If there is only one type of charge in the universe, then

Options

A. `oint E.ds ne 0` on any surface.
B. `ointE.ds=epsilon_(0)//q` if the charge is outside the surface.
C. `ointE.ds` cannot not be defined.
D. `ointE.ds=q//epsilon_(0)` if charges of magnitude q is inside the surface.

Explanation

To solve the question, "If there is only one type of charge in the universe, then...", we can analyze the implications of having only one type of electric charge based on Gauss's Law.

### Step-by-Step Solution:

1. **Understanding Gauss's Law**:
Gauss's Law states that the electric flux (Φ) through a closed surface is proportional to the charge (Q) enclosed within that surface. Mathematically, it is expressed as:
\[ \Phi = \oint \mathbf{E} \cdot d\mathbf{S} = \frac{Q_{\text{inside}}}{\epsilon_0} \]
where \( \epsilon_0 \) is the permittivity of free space.

2. **Identifying the Charge Type**:
If there is only one type of charge in the universe, it means that we can only have either positive charges or negative charges, but not both.

3. **Applying Gauss's Law**:
- If we consider a Gaussian surface that encloses a charge \( Q \):
\[ \oint \mathbf{E} \cdot d\mathbf{S} = \frac{Q}{\epsilon_0} \]
- If \( Q \) is positive, the electric field \( \mathbf{E} \) will radiate outward from the charge.
- If \( Q \) is negative, the electric field will point inward toward the charge.

4. **Considering the Options**:
- **Option 1**: Closed integration \( \oint \mathbf{E} \cdot d\mathbf{S} \neq 0 \) on any surface. This is incorrect because if there are no charges inside the surface, the flux will be zero.
- **Option 2**: Closed integration \( \oint \mathbf{E} \cdot d\mathbf{S} = \frac{\epsilon}{q} \) if the charge is outside the surface. This is also incorrect because the flux through a closed surface only depends on the charge inside it.
- **Option 3**: Closed integration \( \oint \mathbf{E} \cdot d\mathbf{S} \) cannot be defined. This is incorrect as Gauss's Law can still be applied.
- **Option 4**: Closed integration \( \oint \mathbf{E} \cdot d\mathbf{S} = \frac{Q}{\epsilon_0} \) if a charge of magnitude \( Q \) is inside the surface. This is correct according to Gauss's Law.

5. **Conclusion**:
Therefore, if there is only one type of charge in the universe, the correct statement is that the closed integration \( \oint \mathbf{E} \cdot d\mathbf{S} = \frac{Q}{\epsilon_0} \) if a charge of magnitude \( Q \) is inside the surface. This aligns with Gauss's Law.

### Final Answer:
The correct option is:
**Closed integration \( \oint \mathbf{E} \cdot d\mathbf{S} = \frac{Q}{\epsilon_0} \) if a charge of magnitude \( Q \) is inside the surface.**