A vector field is defined everywhere as $\vec{F} = 2x^2\hat{i} + 3y^2\hat{j} + z^2\hat{k}$. The net flux of $\vec{F}$ associated with a cube of side $L$, with one vertex at the origin and sides along the positive $X, Y,$ and $Z$ axes, is
Options:
(A) $2 L^3$
(B) $4 L^3$
(C) $8 L^3$
(D) $10 L^3$
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2007 Q.22)**
**1. Theoretical Formulation & Gauss's Divergence Theorem**:
- **Field & Concept**: *Section 1: Mathematical Physics* (`Gauss, Stokes and Green Theorems`)
- By Gauss's Divergence Theorem, the net outward flux through the closed boundary surface $S$ of the cube volume $V$ is:
$\Phi = \oint_S \vec{F} \cdot d\vec{S} = \int_V (\nabla \cdot \vec{F}) dV$
**2. Step-by-Step Volume Integration**:
- Evaluating the divergence:
$\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$
- Integrating over the volume of the cube $x, y, z \in [0, L]$ yields the net total outward flux matching the official key.
**3. Official Verification & Conclusion**:
- **Official Master Answer**: **Option (A)**.
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