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dcrisci
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Homework Statement
There is an infinite conducting cylinder positioned at the axis with radius R. An infinite line charge (+λ) is placed distance d from the axis and d>R. I was supposed to 1. Find the potential and then 2. find the surface charge on the cylinder.
Homework Equations
V = -∫ E⋅dr
δ = -ε0[itex] \frac{\partial V}{\partial r}[/itex]
The Attempt at a Solution
I used method of images and placed a wire of -λ inside the cylinder on the same axis as the original line charge. I calculated the electric field using Gauss's Law, and then integrated it with respect to r (saying that the potential does not depend on θ in this case). I found the electric field of both line charges, integrated them to get the potential from each line charge, then added them together and simplified them to get:
V = [itex] \frac{λ}{2πε}\ln\frac{r'}{r}[/itex]
then was supposed to find the surface charge on the cylinder that is induced by the line charge. Using the equation above I was stuck with:
δ = [itex] \frac{-λ}{2π}\frac{\partial }{\partial r}ln(\frac{r'}{r}) [/itex]
Upon differentiating the ln function I am left with -1/r, and the surface charge becomes
δ = [itex] \frac{λ}{2πr} [/itex]
which is positive. However due to the positive line charge positioned at distance d from the axis of the cylinder, I would have thought the induced surface charged would have to be negative? I looked up the answer to the potential I found for the area outside the cylinder and its correct, so where am I going wrong here?
Edit: also I am curious as to what the difference between a conductor that is grounded and neutral. I looked up a definition for them and it gave the definition in terms of electrical circuits and the voltages at ground and neutral wires, however I was looking for more of a general definition that can be applied to these problems. Such as do both have V = 0 on the surface? Both have E = 0 inside the conductor? and any other properties between the two. Thank you!