Derive Electric Field of Cylinder Via Gauss' Law

[Jawbreaker]

1. An infinitely long cylinder of radius R contains a uniform charge density Rho. Calculate the electric field using Gauss' law for r> R and R>r



3. I attached a pdf with my attempt at r>R. My answer doesn't agree with those given http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elecyl.html#c3". I noticed that length canceled but I'm not sure if my set up was done correctly. I solved the same problem using direct integration of Coulombs law and there I took the limit of L as it approached infinity. Any suggestions would be great. Thank you :]

 

[G01]

You just dropped the factor of r on the LHS of the second to last line. Other than this algebra error, your answer is correct.

 

[Jawbreaker]

Alright thanks for catching me there. So this is a correct way for setting up a problem for an infinite cylinder? I couldn't wrap my mind around how you could use gauss' law for an infinite surface or volume.

 

[G01]

Yeah, your approach is correct.

If it helps, try to think about it this way: Apply Gauss's law to a finite cylinder, and then take the limit as the length go to infinity. (Since L cancels, you'll get the same answer.)

Of course you have to ignore edge effects on the field with this approach but since you plan on taking the L going to infinity limit anyway, this doesn't matter.