Test: Stokes Theorem - Civil Engineering (CE) MCQ

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.

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.

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 = 

∴ 

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.

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.

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.

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.

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.

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

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.

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.

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

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

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.