Question Bank • QB_EM_148 NAT • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Stokes, Gauss and Green's theorems

Question Statement:

Using Gauss divergence theorem, calculate the outward flux of the vector field $\vec{F} = x\hat{i} + y\hat{j} + z\hat{k}$ through the closed surface of a sphere of radius $R = 2$ centered at the origin (take $\pi = 3.1416$; round off to 1 decimal place).

Answer Key:

100.5

Step-by-Step Solution & Explanation

By Gauss divergence theorem: $\iint_S \vec{F} \cdot \hat{n}\, dS = \iiint_V (\nabla \cdot \vec{F})\, dV$ Here $\nabla \cdot \vec{F} = 1 + 1 + 1 = 3$. Volume of sphere of radius $R = 2$: $V = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi (8) = \frac{32}{3}\pi$ Outward flux: $\text{Flux} = 3 \times V = 3 \times \frac{32}{3}\pi = 32\pi = 32 \times 3.14159 \approx 100.53$ Acceptable range: $99.0 - 102.0$.
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