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

1. Sep 6, 2009

### mjordan2nd

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.

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

The indefinite integral gave me

$$-\frac{o z}{2 e \sqrt{r^2+z^2}}$$

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:

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

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. Sep 6, 2009

### mjordan2nd

Plugging this into mathematica gives

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

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

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