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Hi Jess,

     Gauss's Law is a law of physics, it's always true.  However, as far as usefulness goes, it's only really useful for finding the electric field when you have a charge distribution that exhibits spherical, cylindrical, or planar symmetry.  Some of the examples you'll find in the guide sheets (http://aplusphysics.com/courses/ap-c/tutorials/APC-EField.pdf) or tutorial video (http://aplusphysics.com/courses/ap-c/videos/APC-Gauss/APC-Gauss.html) include a sphere or shell of charge (spherical symmetry), a line of charge (cylindrical symmetry), and a plane of charge.  Hope that helps...  make it a great day!

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Aha.  I understand.  The integration your teacher is showing isn't the integration part of Gauss's Law, she's just using integration to find the charge enclosed by the Gaussian surface.  If she had just told you the charge enclosed was Q=Pi*a*R^3, you could have put that right into Gauss's Law.

 

So to answer your question, when do you have to integrate to find the charge enclosed?  When you're given some sort of charge density function instead of the total enclosed charge.  To use Gauss's Law, you have to know what Qenclosed is.  If it's not given directly, you need to figure out some way to find it, and in many problems, that will involve some amount of integration.

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