The cube shown in the figure has edge length 1.6 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=2.7 and b=1.6. Find the total charge inside the cube. The answer’s unit is pC.

Question

The cube shown in the figure has edge length 1.6 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=2.7 and b=1.6. Find the total charge inside the cube. The answer’s unit is pC.

BlackTom AI · Accepted Answer

To begin, we identify the geometry and the field components given: a cube of edge length L = 1.6 m at the origin with the electric field E = <8, a y + b, c> N/C, where a = 2.7 and b = 1.6, c is not specified but appears only in the z-component.

Next, apply Gauss's law in integral form: ∮ E · dA = Q_enclosed / ε0. The cube has faces perpendicular to x, y, and z. We compute the net flux through all six faces.

- Flux through faces perpendicular to x: Ex = 8 is constant. The two x-faces are at x = 0 and x = L, with normals ±x̂. The flux through the face at x = L is Ex · A = 8 · L^2, and through the face at x = 0 is −8 · L^2. Summing gives zero net flux from the x-faces because the two contributions cancel: 8L^2 − 8L^2 = 0.

- Flux through faces perpendicular to z: Ez = c is constant. The two z-faces at z = 0 and z = L have normals ±ẑ. Each face contributes c · L^2 or −c · L^2, and the sum is again zero: cL^2 − cL^2 = 0. So no net flux from the z-faces, regardless of the value of c.

- Flux through faces perpendicular to y: Ey = a y + b depends on y. The two faces are at y = 0 and y = L with normals +ȳ and −ȳ, respectively. The flux through y = L is Ey(y = L) · L^2 = (aL + b) L^2. The flux through y = 0 is Ey(y = 0) · L^2 = b · L^2 but with a negative sign since the normal points in −ȳ for that face, giving −bL^2. The net flux from the y-faces is [(aL + b) − b] L^2 = (aL) L^2 = a L^3.

Thus, the total net electric flux through the cube is Φ = a L^3, since the x- and z-face contributions cancel out, leaving only the y-direction contribution.

Now relate flux to enclosed charge via Gauss's law: Q = ε0 · Φ = ε0 · a · L^3.

Plug in the numbers: L^3 = (1.6 m)^3 = 4.096 m^3. Therefore a · L^3 = 2.7 × 4.096 ≈ 11.0592. Multiplying by ε0 ≈ 8.854187817×10^−12 F/m gives Q ≈ 11.0592 × 8.854×10^−12 C ≈ 9.80×10^−11 C.

Converting to picocoulombs: 9.80×10^−11 C = 98 pC. The result matches the provided answer of 98 pC.

Note on the role of each component: the Ex contribution cancels because flux through opposite x-faces is equal and opposite; the Ez contribution cancels for the same reason on the z-faces; only Ey contributes to a net flux due to its linear dependence on y across the cube’s extent, leading to a net flux proportional to a and L^3.

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

The cube shown in the figure has edge length 1.6 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=2.7 and b=1.6. Find the total charge inside the cube. The answer’s unit is pC.

View Explanation

Log in for full answers

Similar Questions

To derive the equation you used to solve the first part of this question what law did you use?

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.

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.

Consider a spherical Gaussian surface of radius R centered at the origin. A charge Q is placed inside the sphere. To maximize the magnitude of the flux of the electric field through the Gaussian surface, the charge should be located

Which of the following statement about Gauss's law are correct?

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.

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

More Practical Tools for International Students

Homework AI Solver

Stylized Writing Assistant

Plagiarism Checker Assistant

Citation Assistant

In-Class Translation Assistant

Class Notes Assistant

Quiz Search Assistant

Past Exam Questions by School

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler