Let $\vec{F} = x^3 \hat{i} + y^3 \hat{j} + z^3 \hat{k}$. The outward flux of $\vec{F}$ across the surface of the unit sphere $x^2 + y^2 + z^2 = 1$ is $\frac{a \pi}{5}$, where $a$ is a constant. The value of $a$ is ________ (answer in integer).
Answer Key:
12
Step-by-Step Solution & Explanation
By Gauss's Divergence Theorem:
$\iint_S \vec{F} \cdot \hat{n} \, dS = \iiint_V (\nabla \cdot \vec{F}) \, dV$
Computing the divergence of $\vec{F}$:
$\nabla \cdot \vec{F} = \frac{\partial}{\partial x}(x^3) + \frac{\partial}{\partial y}(y^3) + \frac{\partial}{\partial z}(z^3) = 3(x^2 + y^2 + z^2)$
Using spherical coordinates where $x^2 + y^2 + z^2 = r^2$ and $dV = r^2 \sin\phi \, dr \, d\theta \, d\phi$:
$\iiint_V 3r^2 \, dV = 3 \int_0^{2\pi} d\theta \int_0^\pi \sin\phi \, d\phi \int_0^1 r^4 \, dr$
$= 3 \times (2\pi) \times (2) \times \left[ \frac{r^5}{5} \right]_0^1 = 12\pi \times \frac{1}{5} = \frac{12\pi}{5}$
Comparing with $\frac{a\pi}{5}$, we get $a = 12$.