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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 = r2 sin θ dr dθ dφ. On integrating, r =1->2, φ = 0->2π and θ = 0->π, we get Q = 75 π.
To compute the divergence theorem for a vector field ( mathbf{D} = frac{5}{4} r^2 mathbf{i} ) in spherical coordinates, we follow these steps:
1. Convert the vector field to spherical coordinates:
In spherical coordinates, the relationship between Cartesian and spherical coordinates is given by:
– ( x = r sin theta cos phi )
– ( y = r sin theta sin phi )
– ( z = r cos theta )
The unit vector ( mathbf{i} ) corresponds to ( sin theta cos phi ), thus:
[
mathbf{D} = frac{5}{4} r^2 hat{r}
]
in spherical coordinates, as ( hat{r} ) is the radial unit vector.
2. Compute the divergence of ( mathbf{D} ):
The divergence in spherical coordinates is given by:
[
nabla cdot mathbf{D} = frac{1}{r^2} frac{partial}{partial r}(r^2 D_r) + frac{1}{r sin theta} frac{partial}{partial theta}(D_theta sin theta) + frac{1}{r} frac{partial D_phi}{