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

[EngnrMatt]

Homework Statement

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.

Homework Equations

Q_pul = Q_{along l}/l

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.

 

[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?

 

[EngnrMatt]

ρ*dr I think?

 

[rude man]

ρ*dr I think?

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.

 

[EngnrMatt]

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