Gauss's law and Coulomb's law , although expressed in different forms , are equivalent ways of describing the relation between charge and electric field in static conditions . Gauss's law is `epsilon_(0) phi = q_(encl)`,when `q_(encl)` is the net charge inside an imaginary closed surface called Gaussian surface. The two equations hold only when the net charge is in vaccum or air .
The net flux of the electric field through the surface is

Options

A. due to `q_(1)` and `q_(2)` only
B. due to `q_(3)` and `q_(4)` only
C. equal due to all the four charges
D. cannot say

Explanation

To find the net electric flux through a closed surface using Gauss's law, we can follow these steps:

### Step 1: Understand Gauss's Law
Gauss's law states that the total electric flux (\( \Phi_E \)) through a closed surface is equal to the charge enclosed (\( q_{encl} \)) divided by the permittivity of free space (\( \epsilon_0 \)):
\[ \Phi_E = \frac{q_{encl}}{\epsilon_0} \]

### Step 2: Identify the Charges
In the given problem, we have four charges: \( Q_1 \), \( Q_2 \), \( Q_3 \), and \( Q_4 \). We need to determine which of these charges are enclosed by the Gaussian surface.

### Step 3: Determine the Enclosed Charge
From the information provided:
- \( Q_1 \) and \( Q_2 \) are inside the Gaussian surface.
- \( Q_3 \) and \( Q_4 \) are outside the Gaussian surface.

Thus, the total enclosed charge (\( q_{encl} \)) is:
\[ q_{encl} = Q_1 + Q_2 \]

### Step 4: Calculate the Electric Flux
Using Gauss's law, we can now calculate the electric flux through the surface:
\[ \Phi_E = \frac{Q_1 + Q_2}{\epsilon_0} \]

### Step 5: Conclusion
The net electric flux through the closed surface is given by the expression derived above.

\[ \Phi_E = \frac{Q_1 + Q_2}{\epsilon_0} \]