Originally Posted by

**mathlover10** this is the E field above a square loop with side=a at distance z on the z axis. by symmetry the Ex, Ey field cancel out. it really kind of bothers me that I can't see this substitution, including the sin(tan^-1(u)) portion. I guess it's an easy technique but I wonder how you get the substitution?

$\displaystyle Ez=\frac{4\lambda z}{4\pi \varepsilon o}\int_{-a/2}^{a/2}\frac{ dx}{(z^2+x^2+a^2/4)^{3/2}}, x=\sqrt{a^2/4+z^2}tanu, dx=\sqrt{a^2/4+z^2}sec^{2}udu, I=\frac{1}{a^2/4+z^2}\int cosudu=\frac{1}{a^2/4+z^2}sinu\therefore Ez=\frac{8\lambda az}{4\pi\epsilon o \sqrt{2a^2+4z^2}z^2+a^2/4}$

thanks!