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Force on a charge at the center of a non-uniform charge density cylinder

  1. Sep 6, 2009 #1
    1. The problem statement, all variables and given/known data
    Consider a cylinder, radius R and length L. Suppose the cylinder is charged with a volume charge given by P+P0 + Bz, where p0 and B are constants. Find the force on a charge Q at the center of the cylinder.


    2. Relevant equations

    E=q/4*pi*e*r^2

    3. The attempt at a solution

    I first found the electric field contribution of a disk with uniform charge density on a point z above the center of the disk, where e is epsilon-not and o is the charge density on the disk.

    [tex]\int \frac{z r o 2 \pi }{4 \pi e \left(r^2+z^2\right)^{3/2}} \, dr[/tex]

    The indefinite integral gave me

    [tex]-\frac{o z}{2 e \sqrt{r^2+z^2}}[/tex]

    Evaluating from 0 to R gave me

    (oz/2e)*(1/z-(z^2+r^2)^-1)

    I then used this to evaluate the integral from -z/2 to z/2 on the cylinder, and came up with this expression:

    [tex]\int_{-\frac{L}{2}}^{\frac{L}{2}} \left(\left(\frac{L}{2}+z\right) b+p\right) z \left(\frac{1}{z}-\frac{1}{\sqrt{r^2+z^2}}\right) \, dz[/tex]

    I have two questions:

    Have I gone about this problem correctly so far, and how do I proceed from here. I don't know how to evaluate the integral. I figured I'd calculate the E-field first and then the force.

    Thanks for any help...
     
  2. jcsd
  3. Sep 6, 2009 #2
    Plugging this into mathematica gives

    [tex]\text{If}\left[\text{Im}\left[\frac{r}{L}\right]\geq \frac{1}{2}\left\|\text{Im}\left[\frac{r}{L}\right]\leq -\frac{1}{2}\right\|\text{Re}\left[\frac{r}{L}\right]\neq 0,\frac{-2 b L^3-8 b L r^2+4 b L^2 \sqrt{L^2+4 r^2}+8 L p \sqrt{L^2+4 r^2}-4 b r^2 \sqrt{L^2+4 r^2} \text{Log}[2]+b r^2 \sqrt{L^2+4 r^2} \text{Log}[16]-4 b r^2 \sqrt{L^2+4 r^2} \text{Log}\left[-L+\sqrt{L^2+4 r^2}\right]+4 b r^2 \sqrt{L^2+4 r^2} \text{Log}\left[L+\sqrt{L^2+4 r^2}\right]}{8 \sqrt{L^2+4 r^2}},\text{Integrate}\left[\left(b \left(\frac{L}{2}+z\right)+p\right) \left(1-\frac{z}{\sqrt{z^2+r^2}}\right),\left\{z,-\frac{L}{2},\frac{L}{2}\right\},\text{Assumptions}\to !\left(\text{Im}\left[\frac{r}{L}\right]\geq \frac{1}{2}\left\|\text{Im}\left[\frac{r}{L}\right]\leq -\frac{1}{2}\right\|\text{Re}\left[\frac{r}{L}\right]\neq 0\right)\right]\right][/tex]

    The indefinite integral gives [tex](-(2 p + b (L + z)) (r^2 + z (z - Sqrt[r^2 + z^2])) +
    b r^2 Sqrt[r^2 + z^2] Log[z + Sqrt[r^2 + z^2]])/(2 Sqrt[r^2 + z^2])[/tex]

    I could evaluate the indefinite but I'm not sure if I've set the problem up correctly. Again, any help would be appreciated.
     
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