Mark correct options

Options

A. Gauss's law is valild only for uniform charge distributions
B. Gauss's law is valid only for charges placed in vacuum
C. the electric field calculated by Gauss's law is the field due to al tlhe charges
D. The flux of the electric field through a closed surfasce due to all the charges is equal to the flux due to the charges enclosed by the surface

Explanation

To solve the question regarding Gauss's law and to determine which options are correct, let's analyze each statement step by step.

### Step-by-Step Solution:

1. **Understanding Gauss's Law**:
Gauss's law states that the total electric flux (Φ) through a closed surface is equal to the charge (Q) enclosed by that surface divided by the permittivity of free space (ε₀). Mathematically, it is expressed as:
\[ \Phi = \oint E \cdot dS = \frac{Q_{\text{enclosed}}}{\epsilon_0} \]

2. **Evaluating Option A**:
- **Statement**: "Gauss's law is valid only for the uniform charge distribution."
- **Analysis**: Gauss's law is valid for any charge distribution, not just uniform ones, as long as the symmetry of the charge distribution allows for the calculation of the electric field. Thus, this statement is **incorrect**.

3. **Evaluating Option B**:
- **Statement**: "Gauss's law is valid only for the charge placed in vacuum."
- **Analysis**: Gauss's law applies to charges in any medium, including dielectrics. Therefore, this statement is also **incorrect**.

4. **Evaluating Option C**:
- **Statement**: "The electric field calculated by Gauss's law in the field due to all the charges."
- **Analysis**: This statement is misleading. Gauss's law allows us to calculate the electric field due to the charge enclosed by the Gaussian surface, not all charges. Hence, this statement is **incorrect**.

5. **Evaluating Option D**:
- **Statement**: "The flux of the electric field through a closed surface due to all charges equal to the flux due to the charge enclosed by the surface."
- **Analysis**: This statement is correct as per Gauss's law, which states that the electric flux through a closed surface is determined solely by the charge enclosed within that surface. Thus, this statement is **correct**.

### Conclusion:
- **Correct Option**: D
- **Incorrect Options**: A, B, C