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: Concept Identification:
This problem directly applies Gauss's Law for electrostatics. This law posits that the total electric flux (\(\Phi\)) through any closed surface equals the net electric charge (\(Q_{\text{in}}\)) enclosed by that surface, divided by the permittivity of free space (\(\epsilon_0\)).

Step 2: Governing Equation:
Gauss's Law is formulated as:
\[ \Phi = \frac{Q_{\text{in}}}{\epsilon_0} \]To determine the total enclosed charge, the formula is rearranged:
\[ Q_{\text{in}} = \Phi \times \epsilon_0 \]

Step 3: Calculation:
Given values:
Net electric flux, \( \Phi = 1.05 \, \text{N m}^2 \text{C}^{-1} \)
Permittivity of free space, \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2 \text{N}^{-1} \text{m}^{-2} \)
The total enclosed charge, \(Q_{\text{in}}\), is computed as:
\[ Q_{\text{in}} = (1.05 \, \text{N m}^2 \text{C}^{-1}) \times (8.85 \times 10^{-12} \, \text{C}^2 \text{N}^{-1} \text{m}^{-2}) \]\[ Q_{\text{in}} = 9.2925 \times 10^{-12} \, \text{C} \]Rounded to two decimal places:
\[ Q_{\text{in}} \approx 9.29 \times 10^{-12} \, \text{C} \]

Step 4: Conclusion:
The total charge within the cube is \(9.29 \times 10^{-12}\) C.