GATE Electrical Engineering (EE) 2020 Solved Paper

Question : 48 of 65

Marks: +1, -0

The vector function represented by

F=a_{x}(5 y-k_{1} z)+a_{y}(3 z+k_{2} x)+a_{z}(k_{3} y-4 x)

represents a conservative field, where

a_{x}, a_{y} \& a_{z}

represents unit vectors along the

x, y, \&

z

-axis respectively. Find the values of

k_{1}, k_{2} \& k_{3}=?

Solution:

For a conservative field,

∮ {f}↖{→} ⋅ {d}↖{→} ℓ=0

∇ × {f}↖{→}=0

|\table \;\text " âx "\; , \;\text " ây "\; , \;\text " âz "\; ; \;{d}/{d x} , \;{d}/{d y} , \;{d}/{d z} ; (5 y-K_{1} z) , (3 z+K_{2} x) , (K_{3} y-4 x)|

={a}↖{∧}_{x}(K_{3}-3)-{a}↖{∧} y(-4+K_{1})+a ̂ z(K_{2}-5)

=0

⇒ {K}_{3}=3, {K}_{1}=4, {K}_{2}=5

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