This set of Electromagnetic Theory Multiple Choice Questions & Answers (MCQs) focuses on “Green’s Theorem”.

1. Mathematically, the functions in Green’s theorem will be

a) Continuous derivatives

b) Discrete derivatives

c) Continuous partial derivatives

d) Discrete partial derivatives

View Answer

Explanation: The Green’s theorem states that if L and M are functions of (x,y) in an open region containing D and having continuous partial derivatives then,

∫ (F dx + G dy) = ∫∫(dG/dx – dF/dy)dx dy, with path taken anticlockwise.

2. Find the value of Green’s theorem for F = x^{2} and G = y^{2} is

a) 0

b) 1

c) 2

d) 3

View Answer

Explanation: ∫∫(dG/dx – dF/dy)dx dy = ∫∫(0 – 0)dx dy = 0. The value of Green’s theorem gives zero for the functions given.

3. Which of the following is not an application of Green’s theorem?

a) Solving two dimensional flow integrals

b) Area surveying

c) Volume of plane figures

d) Centroid of plane figures

View Answer

Explanation: In physics, Green’s theorem is used to find the two dimensional flow integrals. In plane geometry, it is used to find the area and centroid of plane figures.

4. The path traversal in calculating the Green’s theorem is

a) Clockwise

b) Anticlockwise

c) Inwards

d) Outwards

View Answer

Explanation: The Green’s theorem calculates the area traversed by the functions in the region in the anticlockwise direction. This converts the line integral to surface integral.

5. Calculate the Green’s value for the functions F = y^{2} and G = x^{2} for the region x = 1 and y = 2 from origin.

a) 0

b) 2

c) -2

d) 1

View Answer

Explanation: ∫∫(dG/dx – dF/dy)dx dy = ∫∫(2x – 2y)dx dy. On integrating for x = 0->1 and y = 0->2, we get Green’s value as -2.

6. If two functions A and B are discrete, their Green’s value for a region of circle of radius a in the positive quadrant is

a) ∞

b) -∞

c) 0

d) Does not exist

View Answer

Explanation: Green’s theorem is valid only for continuous functions. Since the given functions are discrete, the theorem is invalid or does not exist.

7. Applications of Green’s theorem are meant to be in

a) One dimensional

b) Two dimensional

c) Three dimensional

d) Four dimensional

View Answer

Explanation: Since Green’s theorem converts line integral to surface integral, we get the value as two dimensional. In other words the functions are variable with respect to x,y, which is two dimensional.

8. The Green’s theorem can be related to which of the following theorems mathematically?

a) Gauss divergence theorem

b) Stoke’s theorem

c) Euler’s theorem

d) Leibnitz’s theorem

View Answer

Explanation: The Green’s theorem is a special case of the Kelvin- Stokes theorem, when applied to a region in the x-y plane. It is a widely used theorem in mathematics and physics.

9. The Shoelace formula is a shortcut for the Green’s theorem. State True/False.

a) True

b) False

View Answer

Explanation: The Shoelace theorem is used to find the area of polygon using cross multiples. This can be verified by dividing the polygon into triangles. It is a special case of Green’s theorem.

10. Find the area of a right angled triangle with sides of 90 degree unit and the functions described by L = cos y and M = sin x.

a) 0

b) 45

c) 90

d) 180

View Answer

Explanation: dM/dx = cos x and dL/dy = -sin y

∫∫(dM/dx – dL/dy)dx dy = ∫∫ (cos x + sin y)dx dy. On integrating with x = 0->90 and y = 0->90, we get area of right angled triangle as -180 units (taken in clockwise direction). Since area cannot be negative, we take 180 units.

**Sanfoundry Global Education & Learning Series – Electromagnetic Theory.**

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