How do I integrate the electric field of an infinite horizontal line charge at some distance y above the line of charge, using trigonometric substitution? I'm having difficulty defining the limits of integration.

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anonymous · Answer

Using the following diagram |dw:1423494681662:dw|
We can write the  electric field at P due to charge at dx as$$dE = k \lambda dx/r ^{2} $$where lambda is the linear charge density.  Integrating we get$$E =k \lambda \int\limits_{0}^{\infty} dx/r ^{2}$$  but $$x = l*\tan \Theta$$$$dx=l*\sec ^{2}\Theta d \Theta $$ From the diagram we can see in changing the variable from x to theta that the limits will go from 0 to pi/2.  substituting for r using  the trig identity  $$1+ \tan ^{2}\Theta =\sec ^{2} \Theta $$ we complete the final integral.