This set of Electromagnetic Theory Multiple Choice Questions & Answers (MCQs) focuses on “Volume Integral”.

1. The divergence theorem converts

a) Line to surface integral

b) Surface to volume integral

c) Volume to line integral

d) Surface to line integral

View Answer

Explanation: The divergence theorem is given by, ∫∫ D.ds = ∫∫∫ Div (D) dv. It is clear that it converts surface (double) integral to volume(triple) integral.

2. The triple integral is used to compute volume. State True/False

a) True

b) False

View Answer

Explanation: The triple integral, as the name suggests integrates the function/quantity three times. This gives volume which is the product of three independent quantities.

3. The volume integral is three dimensional. State True/False

a) True

b) False

View Answer

Explanation: Volume integral integrates the independent quantities by three times. Thus it is said to be three dimensional integral or triple integral.

4. Find the charged enclosed by a sphere of charge density ρ and radius a.

a) ρ (4πa^{2})

b) ρ(4πa^{3}/3)

c) ρ(2πa^{2})

d) ρ(2πa^{3}/3)

View Answer

Explanation: The charge enclosed by the sphere is Q = ∫∫∫ ρ dv.

Where, dv = r

^{2}sin θ dr dθ dφ and on integrating with r = 0->a, φ = 0->2π and θ = 0->π, we get Q = ρ(4πa

^{3}/3).

5. Evaluate Gauss law for D = 5r^{2}/4 i in spherical coordinates with r = 4m and θ = π/2 as volume integral.

a) 600

b) 588.9

c) 577.8

d) 599.7

View Answer

Explanation: ∫∫ D.ds = ∫∫∫ Div (D) dv, where RHS needs to be computed.

The divergence of D given is, Div(D) = 5r and dv = r

^{2}sin θ dr dθ dφ. On integrating, r = 0->4, φ = 0->2π and θ = 0->π/4, we get Q = 588.9.

6. Compute divergence theorem for D = 5r^{2}/4 i in spherical coordinates between r = 1 and r = 2 in volume integral.

a) 80 π

b) 5 π

c) 75 π

d) 85 π

View Answer

Explanation: D.ds = ∫∫∫ Div (D) dv, where RHS needs to be computed.

The divergence of D given is, Div(D) = 5r and dv = r

^{2}sin θ dr dθ dφ. On integrating, r = 1->2, φ = 0->2π and θ = 0->π, we get Q = 75 π.

7. Compute the Gauss law for D = 10ρ^{3}/4 i, in cylindrical coordinates with ρ = 4m, z = 0 and z = 5, hence find charge using volume integral.

a) 6100 π

b) 6200 π

c) 6300 π

d) 6400 π

View Answer

Explanation: Q = D.ds = ∫∫∫ Div (D) dv, where RHS needs to be computed.

The divergence of D given is, Div(D) = 10 ρ

^{2}and dv = ρ dρ dφ dz. On integrating, ρ = 0->4, φ = 0->2π and z = 0->5, we get Q = 6400 π.

8. Using volume integral, which quantity can be calculated?

a) area of cube

b) area of cuboid

c) volume of cube

d) distance of vector

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Explanation: The volume integral gives the volume of a vector in a region. Thus volume of a cube can be computed.

9. Compute the charge enclosed by a cube of 2m each edge centered at the origin and with the edges parallel to the axes. Given D = 10y^{3}/3 j.

a) 20

b) 70/3

c) 80/3

d) 30

View Answer

Explanation: Div(D) = 10y

^{2}

∫∫∫Div (D) dv = ∫∫∫ 10y

^{2}dx dy dz. On integrating, x = -1->1, y = -1->1 and z = -1->1, we get Q = 80/3.

10. Find the value of divergence theorem for the field D = 2xy i + x^{2} j for the rectangular parallelepiped given by x = 0 and 1, y = 0 and 2, z = 0 and 3.

a) 10

b) 12

c) 14

d) 16

View Answer

Explanation: Div (D) = 2y

∫∫∫Div (D) dv = ∫∫∫ 2y dx dy dz. On integrating, x = 0->1, y = 0->2 and z = 0->3, we get Q = 12.

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

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