Let's break down the problem step by step. --- ## **Step 1: Understanding Gauss's Law** **Gauss's Law** states: \[ \oint_S \vec{E} \cdot d\vec{A} = \frac{q_{\text{enc}}}{\epsilon_} \] where \(S\) is a **closed surface**. **Key Point:** Gauss's Law is valid (can be applied) only for **closed surfaces** (surfaces that fully enclose a volume). --- ## **Step 2: Analyzing Each Surface** Let's look at each surface as described: 1. **Surface (1): Circle** - A flat, 2D surface (not closed). - **Not a closed surface.** 2. **Surface (2): Rectangular Parallelepiped (Box)** - A closed 3D box. - **Closed surface.** 3. **Surface (3): Sphere** - A closed 3D surface. - **Closed surface.** 4. **Surface (4): Cylinder** - A closed 3D surface (assuming it includes the curved surface and both top and bottom bases). - **Closed surface.** --- ## **Step 3: Listing Valid Surfaces** **Gauss's Law is valid for:** - Surface (2): Rectangular parallelepiped (box) - Surface (3): Sphere - Surface (4): Cylinder **Not valid for:** - Surface (1): Circle (not closed) --- ## **Step 4: Final Answer** ### **Final Answer** > **Gauss's Law is valid for surfaces 2 (rectangular parallelepiped), 3 (sphere), and 4 (cylinder).** --- ### **Summary Table** | Surface Number | Shape | Closed Surface? | Gauss's Law Valid? | |----------------|---------------------------|-----------------|--------------------| | 1 | Circle | No | No | | 2 | Rectangular parallelepiped| Yes | Yes | | 3 | Sphere | Yes | Yes | | 4 | Cylinder | Yes | Yes | --- **Only surfaces 2, 3, and 4 are valid for applying Gauss's Law.**