Question

Three point charges \(q+Q,\ q,\ q-Q\) are enclosed by the surface \(S\). What net flux crosses \(S\)?

• A. \(3q\)
• B. \(2q\)
• C. \(3q-Q\)
• D. cannot be determine based on the data given in question

Hint:

In Gauss's law, net flux depends only on total enclosed charge. Opposite charges \(+Q\) and \(-Q\) cancel if both are enclosed.

Answer 1

The Correct Option is A

Concept:
This question is based on Gauss's law. According to Gauss's law: \[ \Phi=\frac{q_{\text{enclosed}}}{\varepsilon_0} \] where, \[ \Phi = \text{net electric flux} \] \[ q_{\text{enclosed}} = \text{total charge enclosed by the closed surface} \] The important point is that the net flux depends only on the total charge enclosed inside the surface.

Step 1: List all charges inside the surface.
The charges enclosed by the surface are: \[ q+Q,\quad q,\quad q-Q \]

Step 2: Add all enclosed charges.
\[ q_{\text{total}}=(q+Q)+q+(q-Q) \] Now simplify: \[ q_{\text{total}}=q+Q+q+q-Q \] Here \(+Q\) and \(-Q\) cancel: \[ Q-Q=0 \] So: \[ q_{\text{total}}=3q \]

Step 3: Apply Gauss's law.
The net flux is: \[ \Phi=\frac{3q}{\varepsilon_0} \] Since the options are written in terms of enclosed charge, the answer is proportional to: \[ 3q \]

Step 4: Check the options.
Option (A) \(3q\) is correct.
Option (B) \(2q\) is incorrect because all three \(q\)-terms are present.
Option (C) \(3q-Q\) is incorrect because \(+Q\) and \(-Q\) cancel.
Option (D) is incorrect because Gauss's law gives a definite answer. Hence, the correct answer is: \[ \boxed{(A)\ 3q} \]