The cube shown in the figure has edge length 2.1 m. There is a non-uniform electric field in the region given by N/C, where a=2.1 and b=2.0. Find the total charge inside the cube. The answer’s unit is pC.

Question

The cube shown in the figure has edge length 2.1 m. There is a non-uniform electric field in the region given by  N/C, where a=2.1 and b=2.0. Find the total charge inside the cube. The answer’s unit is pC.

BlackTom AI · Accepted Answer

To determine the total charge inside the cube when the electric field is non-uniform, use Gauss's law in differential form: ∮ E · dA = Qenc/ε0 and equivalently Qenc = ε0 ∭(∇ · E) dV over the cube.

First, identify the given field form. The problem provides edge length L = 2.1 m, and constants a = 2.1 and b = 2.0. The presence of two constants a and b suggests a linear field components of the form E = (a x) x̂ + (b y) ŷ (a common setup for linear, position-dependent fields). If this is the intended E, then the divergence is ∇ · E = ∂Ex/∂x + ∂Ey/∂y + ∂Ez/∂z. With Ex = a x, Ey = b y, and Ez = 0, we get ∇ · E = a + b = 2.1 + 2.0 = 4.1 (N/C per meter).

Next, compute the cube volume. For a cube with edge length L = 2.1 m, V = L^3 = (2.1 m)^3 = 9.261 m^3.

Now assemble Qenc using Gauss’s law: Qenc = ε0 ∭(∇ · E) dV = ε0 (a + b) ∭ dV = ε0 (a + b) V.

Plug in the constants: ε0 ≈ 8.854187817 × 10^−12 C^2/(N·m^2), a + b = 4.1, and V = 9.261 m^3. Therefore Qenc ≈ (8.854e−12) × 4.1 × 9.261 C ≈ 3.36 × 10^−10 C.

Convert to picocoulombs: 1 C = 10^12 pC, so Qenc ≈ 3.36 × 10^2 pC ≈ 336 pC.

Summary: The total charge inside the cube is approximately 336 pC, given the linear field components Ex = a x and Ey = b y with a = 2.1, b = 2.0 and cube edge 2.1 m. If the field were defined differently (e.g., including Ez or a different dependence), the divergence—and thus the result—would change accordingly.

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

The cube shown in the figure has edge length 2.1 m. There is a non-uniform electric field in the region given by N/C, where a=2.1 and b=2.0. Find the total charge inside the cube. The answer’s unit is pC.

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