GATE 2022 AE – Question 48
For $\vec v=x^3\hat i+y^3\hat j+z^3\hat k$, the outward flux over the unit sphere centered at the origin (one decimal place) is ________.
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 7.46 to 7.54
Explanation
**Divergence theorem.** The outward flux through a closed surface equals the integral of the divergence over the enclosed volume:
$$\oint\vec v\cdot d\vec A=\int_V\nabla\cdot\vec v\,dV .$$
**Divergence.** With $\vec v=(x^3,y^3,z^3)$:
$$\nabla\cdot\vec v=3x^2+3y^2+3z^2=3r^2 .$$
**Integrate over the unit ball** (spherical shells of area $4\pi r^2$):
$$\int_0^13r^2\,(4\pi r^2)\,dr=12\pi\int_0^1r^4\,dr=\frac{12\pi}{5}=7.54 .$$
The flux is **7.5**.