Electric flux through a specific part of a sphere

Question

Take a look at the following problem:

I have solved the problem; the question is related to something else. We can calculate the electric flux by calculating the fraction of solid angle subtended by the center of the sphere on the desired surface, which will also be the fraction of flux. i.e. $$\int d\Omega= \int \frac{\hat{n} \cdot d\vec{S}}{r^2}= \int_{\phi=-\pi/2}^{+\pi/2} \int_{\theta=\pi/2 -\alpha}^{\pi/2} \sin \theta\ d \theta\ d\phi=\pi (\sin \alpha)$$ after which the flux comes out to be $$\frac{Q}{4 \epsilon_0}(\sin \alpha)$$ which corresponds to the second option.

But something regarding this result is bugging me. It is that, when we the angle $\alpha$ is $\pi$, the flux should come out to be $Q/2\epsilon_0$ whereas here it says it is $0$. What exactly is it that I am missing?

Answer 1

The result is not $\sin\alpha$ but $(1-\cos{\alpha})$. It is always good to check as you did with half sphere or something trivial so you can be sure you didn't get the right answer.

You should check the integration boundaries. Just make it simple like

$$ \int\limits_0^\pi \int\limits_0^\alpha \sin\alpha \mathrm{d}\alpha \mathrm{d}\theta = \pi \int\limits_0^\alpha \sin\alpha \mathrm{d}\alpha = \dots $$

and I let you do the rest.