Potential of an infinite cylinder

[phrygian]

Homework Statement

A cylindrical cable with a center rod of diameter a surrounded by a gap and
a thick pipe with inner and outer radii b & c. The charge distribution on the inner rod
is Rho = k cos(s Pi /(2a)), and the outer rod has a uniform charge density such that the total
system is neutral.

Homework Equations

V= - Integral(E.dL)
or
V= 1/(4Pi E0) Integral (Rho(r')/(r-r') dV')

The Attempt at a Solution

When using the first equation which uses the electric field, I end up getting an integral with 1/s Cos(s Pi/2a) which I can't integrate. This makes me think that I am supposed to use the second equation which relates the potential to the charge distribution.

My problem is that in Griffiths the text says that this equation does not work when the charge extends to infinity, because the potential doesn't go to zero at infinity. That explanation was based off an infinite plane. Since here the charge extends to infinity but the potential does go to zero at infinity is the second equation valid?

 

[gabbagabbahey]

Homework Statement

A cylindrical cable with a center rod of diameter a surrounded by a gap and
a thick pipe with inner and outer radii b & c. The charge distribution on the inner rod
is Rho = k cos(s Pi /(2a)), and the outer rod has a uniform charge density such that the total
system is neutral.

I don't see a question here. Are you asked to find the potential per unit length a distance $s$ from the axis of the cable? Is $a$ the radius of the center rod?

When using the first equation which uses the electric field, I end up getting an integral with 1/s Cos(s Pi/2a) which I can't integrate.

You shouldn't be getting an integral like that. Show your calculations.

Since here the charge extends to infinity but the potential does go to zero at infinity is the second equation valid?

What makes you think the potential goes to zero at infinity? (That's a no, by the way )

 

[Bhumble]

The rational of V and E going approaching zero as r approaches infinite is because the general form of the equations are 1/r and 1/r^2 but this is the distance from a charge so if your charge is "at infinite" then you aren't an infinite distance from the charge.

I'm assuming that the problem is looking for V and E everywhere anyway. You want to set the problem up utilizing cylindrical coordinates. V and E will be in the s-hat direction.

The boundary conditions are hinted at by the fact that the cable as a whole is electrically neutral so V = E = 0 s>c. Use this to solve V for a<s<b and b<s<c.