AIIMS 2003 Physics Gauss's Law MCQ Question

To solve this problem, we need to apply Gauss's Law, which is a fundamental principle in electrostatics. Gauss's Law states that the electric flux $\Phi_E$ through a closed surface is proportional to the charge enclosed by that surface. Mathematically, it is expressed as:

$$\Phi_E = \frac{Q_{\text{enc}}}{\varepsilon_0}$$

where:

• $\Phi_E$ is the electric flux,
• $Q_{\text{enc}}$ is the total charge enclosed within the surface,
• $\varepsilon_0$ is the permittivity of free space.

Explanation of the Correct Answer (D: zero)

The problem states that we need to find the electric flux through a surface $S$ due to a distribution of charges. The key to understanding why the correct answer is zero lies in determining whether there are any charges enclosed by surface $S$.

1. Charge Distribution Analysis: If the figure (not visible here) shows that surface $S$ does not enclose any net charge (i.e., the charges might be outside the surface or balanced in a way that their net effect cancels out), then according to Gauss's Law, we have:

   $Q_{\text{enc}} = 0$

   Thus, the electric flux $\Phi_E$ through surface $S$ becomes:

   $$\Phi_E = \frac{0}{\varepsilon_0} = 0$$

   Therefore, the correct answer is D: zero.

Clarification of Incorrect Options

Now let's briefly discuss why the other options (A, B, and C) are incorrect:

• Option A: $\frac{3q}{\varepsilon_0}$
  This option implies that there is a net charge of $3q$ enclosed within the surface $S$. However, if the net charge is zero or not enclosed, this option cannot be correct.

• Option B: $\frac{2q}{\varepsilon_0}$
  Similar to option A, a net charge of $2q$ would lead to a non-zero flux. If there are no charges enclosed, this option cannot be correct.

• Option C: $\frac{q}{\varepsilon_0}$
  This option suggests that there is a net charge of $q$ enclosed within surface $S$. Again, if there is no charge inside, this is not valid.

Conclusion

To summarize, the electric flux through a closed surface is determined by the net charge enclosed by that surface. Since the correct conclusion from the analysis is that there is no net charge enclosed within surface $S$, we find:

$$\Phi_E = \frac{Q_{\text{enc}}}{\varepsilon_0} = 0$$

Thus, the answer is D: zero. This emphasizes the importance of understanding the charge distribution in electrostatic problems and applying Gauss's Law appropriately.