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 \( q \) placed at its center, we can follow these steps:

### Step 1: Understand Gauss's Law
Gauss's Law states that the total electric flux \( \Phi_E \) through a closed surface is equal to the charge \( Q \) enclosed by the surface divided by the permittivity of free space \( \epsilon_0 \):
\[ \Phi_E = \frac{Q}{\epsilon_0} \]

### Step 2: Identify the Charge and the Surface
In this case, we have a charge \( q \) (in microcoulombs) placed at the center of a cube. The cube acts as a closed surface.

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

### Step 4: Calculate the Total Electric Flux
Using Gauss's Law, we can calculate the total electric flux through the entire surface of the cube:
\[ \Phi_E = \frac{q \times 10^{-6}}{\epsilon_0} \]

### Step 5: Determine the Flux Through Each Face
A cube has 6 faces, and since the charge is symmetrically placed at the center, the electric flux will be uniformly distributed across all faces. Therefore, the electric flux through each face \( \Phi_{\text{face}} \) is given by:
\[ \Phi_{\text{face}} = \frac{\Phi_E}{6} = \frac{q \times 10^{-6}}{6 \epsilon_0} \]

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