1. The problem statement, all variables and given/known data(adsbygoogle = window.adsbygoogle || []).push({});

(a) Calculate the electric field at an axial point z of a thin, uniformly charged cylinder of charge density ρ , radius R, and length 2L. z is the distance measured from the center of the cylinder. (b) What becomes of your result in the event z >> L ?

2. Relevant equations

I found the answer to (a) by doing a triple integral and then double checked by integrating the equation for a disk of charge given in my textbook over the z-axis and came up with the same exact result. So I am quite confident this expression is correct. The image linked is my work to find the answer if this doesn't look right.

E_{z}= kq/R^{2}L { 2L + √[ R^{2}+ (z-L)^{2}] - √[ R^{2}+ (z+L)^{2}}

(As a 2nd check...i wrote the expression as kq/z^{2}{ 2L + √[ R^{2}+ (z-L)^{2}] - √[ R^{2}+ (z+L)^{2}} [ z^{2}/R^{2}L ] and in my calculator wrote a quick program to calculate the limit and sure enough...all that garbage on the right goes to 1 as z >> L and R.)

3. The attempt at a solution

This expression should reduce to kq/z^{2}because any charge distribution should mimic a point charge at large distances. The answer to the problem is some type of application of the binomial expansion which I cannot seem to figure out. Everything I try just leads to everything in the bracket except 2L to be zero. Any help is greatly appreciated.

Side Question: This expression is only valid for z outside the cylinder. To calculate z inside the cylinder...is it valid to use the previous integral but changing the bounds from -L to z, and then adding the second integral of the remaining part of the cylinder that is above the charge from z to L and adjusting the radius of dq accordingly, or is there a more simple way to do it?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Electric field due to a FINITE cylinder of charge - Tricky binomial expansion

**Physics Forums | Science Articles, Homework Help, Discussion**