Compute the Gauss law for D = 10ρ3 /4 i, in cylindrical coordinates with ρ = 4m, z = 0 and z = 5, hence find charge using volume integral

To compute Gauss’s law for a given electric displacement field ( mathbf{D} ) and to find the charge enclosed using a volume integral, we work step by step through the problem. Gauss’s law in differential form relates the divergence of the electric displacement field ( mathbf{D} ) to the free charge density ( rho_{free} ) present in the medium:

[

nabla cdot mathbf{D} = rho_{free}

]

Given:

[

mathbf{D} = frac{10rho^3}{4} hat{mathbf{i}}

]

This is given in cylindrical coordinates ((rho, phi, z)) but with a slight confusion in the notation since (hat{mathbf{i}}) is typically used for Cartesian coordinates. Assuming it’s meant to represent the radial component in cylindrical coordinates, it should correctly be (hat{rho}) instead of (hat{mathbf{i}}), so:

[

mathbf{D} = frac{10rho^3}{4} hat{rho}

]

To compute the charge enclosed within a cylindrical volume defined by ( rho = 4m ), between ( z = 0 ) and ( z = 5 ), we first compute the volume integral of the charge density ( rho_{free} ).

Since ( nabla