The cube shown in the figure has edge length 2.0 m. There is a non-uniform electric field in the region given by N/C, where a=2.0 and b=2.8. Find the total charge inside the cube. The answer’s unit is pC.

Question

The cube shown in the figure has edge length 2.0 m. There is a non-uniform electric field in the region given by  N/C, where a=2.0 and b=2.8. Find the total charge inside the cube. The answer’s unit is pC.

BlackTom AI · Accepted Answer

The problem asks for the total charge inside a cube given a non-uniform electric field with a specified divergence. First, recall Gauss's law in differential form: Q_enc = ε0 ∭ (∇·E) dV. This means we need the volume integral of the divergence of E over the cube.

From the information, the divergence of the field is constant and equal to a = 2.0 (in units of N/C per meter, effectively the scalar divergence). Since the cube is axis-aligned with edge length L = 2.0 m, its volume is V = L^3 = 8.0 m^3.

Compute the volume integral: ∭ (∇·E) dV = (a) × V = 2.0 × 8.0 = 16.0 (in the appropriate units for flux, i.e., N/C·m^3).

Apply Gauss's law with ε0 = 8.854187817 × 10^-12 F/m: Q_enc = ε0 × ∭ (∇·E) dV = ε0 × 16.0 = 8.854187817e-12 × 16.0 C.

Carrying out the multiplication: 8.854e-12 × 16 ≈ 1.41664e-10 C ≈ 1.42 × 10^-10 C. Converting to picoCoulombs (1 pC = 10^-12 C) gives Q_enc ≈ 142 pC.

Thus, the total charge inside the cube is approximately 142 pC.

202502PHY1202 Assignment 4 Gauss Law - due February 23, 2025 11:59 pm.

The cube shown in the figure has edge length 2.0 m. There is a non-uniform electric field in the region given by N/C, where a=2.0 and b=2.8. Find the total charge inside the cube. The answer’s unit is pC.

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