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1. Homework Statement
(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. Homework 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 zaxis 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} + (zL)^{2} ]  √[ R^{2} + (z+L)^{2} }
(As a 2nd check...i wrote the expression as kq/z^{2} { 2L + √[ R^{2} + (zL)^{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?
(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. Homework 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 zaxis 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} + (zL)^{2} ]  √[ R^{2} + (z+L)^{2} }
(As a 2nd check...i wrote the expression as kq/z^{2} { 2L + √[ R^{2} + (zL)^{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?
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