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QUESTION: 1

The divergence theorem converts

Solution:

Answer: b

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.

QUESTION: 2

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

Solution:

Answer: a

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.

QUESTION: 3

The volume integral is three dimensional. State True/False

Solution:

Answer: a

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

QUESTION: 4

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

Solution:

Answer: b

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).

QUESTION: 5

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

Solution:

Answer: b

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.

QUESTION: 6

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

Solution:

Answer: c

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.

QUESTION: 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.

Solution:

Answer: d

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.

QUESTION: 8

Using volume integral, which quantity can be calculated?

Solution:

Answer: c

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

QUESTION: 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.

Solution:

Answer: c

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.

QUESTION: 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.

Solution:

Answer: b

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.

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