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Homework Help: Electric potential of cylinder

  1. Oct 1, 2013 #1
    1. The problem statement, all variables and given/known data
    An insulating solid cylinder of radius R, length L carries a uniformly distributed electric charge with density [itex] \rho [/itex]. Chose the z-axis along the axis of the cylinder, z=0 in the middle of the cylinder. the cylinder can be boken down into curcular tabs (disks) of thickness dl and surface charge [itex] \sigma [/itex], the combined slabs integrated over dl make up the cylinder.
    (a)Find the potential on the z axis due to a disk; express [itex] \sigma [/itex] in terms of [itex] \rho [/itex].
    (b) find the potential on the z-axis V(z) for the entire cylinder.
    (c)Calculate the electric field on the z-axis.

    3. The attempt at a solution
    (a) i drew a disk of radius R, and called the point where im calculating the potential at a point P. The disk is the sum of rings (of radius r) from 0 to R, the line from the center of the disk to the point P is z and the line connecting radius r to point P is r'.
    The charge distribution [itex] \sigma =dq/dA [/itex] which turns into [itex] dq=\sigma 2\pi rdr[/itex]

    Potential is:
    [tex] V=k\int \frac{dq}{r'} [/tex]

    Plugging the dq into the potential you get:
    [tex] V=k \int \frac{\sigma 2\pi rdr}{\sqrt{r^2 + z^2}} [/tex]

    Which reduces to:
    [tex] V=\frac{\sigma *\sqrt{R^2 +z^2}}{2 \epsilon_0} [/tex]

    Where [itex] \sigma=\rho dl [/itex]

    Which gives:
    [tex] V=\rho \frac{ \sqrt{R^2 + z^2} dl}{2\epsilon_0} [/tex]

    (b) I know i have to sum the potentials of all the disks to make the cylinder, but idk how to do that.

    is it:
    [tex] V=\int_{-L/2}^{L/2} \rho \frac{\sqrt{R^2 + z^2}dl}{2\epsilon_0} [/tex]

    (c) when i get the answer to (b) i can just take the (-)gradient of it to get E
    Last edited: Oct 1, 2013
  2. jcsd
  3. Oct 1, 2013 #2
    For part (b), is it:
    [tex] V=\int_{-L/2}^{L/2} \rho \frac{\sqrt{R^2 +z^2}dz}{2\epsilon_0} [/tex]
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