Use Gauss' divergence theorem. For F = 3x i + 2y j, ∇·F = 3 + 2 = 5. The sphere x²+y²+z²=9 has radius 3 and volume 36π, so the outward flux is 5×36π = 180π.
Formula / Key Relation
∬_S F·dS = ∭_V (∇·F)dV
∇·F = 3 + 2 = 5
Vsphere = (4/3)π(3³) = 36π
Flux = 5×36π = 180π
Detailed Explanation
Concept & ReasoningA closed-surface flux integral is usually faster through the divergence theorem than by parametrizing the sphere. The divergence of 3x i + 2y j is constant:∂(3x)/∂x + ∂(2y)/∂y + ∂(0)/∂z = 5.The sphere radius is 3, so its volume is (4/3)π(3³)=36π. Integrating constant divergence 5 over that volume gives 180π. This is the net outward flux through the entire spherical surface.How to Solve It in the ExamClosed surface + simple divergence → use divergence theorem immediately.Important Exam PointConfirm the surface is closed before replacing a surface integral with a volume integral.Related Revision PathThis question sits in the revision…
Concept & Reasoning
A closed-surface flux integral is usually faster through the divergence theorem than by parametrizing the sphere. The divergence of 3x i + 2y j is constant: ∂(3x)/∂x + ∂(2y)/∂y + ∂(0)/∂z = 5. The sphere radius is 3, so its volume is (4/3)π(3³)=36π. Integrating constant divergence 5 over that volume gives 180π. This is the net outward flux through the entire spherical surface.
How to Solve It in the Exam
Closed surface + simple divergence → use divergence theorem immediately.
Important Exam Point
Confirm the surface is closed before replacing a surface integral with a volume integral.
Related Revision Path
This question sits in the revision path Electromagnetic Fields → Vector Calculus → Divergence Theorem .
Quick Trick
Closed surface + simple divergence → use divergence theorem immediately.
Common Mistake
Using surface area 4πr² instead of sphere volume is the common mistake in divergence-theorem questions.
Exam Tip
Confirm the surface is closed before replacing a surface integral with a volume integral.
SEO + Knowledge GraphRelated PYQs & Exam ContextExam path, knowledge path and deeper practice links.
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