What is the net flux of the uniform electric field through a cube of side 20 cm oriented so that its faces are parallel to the coordinate planes?

1. Break Down the Problem

To find the net electric flux through the cube, we will need to consider the characteristics of the electric field and how it interacts with the sides of the cube. Since the cube is oriented such that its faces are parallel to the coordinate planes, the electric field will intersect the cube’s faces symmetrically.

2. Relevant Concepts

The electric flux $\Phi_E$ through a surface in an electric field $\vec{E}$ is given by the formula: $\Phi_E = \int \vec{E} \cdot d\vec{A}$ For a uniform electric field, the formula can be simplified to: $\Phi_E = E \cdot A \cdot \cos(\theta)$ where:

• $E$ is the magnitude of the electric field,
• $A$ is the area of the surface,
• $\theta$ is the angle between the electric field and the normal to the surface.

3. Analysis and Detail

In our case:

• The cube has a side length of $0.20 \, \text{m}$.
• The area $A$ of one face of the cube is: $A = \text{side}^2 = (0.20)^2 = 0.04 \, \text{m}^2$
• There are a total of six faces on a cube.

Assuming a uniform electric field $\vec{E}$ is present (the magnitude and direction are not specified but assumed to be constant), the flux through opposite faces will be equal in magnitude but opposite in direction, leading to cancellation. Therefore, the total net flux $\Phi_{\text{net}}$ through the entire cube is zero.

4. Verify and Summarize

Given that the cube has symmetrically opposite faces, the positive flux through one face will cancel with the negative flux through the face directly opposite to it. Consequently, we arrive at the conclusion that the net electric flux through the cube is zero.

Final Answer

$\Phi_{\text{net}} = 0 \, \text{Nm}^2/\text{C}$