The cube shown in the figure has edge length 2.0 m. There is a non-uniform electric field in the region given by
𝐸
→
=
8
𝑖
^
+
(
𝑎
𝑦
+
𝑏
)
𝑗
^
+
𝑐
𝑘
^
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 
𝐸
→
=
8
𝑖
^
+
(
𝑎
𝑦
+
𝑏
)
𝑗
^
+
𝑐
𝑘
^
 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

To find the total charge inside the cube, I’ll use Gauss’s law in differential form: Q_enclosed = ε0 ∭_V (∇·E) dV.

First, compute the divergence of E. Given E = (Ex, Ey, Ez) = (8, a y + b, c) with a = 2.0, b = 2.8, c is a constant. The divergence is ∇·E = ∂Ex/∂x + ∂Ey/∂y + ∂Ez/∂z.
- ∂Ex/∂x = ∂(8)/∂x = 0, since Ex is constant in x.
- ∂Ey/∂y = ∂(a y + b)/∂y = a.
- ∂Ez/∂z = ∂(c)/∂z = 0, since Ez is constant in z.
Thus, ∇·E = a = 2.0 (N/C per meter).

Next, evaluate the volume integral over the cube. The cube has edge length 2.0 m, so its volume V = (2.0 m)^3 = 8.0 m^3.
Therefore ∭_V (∇·E) dV = a × V = 2.0 × 8.0 = 16.0.

Finally, multiply by ε0 to obtain the total enclosed charge: Q = ε0 × 16.0.
Using ε0 ≈ 8.854187817 × 10^-12 C^2/(N·m^2), we get Q ≈ (8.854e-12) × 16 ≈ 1.41664 × 10^-10 C.
Converting to picocoulombs: 1.41664 × 10^-10 C × (10^12 pC/ C) ≈ 142 pC.

Key checks: the divergence depended only on a (not on x or z), so the nonuniformity in Ey with respect to y does not complicate the volume integral beyond the constant a; the cube’s symmetry/size gives the 16 factor when multiplied by ε0.

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 𝐸 → = 8 𝑖 ^ + ( 𝑎 𝑦 + 𝑏 ) 𝑗 ^ + 𝑐 𝑘 ^ 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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