Given three point charges enclosed by a closed surface $a_1$, what is the electric flux through the surface $a_1$ according to Gauss's law?

• A. The electric flux is zero because the surface is closed.
• B. The electric flux is equal to $\frac{q}{{\epsilon }_0}$ where $q$ is the charge outside the surface.
• C. The electric flux depends only on the area of the surface, not the enclosed charge.
• D. The electric flux is equal to $\frac{q_1+q_2+q_3}{{\epsilon }_0}$

Answer: (D)

Verified step by step guidance

1. Recall Gauss's law, which states that the electric flux \( \Phi_E \) through a closed surface is proportional to the net charge enclosed by that surface. Mathematically, it is expressed as: \[ \Phi_E = \frac{Q_{\text{enc}}}{\epsilon_0} \] where \( Q_{\text{enc}} \) is the total charge enclosed and \( \epsilon_0 \) is the permittivity of free space.
2. Identify the charges enclosed by the closed surface \( a_1 \). In this problem, there are three point charges \( q_1 \), \( q_2 \), and \( q_3 \) inside the surface.
3. Sum the enclosed charges to find the total enclosed charge: \[ Q_{\text{enc}} = q_1 + q_2 + q_3 \]
4. Substitute the total enclosed charge into Gauss's law formula to express the electric flux through the surface: \[ \Phi_E = \frac{q_1 + q_2 + q_3}{\epsilon_0} \]
5. Understand that charges outside the closed surface do not contribute to the net electric flux through that surface, and the flux does not depend on the surface area but only on the enclosed charge.