If a vector field $$ vec{V}$$ is related to another vector field $$ vec{A}$$ through $$ vec{V} = abla imes vec{A}$$. Which of the following is true?Note: C and S refer to any closed contour and any surface whose boundary is C.(GATE 2009 || EC || MCQ ||2 MARK)

$$ oint_{c} vec{A} cdot d vec{l} = iint_{s_e} vec{V} cdot d vec{S}$$

$$ oint_{c} vec{V} cdot d vec{l} = iint_{s_e} vec{A} cdot d vec{S}$$

$$ oint_{c} ( abla imes vec{V}) cdot d vec{l} = iint_{s_e} ( abla imes vec{A}) cdot d vec{S}$$

$$ oint_{c} ( abla imes vec{A}) cdot d vec{l} = iint_{s_e} vec{V} cdot d vec{S}$$

A vector $$ overrightarrow{ m P}$$ is given by$$ overrightarrow{ m P} = x^3 y overrightarrow{ m a}_x - x^2 y^2 overrightarrow{ m a}_y - x^2 y z overrightarrow{ m a}_z$$Which of the following statements is TRUE?(GATE 2015|| EC || MCQ ||2 MARK)

P is solenoidal but not irrotational

$$ overrightarrow{ m P}$$ is irrotational but not solenoidal

$$ overrightarrow{ m P}$$ is neither solenoidal nor irrotational

$$ overrightarrow{ m P}$$ is both solenoidal and irrotational

For a vector field $$ vec{A}$$ , which one of the following is FALSE?(GATE 2020|| EC || MCQ ||1 MARK)

$$ vec{A} ext{ is irrotational if } abla^2 vec{A} = 0$$

$$ vec{A} ext{ is solenoidal if } abla cdot vec{A} = 0$$

$$ abla imes vec{A} ext{ is also another vector field}$$

$$ abla imes ( abla imes vec{A}) = abla ( abla cdot vec{A}) - abla^2 vec{A}$$

GATE EC||ELECTROMAGNECTIC||VECTOR||PYQS(2000-2025)

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Question 1

If a vector field [Tex]\vec{V}[/Tex] is related to another vector field [Tex]\vec{A}[/Tex] through [Tex]\vec{V} = \nabla \times \vec{A}[/Tex]. Which of the following is true?

Note: C and S refer to any closed contour and any surface whose boundary is C.

(GATE 2009 || EC || MCQ ||2 MARK)

[Tex]\oint_{c} \vec{V} \cdot d\vec{l} = \iint_{s_e} \vec{A} \cdot d\vec{S}[/Tex]
[Tex]\oint_{c} \vec{A} \cdot d\vec{l} = \iint_{s_e} \vec{V} \cdot d\vec{S}[/Tex]
[Tex]\oint_{c} (\nabla \times \vec{V}) \cdot d\vec{l} = \iint_{s_e} (\nabla \times \vec{A}) \cdot d\vec{S}[/Tex]
[Tex]\oint_{c} (\nabla \times \vec{A}) \cdot d\vec{l} = \iint_{s_e} \vec{V} \cdot d\vec{S}[/Tex]

Question 2

Consider a closed surface S surrounding a volume V. If [Tex]\overrightarrow{\rm r}[/Tex] is the position vector of a point inside S, with [Tex]\hat{n}[/Tex] the unit normal on S, the value of the integral [Tex]\iint_{S} 5\overrightarrow{\rm \vec{r}} \cdot \hat{n} \, dS[/Tex] is

(GATE 2011 || EC || MCQ ||2 MARK)

3 V
5 V
10 V
15 V

Question 3

The direction of vector [Tex]\vec{A}[/Tex] is radially outward from the origin, with [Tex]|\vec{A}| = kr^n[/Tex] where [Tex]r^2 = x^2 + y^2 + z^2[/Tex] and k is a constant. The value of $n$ for which [Tex]\nabla \cdot \vec{A} = 0[/Tex] is

(GATE 2012|| EC || MCQ ||1 MARK)

-2
2
1
0

Question 4

A vector [Tex]\overrightarrow{\rm P}[/Tex] is given by

[Tex]\overrightarrow{\rm P} = x^3 y\overrightarrow{\rm a}_x - x^2 y^2 \overrightarrow{\rm a}_y - x^2 y z \overrightarrow{\rm a}_z[/Tex]

Which of the following statements is TRUE?

(GATE 2015|| EC || MCQ ||2 MARK)

P is solenoidal but not irrotational
[Tex]\overrightarrow{\rm P}[/Tex] is irrotational but not solenoidal
[Tex]\overrightarrow{\rm P}[/Tex] is neither solenoidal nor irrotational
[Tex]\overrightarrow{\rm P}[/Tex] is both solenoidal and irrotational

Question 5

(GATE 2015|| EC || NAT ||2 MARK)

78.53

Question 6

If the vector function [Tex]\vec{F} = (3y - k_1 z)\vec{a}_x + (k_2 x - 2z)\vec{a}_y - (k_3 y + z)\vec{a}_z[/Tex] is irrotational, then the values of the constants [Tex]k_1, k_2[/Tex] and [Tex]k_3[/Tex] respectively, are

(GATE 2017|| EC || MCQ ||2 MARK)

0.3, –2.5, 0.5
0.0, 3.0, 2.0
0.3, 0.33, 0.5
4.0, 3.0, 2.0

Question 7

For a vector field [Tex]\vec{A}[/Tex] , which one of the following is FALSE?

(GATE 2020|| EC || MCQ ||1 MARK)

[Tex]\vec{A} \text{ is solenoidal if } \nabla \cdot \vec{A} = 0[/Tex]
[Tex]\nabla \times \vec{A} \text{ is also another vector field}[/Tex]
[Tex]\vec{A} \text{ is irrotational if } \nabla^2 \vec{A} = 0[/Tex]
[Tex]\nabla \times (\nabla \times \vec{A}) = \nabla (\nabla \cdot \vec{A}) - \nabla^2 \vec{A}[/Tex]

Question 8

(GATE 2021|| EC || NAT ||1 MARK)

56.55

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