# Finding Charge p.u.l. Along Infinitely Long Cylinder

1. Sep 16, 2014

### EngnrMatt

1. The problem statement, all variables and given/known data

An infinitely long cylinder of radius a in free space is charged with a volume charge density ρ(r) = ρ0*(a-r)/a (0 ≤ r ≤ a), where ρ0 is a constant and r the radial distance from the cylindrical axis. Find the charge per unit length of the cylinder.

2. Relevant equations

Qpul = Qalong l/l

3. The attempt at a solution

I'm pretty sure I'm supposed to integrate in cylindrical coordinates, however, it has been a while since I have done so. The limits of integration should be 0 to a. The equation for ρ0 is being integrated. But I thought there was something you're supposed to do when integrating in cylindrical. Or maybe it would actually be better in rectangular? Though I doubt that.

2. Sep 17, 2014

### rude man

harge per unit length = charge inside volume of unit length.

So you want to find the total charge in a unit length of the cylinder.
What is the charge in a cylindrical shell section of the unit length cylinder between radii r and r+dr?

3. Sep 17, 2014

### EngnrMatt

ρ*dr I think?

4. Sep 17, 2014

### rude man

that can't be right since ρ has units of QL-3 so ρ*dr would have units of QL-3L2 including the fact that we assume unit length. But we need units of Q.

To find the volume of a cylindrical shell, subtract a slightly larger shell volume from a slightly smaller volume. Make the outer radius r + dr and the inner radius r, then subtract and drop any terms of order dr2.

Or, take the area of the shell and multiply by the thickness dr.

5. Sep 19, 2014

### EngnrMatt

I actually figured it out on my own finally. Thanks for your time though.