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 due to `q_(3)` and `q_(4)` is

Options

A. zero
B. positive
C. negative
D. cannot say

Answer

To solve the problem, we need to apply Gauss's law, which relates the electric flux through a closed surface to the charge enclosed by that surface. Here’s a step-by-step solution:

### Step 1: Understand Gauss's Law
Gauss's law states that the electric flux (Φ) through a closed surface is proportional to the charge (q_encl) enclosed within that surface. Mathematically, it is expressed as:
\[ \Phi = \frac{q_{encl}}{\epsilon_0} \]
where \( \epsilon_0 \) is the permittivity of free space.

### Step 2: Identify the Charges
In the given scenario, we have charges \( q_3 \) and \( q_4 \) which are located outside the Gaussian surface, while charges \( q_1 \) and \( q_2 \) are inside the Gaussian surface.

### Step 3: Determine the Enclosed Charge
Since \( q_3 \) and \( q_4 \) are outside the Gaussian surface, they do not contribute to the enclosed charge. Therefore, the total enclosed charge \( q_{encl} \) is solely due to \( q_1 \) and \( q_2 \).

### Step 4: Calculate the Electric Flux Due to \( q_3 \) and \( q_4 \)
According to Gauss's law, since \( q_3 \) and \( q_4 \) are outside the Gaussian surface, their contribution to the electric flux through the surface is zero. Thus:
\[ \Phi_{q3, q4} = 0 \]

### Step 5: Conclusion
The net electric flux through the Gaussian surface due to charges \( q_3 \) and \( q_4 \) is zero because they are not enclosed by the surface.

### Final Answer
The net flux of the electric field through the surface due to \( q_3 \) and \( q_4 \) is:
\[ \Phi = 0 \]

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