Problem 13 - University Physics with Modern Physics

Question

22\. 13. A \(9.60-\mu\) C point charge is at the center of a cube with sides of length 0.500 \(\mathrm{m}\) (a) What is the electric flux through one of the six faces of the cube? (b) How would your answer to part (a) change if the sides were 0.250 \(\mathrm{m}\) long? Explain.

Step-by-Step Solution

Answer

Electric flux per face is independent of cube size; it's \( \frac{9.60 \times 10^{-6}}{6 \times 8.85 \times 10^{-12}} \).

Step 1: Apply Gauss's Law

Gauss's Law states that the total electric flux \( \Phi \) through a closed surface is given by \( \Phi = \frac{Q_{ ext{enc}}}{\varepsilon_0} \), where \( Q_{ ext{enc}} \) is the charge enclosed and \( \varepsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \, \text{C}^2/\text{N} \cdot \text{m}^2 \). In this case, the charge enclosed is \( 9.60 \, \mu \text{C} \), or \( 9.60 \times 10^{-6} \, \text{C} \).

Step 2: Calculate Total Electric Flux

Substitute the known values into the formula: \( \Phi = \frac{9.60 \times 10^{-6} \, \text{C}}{8.85 \times 10^{-12} \, \text{C}^2/\text{N} \cdot \text{m}^2} \). Calculate to determine the total electric flux through the cube.

Step 3: Find Electric Flux Through One Face

Since the charge is at the center of a symmetric cube, the electric flux is evenly distributed through each of the six faces. Therefore, divide the total flux by 6: \( \Phi_{ ext{face}} = \frac{\Phi}{6} \).

Step 4: Evaluate Effect of Cube Side Length Change

Gauss's law indicates that the total flux depends only on the enclosed charge and not on the cube's dimensions. Since the charge is still at the center regardless of the cube size, changing the cube's side length does not affect the flux through each face.

Key Concepts

Gauss's Law

Gauss's Law forms a central part of electromagnetism. This powerful law relates the electric flux passing through a closed surface to the electric charge enclosed by that surface.
Gauss's Law equation is expressed as:

• \( \Phi = \frac{Q_{\text{enc}}}{\varepsilon_0} \)

where:

• \( \Phi \) is the total electric flux through the closed surface.
• \( Q_{\text{enc}} \) is the total charge enclosed within the surface.
• \( \varepsilon_0 \) is the permittivity of free space, a constant value.

Gauss's Law is particularly helpful when dealing with symmetrical charge distributions, like spheres or cylindrical shapes.
In our problem, a point charge is placed at the center of a cube. Hence, the flux through each face of the cube can easily be determined. This law demonstrates that the shape or size of the surface doesn't affect the flux - only the amount of enclosed charge matters.

Permittivity of Free Space

The permittivity of free space, noted as \( \varepsilon_0 \), is a key constant in electromagnetism. It quantifies the ability of the vacuum to permit electric field lines. Essentially, it serves as a measure of how much resistance is encountered when forming an electric field in a vacuum.
The standard value is:

• \( \varepsilon_0 \approx 8.85 \times 10^{-12} \, \text{C}^2/\text{N} \cdot \text{m}^2 \)

This constant appears in numerous fundamental equations, such as Gauss's Law and Coulomb's Law, where it influences how electric force and field interactions are calculated.
In the original exercise, \( \varepsilon_0 \) factors into calculating the total electric flux through the cube. Understanding this constant helps determine how electric fields interact in a vacuum environment, which can be computed directly by using known charges and spatial properties.

Symmetric Distribution

Symmetry plays a vital role in simplifying the analysis of electric fields in physical systems. A symmetrical distribution means that the system looks the same from multiple perspectives or orientations.
In electromagnetism, symmetry often helps to distribute quantities like electric fields or flux evenly, making calculations far easier.
Taking our cube example:

• With the charge at the center, and identical cube faces, the system displays perfect symmetry.
• This means the electric flux is evenly split among all six faces of the cube. Thus, we divide the total flux by 6 to find the flux through each face.
• The symmetry ensures that altering the cube's side length does not change how the flux distributes as long as the charge remains centralized.

Symmetrical systems like this allow us to use Gauss's Law more efficiently, leading to straightforward solutions for electric field problems.