Question

The inward and outward electric flux from a closed surface are \( 6\times10^4 \,\text{Nm}^2\text{C}^{-1} \) and \( 3\times10^4 \,\text{Nm}^2\text{C}^{-1} \). Then the net charge (in coulomb) inside the closed surface is

• A. \( -6\times10^4 \varepsilon_0 \)
• B. \( 6\times10^4 \varepsilon_0 \)
• C. \( 3\times10^4 \varepsilon_0 \)
• D. \( 9\times10^4 \varepsilon_0 \)
• E. \( -3\times10^4 \varepsilon_0 \)

Hint:

Outward flux positive, inward flux negative.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This problem applies Gauss's Law for electrostatics. Gauss's Law states that the net electric flux through any closed surface (a Gaussian surface) is directly proportional to the net electric charge enclosed within that surface.
Step 2: Key Formula or Approach:
1. Gauss's Law is given by the equation: \(\Phi_{net} = \frac{Q_{enclosed}}{\epsilon_0}\), where \(\Phi_{net}\) is the net electric flux, \(Q_{enclosed}\) is the net charge inside the surface, and \(\epsilon_0\) is the permittivity of free space. 2. By convention, electric flux is considered positive if it is directed outward from the closed surface and negative if it is directed inward. 3. Calculate the net flux by summing the outward (positive) and inward (negative) fluxes. 4. Use Gauss's Law to solve for \(Q_{enclosed}\).
Step 3: Detailed Explanation:
We are given:
Outward flux, \(\Phi_{out} = +3 \times 10^4\) NM\(^2\)C\(^{-1}\)
Inward flux, \(\Phi_{in} = -6 \times 10^4\) NM\(^2\)C\(^{-1}\)
The net electric flux is the algebraic sum of the inward and outward fluxes: \[ \Phi_{net} = \Phi_{out} + \Phi_{in} \] \[ \Phi_{net} = (3 \times 10^4) + (-6 \times 10^4) = -3 \times 10^4 \text{ NM}^2\text{C}^{-1} \] Now, we apply Gauss's Law to find the enclosed charge: \[ \Phi_{net} = \frac{Q_{enclosed}}{\epsilon_0} \] Rearranging to solve for \(Q_{enclosed}\): \[ Q_{enclosed} = \Phi_{net} \cdot \epsilon_0 \] \[ Q_{enclosed} = (-3 \times 10^4) \epsilon_0 \] Step 4: Final Answer:
The net charge inside the closed surface is \(-3 \times 10^4 \epsilon_0\) coulombs.