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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.
The triple integral is used to compute volume. State True/False
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.
Answer: a
Explanation: Volume integral integrates the independent quantities by three times. Thus it is said to be three dimensional integral or triple integral.
Find the charged enclosed by a sphere of charge density ρ and radius a.
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).
Evaluate Gauss law for D = 5r^{2}/2 i in spherical coordinates with r = 4m and θ = π/2 as volume integral.
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.
Compute divergence theorem for D = 5r^{2}/4 i in spherical coordinates between r = 1 and r = 2 in volume integral.
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.
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.
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.
Answer: c
Explanation: The volume integral gives the volume of a vector in a region. Thus volume of a cube can be computed.
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.
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.
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.
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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