GATE 2020 EE Q.27: solution and answer

GATE 2020 · EEQ.27 · 2 marks · MCQ

The vector function expressed 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 $a_{x}, a_{y}, a_{z}$ are unit vectors along x, y and z directions, respectively. The values of constants $k_{1}, k_{2}, k_{3}$ are given by:

(A)
$k_{1} = 3, k_{2} = 3, k_{3} = 7$
(B)
$k_{1} = 3, k_{2} = 8, k_{3} = 5$
(C)
$k_{1} = 4, k_{2} = 5, k_{3} = 3$
(D)
$k_{1} = 0, k_{2} = 0, k_{3} = 0$

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Answer and solution

ANSWER

$k_{1} = 4, k_{2} = 5, k_{3} = 3$ — option (C).

Official answer: CSource: Official answer key, GATE 2020 EE (IIT Delhi)

Conservative field condition: on all of space, $F$ is conservative exactly when $\nabla \times F = 0$ . Components: $F_{x} = 5 y - k_{1} z$ , $F_{y} = 3 z + k_{2} x$ , $F_{z} = k_{3} y - 4 x$ .
$x$ -component of the curl: $\frac{\partial F _{z}}{\partial y} - \frac{\partial F _{y}}{\partial z} = k_{3} - 3 = 0 \Rightarrow k_{3} = 3$ .
$y$ -component of the curl: $\frac{\partial F _{x}}{\partial z} - \frac{\partial F _{z}}{\partial x} = - k_{1} - (- 4) = 4 - k_{1} = 0 \Rightarrow k_{1} = 4$ .
$z$ -component of the curl: $\frac{\partial F _{y}}{\partial x} - \frac{\partial F _{x}}{\partial y} = k_{2} - 5 = 0 \Rightarrow k_{2} = 5$ .
Check with a potential: $ϕ = 5 x y - 4 x z + 3 y z$ gives $\nabla ϕ = (5 y - 4 z) a_{x} + (5 x + 3 z) a_{y} + (3 y - 4 x) a_{z}$ , which is $F$ with $k_{1} = 4, k_{2} = 5, k_{3} = 3$ .
So $k_{1} = 4, k_{2} = 5, k_{3} = 3$ , which is option (C).

KEY IDEA

A field defined on all of space is conservative exactly when its curl is zero, so it is the gradient of a scalar potential. Set each of the three curl components to zero. Each one fixes one constant: the $x$ -component gives $k_{3}$ , the $y$ -component gives $k_{1}$ and the $z$ -component gives $k_{2}$ . Then check by building the potential $ϕ = 5 x y - 4 x z + 3 y z$ .

WATCH OUT FOR

The usual slip is the sign or order in the $y$ -component of the curl. It is $\partial F_{x} / \partial z - \partial F_{z} / \partial x$ , and reversing it gives $k_{1} = - 4$ , which matches no option. The other mistake is testing divergence (solenoidal) instead of curl. Here $\nabla \cdot F = 0$ for every choice of $k_{1}, k_{2}, k_{3}$ , because no component depends on its own coordinate, so a divergence test says nothing. Option (D) is a trap: with every $k = 0$ the curl is $(- 3, 4, - 5) \neq = 0$ .

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GATE 2020 EE Q.27: solution and answer

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