Why does Coulomb's Law have a pi in the denominator while Gauss' Law does not?

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The discussion centers on the discrepancy between the denominators in the electric field calculations for a uniform straight infinite line charge using Gauss' Law and Coulomb's Law. While both methods yield the same numerator related to linear charge density, Gauss' Law results in a denominator of 2r times epsilon, whereas Coulomb's Law includes an additional factor of 2(pi)r times epsilon. This difference raises questions about the inclusion of pi in Coulomb's Law. The original poster expresses confusion over this aspect but later acknowledges posting in the wrong forum. Understanding the geometric considerations in each law clarifies the differing denominators.
Old Guy
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I'm sure I'm missing something simple here, but to the point:

I calculate the electric field a distance r from a uniform straight infinite line charge using Gauss' Law and get an answer; I do the same calculation using Coulomb's law and get the same answer but a pi remains in the denominator (that was not there in the Gauss' Law solution). I don't know how to enter the equation here, but in both answers the numerator is the linear charge density. The Gauss' Law denonminator is 2r times epsilon; Coulomb's law denominator is 2(pi)r times epsilon. Help!
 
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oops!

Sorry, all; I just realized I posted this in the wrong forum.
 

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