Gauss theorem:
Gauss theorem is mainly used to find out the electric flux linked to a closed surface. It does not depend upon the shape or size of surface. According to this theorem, the electric flux linked to a closed surface is equal to `((1)/(epsilon_(0)))` times the charge enclosed by the surface.
Let we have a charge q, now if we want to find out the net flux linked to a closed surface around it them,
Electric flux `phi = oint_(s) vecE. vecds = (q)/(epsilon_(0))`
Net flux linked to a closed surface around a charge particle is...........times the charge.

Options

A. `epsilon_(0)`
B. `(1)/(epsilon_(0))`
C. `epsilon_(0)^(2)`
D. None of these.

Explanation

To solve the problem using Gauss's theorem, we will follow these steps:

### Step 1: Understand Gauss's Theorem
Gauss's theorem states that the electric flux (Φ) linked to a closed surface is directly proportional to the charge (q) enclosed within that surface. Mathematically, it is expressed as:

\[ \Phi = \oint_{S} \vec{E} \cdot d\vec{s} = \frac{q}{\epsilon_0} \]

where:
- \(\Phi\) is the electric flux,
- \(\vec{E}\) is the electric field,
- \(d\vec{s}\) is the differential area vector,
- \(q\) is the total charge enclosed by the surface,
- \(\epsilon_0\) is the permittivity of free space.

### Step 2: Identify the Charge Enclosed
In this scenario, we have a charge \(q\) placed inside a closed surface (Gaussian surface). According to Gauss's theorem, the total charge enclosed by this surface is simply \(q\).

### Step 3: Apply Gauss's Theorem
Using Gauss's theorem, we can express the net electric flux linked to the closed surface around the charge \(q\):

\[ \Phi = \frac{q}{\epsilon_0} \]

### Step 4: Relate Electric Flux to Charge
From the equation derived from Gauss's theorem, we can see that the net flux linked to a closed surface around a charge particle is equal to \(\frac{1}{\epsilon_0}\) times the charge \(q\).

### Conclusion
Thus, the net flux linked to a closed surface around a charge particle is:

\[ \text{Net flux} = \frac{1}{\epsilon_0} \times q \]

### Final Answer
The net flux linked to a closed surface around a charge particle is \(\frac{1}{\epsilon_0}\) times the charge.

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