The problem is: Two 1.20m nonconducting wires meet at a right angle. One segment carries 2.50 of charge distributed uniformly along its length and the other carries 2.50C also distributed
uniformly along its length.
Find the magnitude and direction of the electric field these wires produce at point, which is 60.0cm from each wire.
So, I know this is a math community but I reduced the problem of the electric field at the point due to one of these wires to the following integral:
$$2\cdot\dfrac{\lambda D}{4\pi\epsilon_{0}}\int_{-x}^0 \dfrac{1}{(x^2+D^2)^{3/2}} dx$$ where $D$ is 60.0cm. I know the magnitude of the electric field due to both of them are just $\sqrt{2}$ of the value of the inverse.
I've tried $u$ substitution and trig integrals, but it doesn't give me anything helpful. What steps do I need to use?
Best Answer
Hint: Substitute $x=Dtan(u)\implies dx=D{sec}^2(u)du$:$$\int \frac{1}{(x^2+D^2)^{\frac{3}{2}}}dx=\frac{1}{D^2}\int cos(u)=\frac{x}{D^2\sqrt{x^2+D^2}}+C$$