The equation out of four maxweel's equations which show (s) electric field lines do not form closed loops is/are

Options

A. `oint _(s) vecE.bar(ds)=q// epsi _(0)`
B. `oint _(s) barB.bards=0`
C. `oint barE.dbart= (d phi_(B))/(dt)`
D. None of these

Explanation

To determine which of Maxwell's equations indicates that electric field lines do not form closed loops, we can analyze each of the four equations.

### Step-by-Step Solution:

1. **Identify Maxwell's Equations**:
The four Maxwell's equations are:
- Gauss's Law: \(\oint \mathbf{E} \cdot d\mathbf{s} = \frac{Q_{\text{enc}}}{\epsilon_0}\)
- Gauss's Law for Magnetism: \(\oint \mathbf{B} \cdot d\mathbf{s} = 0\)
- Faraday's Law of Induction: \(\oint \mathbf{E} \cdot d\mathbf{s} = -\frac{d\Phi_B}{dt}\)
- Ampère-Maxwell Law: \(\oint \mathbf{B} \cdot d\mathbf{s} = \mu_0 I_{\text{enc}} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}\)

2. **Analyze Gauss's Law**:
Gauss's Law states that the electric flux through a closed surface is proportional to the charge enclosed. This implies that electric field lines originate from positive charges and terminate at negative charges, indicating that they do not form closed loops.

3. **Analyze Gauss's Law for Magnetism**:
This law states that the magnetic flux through a closed surface is zero, indicating that magnetic field lines do form closed loops.

4. **Analyze Faraday's Law of Induction**:
This law indicates that a changing magnetic field induces an electric field. The line integral of the electric field around a closed loop is related to the rate of change of magnetic flux, which suggests that electric field lines can be induced but do not form closed loops on their own.

5. **Analyze Ampère-Maxwell Law**:
This law relates the magnetic field to the current and the rate of change of electric field. It also implies that magnetic field lines can form closed loops.

6. **Conclusion**:
From the analysis, we can conclude that **Gauss's Law** is the equation that shows that electric field lines do not form closed loops. Therefore, the correct answer is:

**Answer**: Gauss's Law (\(\oint \mathbf{E} \cdot d\mathbf{s} = \frac{Q_{\text{enc}}}{\epsilon_0}\))