Final answer:
To calculate the D-field inside a cylindrical region with a given volume charge density using Gauss's law, one must integrate the charge density over the cylinder's volume and apply the law to a cylindrical Gaussian surface.
Step-by-step explanation:
To find the D-field as a function of ρ (rho) in the region ρ < 3 for a given volume charge density ρᵥ(ρ) using the integral form of Gauss's law, we must integrate the charge density over the volume enclosed by the Gaussian surface. Since the given charge density in the cylindrical coordinate is defined as ρᵥ = 2ρ² for ρ < 3, we will consider a cylindrical Gaussian surface with radius ρ and length L.
The amount of charge dQ contained in a differential volume element dV within the cylinder is given by:
dQ = ρᵥ dV = 2ρ² dV
The differential volume element in cylindrical coordinates is dV = ρ dρ dθ dz. Integrating over the volume of the cylinder we get:
Qᵢᵣᵧ = ∫∫∫ ρᵥ dV = ∫_{0}^{L} dz ∫_{0}^{2π} dθ ∫_{0}^{ρ2} 2ρ² ρ dρ
Calculate the integral to find the total enclosed charge Qᵢᵣᵧ. Apply Gauss's Law, ΨD · dA = Qᵢᵣᵧ/ε₀, over the cylindrical surface to find D, the electric displacement field. Keep in mind that the displacement field will be directed radially outwards due to the symmetry of the charge distribution in a cylinder.