Derive Electric Field of Cylinder Via Gauss' Law

Jawbreaker
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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 :]
 

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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.
 
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.
 
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.
 
To solve this, I first used the units to work out that a= m* a/m, i.e. t=z/λ. This would allow you to determine the time duration within an interval section by section and then add this to the previous ones to obtain the age of the respective layer. However, this would require a constant thickness per year for each interval. However, since this is most likely not the case, my next consideration was that the age must be the integral of a 1/λ(z) function, which I cannot model.
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