Under what conditions can the electric flux `phi_E` be found through a closed surface?

Options

A. If the magnitude of the electric field is known everywhere on the surface.
B. If the total charge inside the surface is specified.
C. If the total charge outside the surface is specified.
D. Only if the location of each point charge inside the surface is specified.

Explanation

To determine the conditions under which the electric flux \( \phi_E \) can be found through a closed surface, we can analyze the options provided in the question.

### Step-by-Step Solution:

1. **Understanding Electric Flux**:
Electric flux \( \phi_E \) through a closed surface is defined as the integral of the electric field \( \vec{E} \) over the surface area \( A \):
\[ \phi_E = \int \vec{E} \cdot d\vec{A} \]
According to Gauss's Law, the electric flux through a closed surface is directly related to the charge enclosed within that surface.

2. **Analyzing the Options**:
- **Option A**: If the magnitude of the electric field is known everywhere on the surface.
- While knowing the electric field everywhere allows for the calculation of flux, it is not a necessary condition according to Gauss's Law. The flux can still be determined if the enclosed charge is known.

- **Option B**: The total charge inside the surface is specified.
- This option aligns with Gauss's Law, which states that the electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space (\( \epsilon_0 \)):
\[ \phi_E = \frac{Q_{\text{enclosed}}}{\epsilon_0} \]
Therefore, knowing the total charge inside the surface is sufficient to find the electric flux.

- **Option C**: The total charge outside the surface is specified.
- The charge outside the surface does not affect the electric flux through the closed surface. According to Gauss's Law, only the charge enclosed matters.

- **Option D**: Only if the location of each point charge inside is specified.
- While knowing the location of charges can help in calculating the electric field, it is not necessary to specify the locations of charges to find the electric flux, as long as the total enclosed charge is known.

3. **Conclusion**:
The correct condition under which the electric flux \( \phi_E \) can be found through a closed surface is:
- **Option B**: The total charge inside the surface is specified.