Question

* The following surface integral is to be evaluated over a sphere for the given steady velocity vector field $F=x\hat{i}+y\hat{j}+z\hat{k}$ defined with respect to a Cartesian coordinate system having i, j and k as unit base vectors.
${\iint }_s\frac{1}{4}(F.n)dA$
Where S is the sphere, $x^2+y^2+z^2=1$ and n is the outward unit normal vector to the sphere. The value of the surface integral is

• A. $\pi$
• B. 2$\pi$
• C. 3$\pi$/4
• D. 4$\pi$

Answer 1

The correct option is A $\pi$

$\overrightarrow{F}=x\hat{i}+y\hat{j}+z\hat{k}$
$\overrightarrow{\nabla }.\overrightarrow{F}=3$
By Guass Divergence Theorem
$\frac{1}{4}{\iint }_s\left(\overrightarrow{F.\hat{n}}\right)dA=\frac{1}{4}{\iiint }_{v}3dv=\frac{3}{4}\left(V\right)$
$=\frac{3}{4}\left(\frac{4}{3}\pi {\left(1\right)}^3\right)=\pi$