Question

Let \( \vec{F} = x\hat{i} + y\hat{j} + z\hat{k} \) and \( S \) be the sphere given by \( (x - 2)^2 + (y - 2)^2 + (z - 2)^2 = 4. \) If \( \hat{n} \) is the unit outward normal to \( S \), then \( \dfrac{1}{\pi} \iint_S \vec{F} \cdot \hat{n} \, dS \) is .............

Hint:

When you see a flux integral over a closed surface, try applying the Divergence Theorem immediately.

Answer 1

Correct Answer: 32

Step 1: Use the Divergence Theorem.
\[ \iint_S \vec{F} \cdot \hat{n} \, dS = \iiint_V \nabla \cdot \vec{F} \, dV. \] Since \( \vec{F} = x\hat{i} + y\hat{j} + z\hat{k} \), \[ \nabla \cdot \vec{F} = 3. \]

Step 2: Find volume of the sphere.
Sphere radius \( r = 2 \), so \[ V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (2)^3 = \frac{32\pi}{3}. \]

Step 3: Compute flux.
\[ \iint_S \vec{F} \cdot \hat{n} \, dS = 3V = 3 \times \frac{32\pi}{3} = 32\pi. \]

Step 4: Simplify.
\[ \frac{1}{\pi} \iint_S \vec{F} \cdot \hat{n} \, dS = \frac{32\pi}{\pi} = 32. \]

Final Answer: \[ \boxed{32} \]