Volume charge density is located in free space as \(\rho_{v}=2 e^{-1000 r} \mathrm{nC} / \mathrm{m}^{3}\) for \(0

To find the total charge enclosed by a sphere of radius r = 1mm, let's first define the given volume charge density: Ρv(r) = 2e^{-1000r} nC/m³ (for 0 < r < 1mm) Now, we need to integrate this charge density over the whole volume of the sphere: Total charge (Q) = ∫∫∫Ρv(r) dv Where the limits of integration are from r = 0 to r = 1mm, and we need to convert the radius into meters (0 to 0.001m). And dv = r²sinθ dr dθ dϕ where θ is from 0 to π, and ϕ is from 0 to 2π. Q = ∫(0 to 0.001) ∫(0 to π) ∫(0 to 2π) 2e^{-1000r} r² sinθ dr dθ dϕ

Let's solve the volume integral one variable at a time. First, we perform the integral with respect to ϕ: Q = ∫(0 to 0.001) ∫(0 to π) [2π * 2e^{-1000r} r² sinθ] dr dθ Now, we perform the integral with respect to θ: Q = ∫(0 to 0.001) [2π * -2e^{-1000r} r² (cosθ from 0 to π)] dr Q = ∫(0 to 0.001) [4π * 2e^{-1000r} r²] dr Finally, we perform the integral with respect to r: Q = -4π * 2 * 10^{-6} [e^{-1000r} r²/(-1000) + 2e^{-1000r} r/(-1000)^2 + 2e^{-1000r}/(-1000)^3] from r=0 to r=(1mm = 0.001m) By plugging r = 0.001m and r = 0 into the above expression and calculating we get, Q ≈ 7.84 * 10^{-12} C The total charge enclosed by the sphere is approximately 7.84 pC.

Now that we have the total enclosed charge, we can use Gauss's Law to find the electric flux density at r=1mm. Gauss's Law states that the total electric flux through a closed surface is equal to the total enclosed charge divided by the permittivity of free space (ε0). Flux = Q / ε0 The electric flux density on the surface (Dr) is equal to the total electric flux divided by the surface area of the sphere (A). Surface area of the sphere A = 4πr². In this case, r = 1mm = 0.001m. Dr = Flux / A = (Q / ε0) / (4πr²) By plugging the known values into the equation: ε0 = 8.85 * 10^{-12} C²/Nm² Q ≈ 7.84 * 10^{-12} C r = 0.001m Dr ≈ ((7.84 * 10^{-12} C) / (8.85 * 10^{-12} C²/Nm²)) / (4π(0.001m)²) Dr ≈ 140 N/C The electric flux density on the surface r = 1mm is approximately 140 N/C.