##.

##\displaystyle\oint\limits\vec{E}\cdot d\vec{A} = \frac{Q}{\epsilon_0}##

Since ##E=0## everywhere, ##Q=0##

##\lambda L+\sigma\cdot 2\pi R L=0##

##\lambda+2\pi R\sigma=0##

##\displaystyle\sigma=\rm-\frac{\lambda}{2\pi R}##

Upon substituting the values, we get ##\rm\sigma=-2.6\times 10^{-7}\ C/m^2##

##\sigma_{outside}=\rm+2.6\times 10^{-7}\ C/m^2=\rm +260\ nC/m^2##

But the correct answer is ##\rm +26\ nC/m^2##. I'm off by a factor of ##10##; where have I gone wrong?

 

[TSny]

Welcome to PF!

##\displaystyle\sigma=\rm-\frac{\lambda}{2\pi R}##

Upon substituting the values, we get ##\rm\sigma=-2.6\times 10^{-7}\ C/m^2##

Your formula is correct. When I substitute the values, I get a result that is about 1/10 of your value. Check your work. If you still aren't getting the correct value, show the numerical values that you used in the formula.

 

[Meow12]

Welcome to PF!Your formula is correct. When I substitute the values, I get a result that is about 1/10 of your value. Check your work. If you still aren't getting the correct value, show the numerical values that you used in the formula.

Yeah, I had made a silly calculation mistake. Thanks for your post.