A vector field is defined as \overrightarrow{ {f}( {x}, {y}, {z})= { {x}}/{[ {x}^{2}+ {y}^{2}+ {z}^{2}]^{\frac{3}{2}}} \hat{ {i}+ { {y}}/{[ {x}^{2}+ {y}^{2}+ {z}^{2}]^{\frac{3}{2}}} \hat{ {j}+ { {z}}/{[ {x}^{2}+ {y}^{2}+ {z}^{2}]^{\frac{3}{2}}} \hat{ {k} where, \hat{ {i}, \hat{ {j}, \hat{ {k} are unit vectors along the axes of a right-handed rectangular/Cartesian coordinate system. The surface integral \iint \overrightarrow{f} . d \overrightarrow{S} (where d \overrightarrow{S} is an elemental surface area vector) evaluated over the inner and outer surfaces of a spherical shell formed by two concentric spheres with origin as the center, and internal and external radii of 1 and 2 , respectively, is

GATE Mechanical Engineering (ME) 2020 Shift 1 Solved Paper

Question : 48 of 65

Marks: +1, -0

A vector field is defined as

{ {f}↖{→}( {x}, {y}, {z})=\;{ {x}}/{[ {x}^{2}+ {y}^{2}+ {z}^{2}]^{\;{3}/{2}}} { {i}↖{∧}+\;{ {y}}/{[ {x}^{2}+ {y}^{2}+ {z}^{2}]^{\;{3}/{2}}} { {j}↖{∧}+\;{ {z}}/{[ {x}^{2}+ {y}^{2}+ {z}^{2}]^{\;{3}/{2}}} { {k}↖{∧}

where,

{ {i}↖{∧}, { {j}↖{∧}, { {k}↖{∧}

are unit vectors along the axes of a right-handed rectangular/Cartesian coordinate system. The surface integral

\iint {f}↖{→} . d {S}↖{→}

(where

d {S}↖{→}

is an elemental surface area vector) evaluated over the inner and outer surfaces of a spherical shell formed by two concentric spheres with origin as the center, and internal and external radii of 1 and 2 , respectively, is

$2 π$
$0$
$4 π$
$8 π$

Validate

Solution:

div F=\;{(x^{2}+y^{2}+z^{2})^{{1}/{2}}}/{(x^{2}+y^{2}+z^{2})^{3}}[\table -2 x^{2}+y^{2}+z^{2}+x^{2}-2 y^{2}+z^{2} ; +x^{2}+y^{2}-2 z^{2}]

=0

∫∫ {F}↖{→} ⋅ d {s}↖{→}=∫∫∫ div F d v=0

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