A charge q is placed at the point of intersection of body diagonals of a cube. The electric flux passing through any one of its face is

Answer

To find the electric flux passing through one face of a cube when a charge \( q \) is placed at the intersection of the body diagonals, we can use Gauss's Law. Here’s a step-by-step solution:

### Step 1: Understand the Geometry of the Cube
The charge \( q \) is located at the center of the cube, which is the point of intersection of the body diagonals. A cube has 6 faces.

**Hint:** Visualize the cube and the position of the charge at its center.

### Step 2: Apply Gauss's Law
Gauss's Law states that the total electric flux \( \Phi \) through a closed surface is equal to the charge enclosed \( Q \) divided by the permittivity of free space \( \epsilon_0 \):
\[ \Phi = \frac{Q}{\epsilon_0} \]

**Hint:** Remember that Gauss's Law applies to closed surfaces.

### Step 3: Calculate the Total Electric Flux
Since the charge \( q \) is enclosed within the cube, the total electric flux through the entire surface of the cube is:
\[ \Phi_{\text{total}} = \frac{q}{\epsilon_0} \]

**Hint:** Consider the symmetry of the cube; the charge is at the center.

### Step 4: Determine the Flux Through One Face
Since the cube has 6 identical faces, the electric flux through one face can be found by dividing the total flux by the number of faces:
\[ \Phi_{\text{one face}} = \frac{\Phi_{\text{total}}}{6} = \frac{q}{6\epsilon_0} \]

**Hint:** Think about how the total flux is evenly distributed among all faces of the cube.

### Final Answer
Thus, the electric flux passing through any one of the faces of the cube is:
\[ \Phi = \frac{q}{6\epsilon_0} \]