Flux through the sphere

Question

I have the vector field in spherical coordinates

$$ \mathbf{F} (r, \theta, \phi) = r^2\cos(\phi) \, \mathbf{e}_{\theta} $$

Why is the flux through the sphere zero?

Answer 1

Your field is

$$\vec{F} = r^2 \cos{\phi} \vec{e_{\theta}}$$

As there is only a $\theta$ component of the field, then the divergence of this field is

$$\vec{\nabla}\cdot \vec{F} = \frac{1}{r \sin{\theta}} \frac{\partial}{\partial \theta} (\sin{\theta} F_{\theta}) = r \cos{\phi} \cot{\theta}$$

The net flux through the sphere is the integral of the divergence of the field through the volume of the sphere:

$$\int_0^R dr \: r^3 \int_0^{\pi} d\theta \: \cos{\theta} \int_0^{2 \pi} d\phi \: \cos{\phi}$$

This integral, however, is zero because the integral over $\phi$ is zero.

Answer 2

It is known as Gauss's Law and it holds for any closed surface.