Consider three different closed surfaces, Surface A is a sphere of radius R, Surface B is a sphere of radius 2R, and surface C is a cube with side length R. At the geometric center of each closed surface is a small ball that is completely enclosed by the surface and carries electrical charge +q. Assuming that there are no other charged objects inside any of the surfaces, rank the surfaces based on the field line flux through them, from largest to smallest.

Options

A. `A gt B gtC`
B. `A gt C gt B`
C. `B gt C gt A`
D. All are equal

Answer

To solve the problem, we will apply Gauss's Law, which states that the electric flux (Φ) through a closed surface is proportional to the net charge (Q) enclosed by that surface. Mathematically, Gauss's Law is expressed as:

\[ \Phi = \frac{Q}{\epsilon_0} \]

where:
- \(\Phi\) is the electric flux through the closed surface,
- \(Q\) is the total charge enclosed within the surface,
- \(\epsilon_0\) is the permittivity of free space.

### Step-by-step Solution:

1. **Identify the Charge Enclosed**:
- Each surface (A, B, and C) has a charge \(+q\) at its center. Therefore, the charge enclosed by each surface is the same: \(Q = +q\).

2. **Apply Gauss's Law**:
- According to Gauss's Law, the electric flux through each surface is given by:
\[ \Phi = \frac{Q}{\epsilon_0} = \frac{q}{\epsilon_0} \]
- Since \(Q\) is the same for all three surfaces, we can conclude that the flux through each surface will also be the same.

3. **Calculate the Flux for Each Surface**:
- For Surface A (sphere of radius R):
\[ \Phi_A = \frac{q}{\epsilon_0} \]
- For Surface B (sphere of radius 2R):
\[ \Phi_B = \frac{q}{\epsilon_0} \]
- For Surface C (cube with side length R):
\[ \Phi_C = \frac{q}{\epsilon_0} \]

4. **Compare the Fluxes**:
- Since \(\Phi_A = \Phi_B = \Phi_C = \frac{q}{\epsilon_0}\), the flux through all three surfaces is equal.

5. **Rank the Surfaces**:
- Therefore, the ranking of the surfaces based on the electric field line flux through them is:
\[ \Phi_A = \Phi_B = \Phi_C \]
- All surfaces have the same flux.

### Final Answer:
The ranking of the surfaces based on the field line flux through them, from largest to smallest, is:
- All are equal (D).