Question

If the net flux through a cube is 1.05 N m\(^2\) C\(^{-1}\), what will be the total charge inside the cube? (Given: The permittivity of free space is \(8.85 \times 10^{-12}\) C\(^2\) N\(^{-1}\) m\(^{-2}\)).

• A. \(9.29 \times 10^{-11}\) C
• B. \(9.27 \times 10^{-10}\) C
• C. \(9.22 \times 10^{-6}\) C
• D. \(9.29 \times 10^{-12}\) C

Hint:

Gauss's Law provides a powerful link between electric flux and the enclosed charge. The shape of the closed surface (a cube, sphere, etc.) does not matter for the total flux, only the net charge inside does. If you know the flux, you can find the charge, and vice versa.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem is a direct application of Gauss's Law of electrostatics. Gauss's Law states that the total electric flux (\(\Phi\)) through any closed surface is equal to the total net electric charge (\(Q_{\text{in}}\)) enclosed within that surface, divided by the permittivity of free space (\(\epsilon_0\)).

Step 2: Key Formula or Approach:
According to Gauss's Law:
\[ \Phi = \frac{Q_{\text{in}}}{\epsilon_0} \] To find the total charge inside the cube, we can rearrange the formula:
\[ Q_{\text{in}} = \Phi \times \epsilon_0 \]

Step 3: Detailed Explanation:
Given:
Net electric flux, \( \Phi = 1.05 \, \text{N m}^2 \text{C}^{-1} \)
Permittivity of free space, \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2 \text{N}^{-1} \text{m}^{-2} \)
Now, we calculate the total charge enclosed, \(Q_{\text{in}}\):
\[ Q_{\text{in}} = (1.05 \, \text{N m}^2 \text{C}^{-1}) \times (8.85 \times 10^{-12} \, \text{C}^2 \text{N}^{-1} \text{m}^{-2}) \] \[ Q_{\text{in}} = 9.2925 \times 10^{-12} \, \text{C} \] Rounding to two decimal places, we get:
\[ Q_{\text{in}} \approx 9.29 \times 10^{-12} \, \text{C} \]

Step 4: Final Answer:
The total charge inside the cube will be \(9.29 \times 10^{-12}\) C.