Question
Download Solution PDFIf \({\rm{\vec F}} = {\rm{x\;\vec i}} + {\rm{y\;\vec j}} + {\rm{z\;\vec k}}\) and s is the closed surface of x2 + y2 + z2 = a2. Then \(\mathop \int\!\!\!\int \nolimits_{\rm{s}}^{} {\rm{\vec F}} \cdot {\rm{\hat n\;ds}}\) is
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
According to Gauss Divergence theorem:
\(\oint A.ds=\iiint{\left( \nabla .A \right)dv}\)
\(\oint \overrightarrow{F.}\hat{n}ds=\iiint{\left( \nabla .F \right)dv}\)
Calculation:
Given:
\({\rm{\vec F}} = {\rm{x\;\vec i}} + {\rm{y\;\vec j}} + {\rm{z\;\vec k}}\)
\(\nabla {\rm{\;}}.{\rm{\vec F\;}} = \frac{{\partial \left( {\rm{x}} \right)}}{{\partial {\rm{x}}}} + \frac{{\partial \left( {\rm{y}} \right)}}{{\partial {\rm{y}}}} + \frac{{\partial \left( {\rm{z}} \right)}}{{\partial {\rm{z}}}} = 3\)
x2 + y2 + z2 = a2
∴ closed surface is a sphere of radius r
∴ \(\iiint{3\text{ }\!\!~\!\!\text{ dv}=3\text{V}=3\times \frac{4}{3}\text{ }\!\!\pi\!\!\text{ }{{\text{a}}^{3}}=4\text{ }\!\!\pi\!\!\text{ }{{\text{a}}^{3}}}\)Last updated on Jul 9, 2025
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