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## Homework Statement

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.

## 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

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