A charge `q muC` is placed at the centre of a cube of a side `0.1m`, then the electric flux diverging from each face of the cube is

Options

A. `(q xx 10^(-6))/(24 epsilon_(0))`
B. `(q xx 10^(-4))/(epsilon_(0))`
C. `(q xx 10^(-6))/(6 epsilon_(0))`
D. `(q xx 10^(-4))/(12 epsilon_(0))`

Answer

To solve the problem of finding the electric flux diverging from each face of a cube with a charge \( q \) (in microcoulombs) placed at its center, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Electric Flux**:
Electric flux (\( \Phi_E \)) through a surface is defined as the total electric field (\( E \)) passing through that surface. According to Gauss's Law, the total electric flux through a closed surface is proportional to the charge enclosed by that surface.

2. **Apply Gauss's Law**:
Gauss's Law states that:
\[ \Phi_E = \frac{Q}{\epsilon_0} \]
where \( Q \) is the total charge enclosed and \( \epsilon_0 \) is the permittivity of free space (\( \epsilon_0 \approx 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \)).

3. **Convert the Charge to Coulombs**:
Since the charge \( q \) is given in microcoulombs (µC), we need to convert it to coulombs:
\[ Q = q \times 10^{-6} \, \text{C} \]

4. **Calculate the Total Electric Flux**:
Substitute the value of \( Q \) into Gauss's Law:
\[ \Phi_E = \frac{q \times 10^{-6}}{\epsilon_0} \]

5. **Determine the Flux Through Each Face of the Cube**:
The cube has 6 faces, and by symmetry, the electric flux is equally distributed through each face. Therefore, the electric flux through each face (\( \Phi_{\text{face}} \)) is:
\[ \Phi_{\text{face}} = \frac{\Phi_E}{6} = \frac{q \times 10^{-6}}{6 \epsilon_0} \]

6. **Final Expression**:
Thus, the electric flux diverging from each face of the cube is:
\[ \Phi_{\text{face}} = \frac{q \times 10^{-6}}{6 \epsilon_0} \]