Evaluate Gauss law for D = 5r2 /4 i in spherical coordinates with r = 4m and θ = π/2

To evaluate Gauss’s Law using the provided electric flux density (D = frac{5r^2}{4} hat{i}) in spherical coordinates for (r = 4 ,m) and (theta = frac{pi}{2}), let’s first clarify the given parameters and the application of Gauss’s Law within this context.

First, it’s important to note that the electric flux density (mathbf{D}) given as (D = frac{5r^2}{4} hat{i}) seems to be expressed in a form that mixes variables from Cartesian coordinates ((hat{i}) is a unit vector in the direction of the x-axis in Cartesian coordinates) with those of spherical coordinates ((r) is the radial distance in spherical coordinates). For a more consistent application related to Gauss’s Law in spherical coordinates, it’s essential to correctly interpret or transform this expression into spherical coordinates fully. However, assuming the intent was to express (mathbf{D}) in a purely spherical context and aiming to perform an evaluation as requested, one needs to correct the misunderstanding about the vector notations and how Gauss’s Law applies.

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Given the context, let’s reframe the statement into a scenario Gauss’s Law could be applied to directly:

If by (D = frac{5r^2}{4}) you meant the magnitude of the electric flux density in spherical coordinates without specifying a direction, and considering (