Question

Four closed surfaces and corresponding charge distributions are shown below. The respective electric fluxes through the surfaces \( S_1, S_2, S_3, S_4 \) are \( \Phi_1, \Phi_2, \Phi_3, \Phi_4 \). Then, which of the following is correct?

• A. \( \Phi_1>\Phi_2>\Phi_3>\Phi_4 \)
• B. \( \Phi_1 = \Phi_2>\Phi_3>\Phi_4 \)
• C. \( \Phi_1>\Phi_2 = \Phi_3>\Phi_4 \)
• D. \( \Phi_1>\Phi_2 = \Phi_3<\Phi_4 \)

Hint:

Gauss's law states that the electric flux through a closed surface depends only on the enclosed charge. The greater the enclosed charge, the greater the flux.

Answer 1

The Correct Option is B

The electric flux \( \Phi \) through a surface is related to the total charge enclosed by the surface, according to Gauss’s law. Mathematically, it is expressed as: \[ \Phi = \frac{Q_{enc}}{\varepsilon_0} \] Where:
- \( \Phi \) is the electric flux,
- \( Q_{enc} \) is the total charge enclosed by the surface,
- \( \varepsilon_0 \) is the permittivity of free space.
According to Gauss's law, the electric flux depends on the enclosed charge and not on the distribution of the charge or the shape of the surface, as long as the surface encloses the same amount of charge.
Let’s evaluate each surface based on the charges:
- Surface \( S_1 \) has an enclosed charge of \( +2q \),
- Surface \( S_2 \) has an enclosed charge of \( -2q \),
- Surface \( S_3 \) has an enclosed charge of \( +3q \),
- Surface \( S_4 \) has an enclosed charge of \( +5q \).
Since the electric flux depends on the total enclosed charge, we have: \[ \Phi_1 = \Phi_2 = \frac{2q}{\varepsilon_0} \quad \text{(as they enclose equal charge)} \] \[ \Phi_3 = \frac{3q}{\varepsilon_0} \quad \text{(enclosing a larger charge)} \] \[ \Phi_4 = \frac{5q}{\varepsilon_0} \quad \text{(enclosing the largest charge)} \] Thus, the fluxes follow the relationship: \[ \Phi_1 = \Phi_2>\Phi_3>\Phi_4 \]