A charge of uniform volume density (40nC/m^3) fills a cube with 8cm edges. What is the net flux through the surface of the cube?
To find the net flux through the surface of the cube, we need to use Gauss's Law. Gauss's Law states that the electric flux through a closed surface is proportional to the charge enclosed by the surface. In this case, we have a cube with a charge of uniform volume density.
Let's calculate step by step:
Step 1: Determine the charge enclosed by the cube.
Since the charge has a uniform volume density, we can determine the total charge by multiplying the charge density by the volume of the cube.
The volume of the cube can be calculated by finding the cube of its edge length.
Given that the cube has 8 cm edges, the volume of the cube is (8 cm)^3 = 512 cm³.
Since we are working with the charge density in terms of nC/m³, we need to convert the volume to cubic meters.
1 m = 100 cm, so the volume becomes (8 cm / 100 cm/m)^3 = (0.08 m)^3 = 0.000512 m³.
Now we can calculate the charge enclosed by the cube by multiplying the volume by the charge density:
Charge enclosed = Charge density × Volume
Charge enclosed = (40 nC/m³) × (0.000512 m³)
Charge enclosed = 0.02048 nC
Step 2: Calculate the net flux through the surface of the cube.
Since the charge enclosed is known, we can use Gauss's Law to find the net flux through the surface of the cube.
Gauss's Law states that the electric flux (Φ) through a closed surface is equal to the charge enclosed (Q) divided by the electric constant (ε₀).
The electric constant, ε₀, is approximately equal to 8.854 × 10⁻¹² C²/N∙m².
Net flux through the surface = Charge enclosed / ε₀
Net flux through the surface = 0.02048 nC / (8.854 × 10⁻¹² C²/N∙m²)
Net flux through the surface = 2.31 × 10⁹ N∙m²/C
So, the net flux through the surface of the cube is 2.31 × 10⁹ N∙m²/C.