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Uniformly Charged Cylinder - Potential at distance d?

  1. Oct 12, 2011 #1
    1. The problem statement, all variables and given/known data
    Consider a uniformly charged solid cylinder of radius R, length L and charge density ρ. Find the
    potential at a distance d (> L/2) from the centre of the object, along the axis of the cylinder.

    2. Relevant equations
    V=∫kdq/r



    3. The attempt at a solution
    For me, it makes most sense to express this integral in cylindrical coordinates, seeing as the object is a cylinder. Also, since the axis of which the cylinder is on is not specified. I chose the z axis on a (x,y,z).

    -stuff used for integration, respectively.
    s[0,R]
    Ø[0,2π]
    z[0,L]

    dq=sdsdØdz


    I'm not too good at expressing notation on the computer so this is the basics of what i tried:

    1st attempt: I know V(z)=∫E.dl , so, I tried to solve for E by using
    E=∫kdq/r^2
    E=∫∫∫(kρ/z^2)sdsdØdz

    after I finished this integral, I lost confidence when doing the integral for V(z) because It didn't seem right to integrate over the same limits of integrations seeing as dl would be expressed as sdsdØdz - please tell me if i'm wrong.

    2nd attempt: I just started with V=∫kdq/r.
    V=∫∫∫(kρ/z)sdsdØdz
    giving me
    V(z)=kρπ*ln(z)*R^2

    - I'm not sure which method is correct, if either. Help would be greatly appreciated.
    - Also, since were just looking for the function with respect to a distance and because of the way the question was stated, I felt that it was okay to express r as just z rather than (L/2 + z)
     
  2. jcsd
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