Electric field of a uniform finite cylinder

[dipole]

Homework Statement

I have a solid cylinder of uniform charge density whose axis is centered along the z-axis. I am trying to calculate the electric field at a point on the z-axis.

What I'm trying to do is to start by first calculating the field of a disk centered on the z-axis at a point on the z-axis, then sum up a bunch of disks to find the field of the cylinder.

Homework Equations

I find that the electric field of a uniform disk at a point on the z-axis is given by:

$2\pi\rho[1-\frac{z}{\sqrt{z^{2}+R^{2}}}]$

Where R is the radius of the disk and rho is the charge density.

Now I want to write this as:

$2\pi\rho[1-\frac{z-z'}{\sqrt{(z-z')^{2}+R^{2}}}]$

Where z' is the position on the z-axis of the nth disk, and z-z' is the distance between the disk and the point of interest. Integrating the above expression over the length of the cylinder, however, leads to an electric field which increases with z and approaches a constant value, rather than decreasing and dropping off to zero at infinity.

I'm not sure what I'm doing wrong, but I know that I'm evaluating the above integral correctly, and I know the expression for the field of uniform disk is correct, so I must not be summing the disks properly... any help?

 

[rude man]

Inside and outside the cylinder?

 

[PhysicsGente]

I think it'll be easier if you start by finding the potential at z and checking it for consistency. As for summing the disks, you can choose easy points as the limits for your integral. For example, you can set the point z where you want to know the E-field at the origin and and say that the left end of the cylinder is at z. Where would the right end then be? ;)

 

[Mindscrape]

I remember doing this problem at one point or another, and I remember that the integrals had the potential to be pretty nasty. What answer are you getting? Explicitly, what integral are you doing?

You may be getting the right answer, just in a different form.

 

[paranormal]

Hello,

Thank you for sharing your calculations and concerns regarding the electric field of a uniform finite cylinder. From your description, it seems like you are on the right track, but there may be a small error in your integration or summation of the disks.

One potential issue could be that you are not taking into account the varying distances of the disks from the point of interest on the z-axis. As you move further away from the center of the cylinder, the disks will have a smaller contribution to the overall electric field. Therefore, the integral and summation should also take into account the distance between each disk and the point of interest.

Additionally, it may be helpful to double check your integration limits and make sure they are properly accounting for the entire length of the cylinder. It is also worth considering any simplifications or assumptions you may have made in your calculations.

I hope this helps guide you in finding the error in your calculations. Remember to always double check your work and make sure all variables and assumptions are properly accounted for. Best of luck with your homework!