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Charge density in a cylinder

  1. May 28, 2015 #1
    1. The problem statement, all variables and given/known data
    A cylinder of radius a and length l has charge distribution

    ρ=Cr2
    where C is a constant and r is radial distance in cylindrical coordinates.
    Derive an expression for the average charge density within the cylinder.

    2. Relevant equations
    Well, charge density given is within the volume, I think.
    So for a point on the axis of the cylinder should be ρ divided by the length shouldn't it? Or is that being too simple?

    3. The attempt at a solution

    ρ = Cr2 / l

    Can someone confirm this or point me in the right direction, please.
     
  2. jcsd
  3. May 28, 2015 #2
    First you need to integrate the distribution over the cylinder to find the total charge.
     
  4. May 28, 2015 #3
    ∫ Cr2 dr

    C ∫ r2 dr (0 < l < L)

    CL3/3
     
  5. May 28, 2015 #4
    The radius 0 < r < a .
    The length L.

    You need to integrate over the VOLUME of the cylinder to find the total charge in the volume. I advise working in cylindrical co-ordinates. where [tex] dV = rdrd\theta dz[/tex]
     
  6. May 29, 2015 #5
    s ρ(r) dV = Q

    ∫ Cr2 * 4πr2 dr = 4Cπ ∫ r4 dr

    so
    Q = 4CπR5 / 5
     
  7. May 29, 2015 #6
    You need to integrate over the cylinder.

    [tex] \int dV \ = \iiint rdrd\theta dz\ = \int_{0}^{L} dz \ \int_{0}^{2\pi}d\theta \int_{0}^{a}rdr [/tex] This is the volume integral for a cylinder and as you can see, doing the integral gives the volume of a cylinder of radius, a, and length, L. But since your integrating a function over this volume, you want
    [tex] \int \rho(r) dV [/tex] You can split the integral similarly to find the total charge.
     
    Last edited: May 29, 2015
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