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Test: Stokes Theorem - Civil Engineering (CE) MCQ


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15 Questions MCQ Test Engineering Mathematics - Test: Stokes Theorem

Test: Stokes Theorem for Civil Engineering (CE) 2024 is part of Engineering Mathematics preparation. The Test: Stokes Theorem questions and answers have been prepared according to the Civil Engineering (CE) exam syllabus.The Test: Stokes Theorem MCQs are made for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Stokes Theorem below.
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Test: Stokes Theorem - Question 1

Stokes theorem is used to convert __________ into _________.

Detailed Solution for Test: Stokes Theorem - Question 1

Stokes theorem:

(i) Stoke's theorem enables us to transform the surface integral of the curl of the vector field A into the line integral of that vector field A over the boundary C of that surface and vice-versa. The theorem states.

(ii) The flux of the curl of a vector function A over any surface S of any shape is equal to the line integral of the vector field A over the boundary C of that surface i.e.

Stokes Theorem is given as:

It converts a line integral to a surface integral and uses the curl operation.

Test: Stokes Theorem - Question 2

Which of the following is related with Stoke's Theorem?

Detailed Solution for Test: Stokes Theorem - Question 2

Stokes theorem:

(i) Stoke's theorem enables us to transform the surface integral of the curl of the vector field A into the line integral of that vector field A over the boundary C of that surface and vice-versa. The theorem states.

(ii) The flux of the curl of a vector function A over any surface S of any shape is equal to the line integral of the vector field A over the boundary C of that surface i.e.

Stokes Theorem is given as:

It converts a line integral to a surface integral and uses the curl operation.

Hence Stokes theorem uses the curl operation.

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Test: Stokes Theorem - Question 3

By applying Stokes theorem, the value of   where C is the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0) and (0, 0, 6), is

Detailed Solution for Test: Stokes Theorem - Question 3

Concept:

By stokes theorem:

Calculation:

Given:

Equation of the plane through A, B, C is

∴ 3x + 2y + z = 6

∴ ϕ = 3x + 2y + z - 6 represents equation of the surface.

Vector normal to the surface is given by gradient:

Unit vector n̂: 

A (2, 0, 0), B (0, 3, 0) and C (0, 0, 6)

Area of ΔABC = magnitude of 1/2 x 

∴ 

∴ 

∴ 

Area = 

∴ 

Test: Stokes Theorem - Question 4

Consider a vector field . The closed loop line integral can be expressed as

Detailed Solution for Test: Stokes Theorem - Question 4

Stoke’s theorem: The line integral of a vector  around closed path L is equal to the integral of curl over the open surface is enclosed by the closed path L.

Test: Stokes Theorem - Question 5

Stokes' theorem is valid irrespective of 

1. Shape of closed curve C

2. Type of vector A

3. Type of coordinate system

4. Whether the surface is closed or open

Which of the above statements are correct?

Detailed Solution for Test: Stokes Theorem - Question 5

Concept:

Stokes’s theorem:

: It states that the circulation of a vector field  around a (closed) path L is equal to the surface integral of the curl of  over the open surface, S bounded by L (fig).

 provided  and

∇ ×  are continuous on S.

Fig: Determining the sense of   and  involved in Stokes’s theorem

The curl of  is an axial (or rotational) vector whose magnitude is the maximum circulation of  per unit area as the area tends to zero and whose direction is the normal direction of the area is oriented to make the circulation maximum.

That is the curl of the vector defined as:

With the stokes theorem, we can convert closed line integral into surface integral.

Stokes’s theorem does not apply to the closed surface.

Test: Stokes Theorem - Question 6

Find the value of Stoke’s theorem for y i + z j + x k.

Detailed Solution for Test: Stokes Theorem - Question 6

The curl of y i + z j + x k is i(0 - 1) – j(1 - 0) + k(0 - 1) =
-i –j –k. Since the curl is zero, the value of Stoke’s theorem is zero.

The function is said to be irrotational.

Test: Stokes Theorem - Question 7

Which of the following theorem convert line integral to surface integral?

Detailed Solution for Test: Stokes Theorem - Question 7

The Stoke’s theorem is given by ∫A.dl = ∫∫ Curl (A..ds. Green’s theorem is given by, ∫ F dx + G dy = ∫∫ (dG/dx – dF/dy) dx dy. It is clear that both the theorems convert line to surface integral.

Test: Stokes Theorem - Question 8

The Stoke’s theorem can be used to find which of the following?

Detailed Solution for Test: Stokes Theorem - Question 8

It states that the line integral of a function gives the surface area of the function enclosed by the given region. This is computed using the double integral of the curl of the function.

Test: Stokes Theorem - Question 9

The conductivity of a material with current density 1 unit and electric field 200 μV is

Detailed Solution for Test: Stokes Theorem - Question 9

The current density is given by, J = σE. To find conductivity, σ = J/E = 1/200 X 10-6 = 5000.

Test: Stokes Theorem - Question 10

The voltage of a capacitor 12F with a rating of 2J energy is

Detailed Solution for Test: Stokes Theorem - Question 10

We can compute the energy stored in a capacitor from Stoke’s theorem as 0.5Cv2. Thus given energy is 0.5 X 12 X v2.

We get v = 0.57 volts.

Test: Stokes Theorem - Question 11

Which among the following theorems uses the curl operation?

Detailed Solution for Test: Stokes Theorem - Question 11

Concept:

1. The curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional space.

2. The curl of a scalar field is undefined. It is defined only for 3D vector fields.

3. For a vector F = F1i + F2j + F3k

Stokes theorem:

(i) Stoke's theorem enables us to transform the surface integral of the curl of the vector field A into the line integral of that vector field A over the boundary C of that surface and vice-versa. The theorem states.

(ii) The flux of the curl of a vector function A over any surface S of any shape is equal to the line integral of the vector field A over the boundary C of that surface i.e.

Stokes Theorem is given as:

It converts a line integral to a surface integral and uses the curl operation.

Hence Stokes theorem uses the curl operation.

So option (2) is the correct answer.

Test: Stokes Theorem - Question 12

 The Stoke’s theorem uses which of the following operation?

Detailed Solution for Test: Stokes Theorem - Question 12

∫A.dl = ∫∫ Curl (A..ds is the expression for Stoke’s theorem. It is clear that the theorem uses curl operation.

Test: Stokes Theorem - Question 13

Find the value of Stoke’s theorem for A = x i + y j + z k. The state of the function will be

Test: Stokes Theorem - Question 14

The energy stored in an inductor 2H and current 4A is

Detailed Solution for Test: Stokes Theorem - Question 14

From Stoke’s theorem, we can calculate energy stored in an inductor as 0.5Li2. E = 0.5 X 2 X 42 
= 16 units.

Test: Stokes Theorem - Question 15

Stoke's theorem is primarily concerned with the relationship between:

Detailed Solution for Test: Stokes Theorem - Question 15

Stoke's theorem establishes a connection between surface integrals and line integrals. It states that the surface integral of the curl of a vector field over a surface is equal to the line integral of the vector field around the closed curve bounding the surface, and vice versa. Therefore, the correct option is C, which corresponds to the relationship between surface integrals and volume integrals.

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