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.

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.简答题

类似问题

A point charge q is located at the center of a cube of edge L. What is the flux of the electric field of q through one face of the cube?

Five identical point charges q are located at a distance 2R from the center of a sphere of radius R. The total electric flux through the surface of the sphere is

Using the Divergence Theorem, which of the following is Gauss' Law in differential form? 1) ∇ × 𝐵 → = 1 𝜖 0 𝑞 2) ∇ × 𝐸 → = 1 𝜖 0 𝑞 3) ∇ ∙ 𝐵 → = 1 𝜖 0 𝑞 4) ∇ ∙ 𝐸 → = 1 𝜖 0 𝑞

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?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单