Finding current in a sphere while given a changing current density

Question

im trying to calculate the current running through a sphere while given a current density $$J=Jcos(2\theta)e^{-t/r} \hat r $$ J is given in spherical coordinates. I know that: $$I=\int J\,da $$ But I can't seem to understand how to "build" my integral. I was thinking that i need to do something like $$ \int r\,dr\,sin\theta\,d\theta*J $$ but this gets very messy because of the exponent containing expression with "r". Any tips/help please? Thanks

Answer 1

I am not sure what is exactly your problem with evaluating the integral. Here is what I would do. We have hat $J$ points radially, so the total current entering or exciting a sphere of radius $R$ is simply given by $$ I= \int |J| |da|=\int |J| R^2 d\Omega = \int |J| R^2 d\varphi \sin\theta d\theta.$$ as the normal of the area element $da$ is always parallel to the current. The resulting integrals are not at all difficult to evaluate. We have (evaluating first the integral over $\varphi$) $I = JR^2 \int \cos(2\theta) d\varphi \sin\theta d\theta = 2\pi J R^2 e^{-t/R} \int_{0}^\pi \cos(2\theta) \sin\theta d\theta = - \frac{4}{3} J R^2 e^{-t/R}$$