If charges q/2 and 2q are placed at the centre of face and at the corner of a cube, then the total flux through the cube will be

Options

A. `(q)/(2epsilon_(@))`
B. `(q)/(epsilon_(@))`
C. `(q)/(6piepsilon_(0))`
D. `(q)/(8epsilon_(@))`

Answer

To solve the problem of finding the total electric flux through a cube when charges \( \frac{q}{2} \) and \( 2q \) are placed at the center of a face and at a corner of the cube respectively, we can follow these steps:

### Step 1: Identify the Charges and Their Positions
- Charge \( \frac{q}{2} \) is placed at the center of one face of the cube.
- Charge \( 2q \) is placed at one of the corners of the cube.

### Step 2: Calculate the Contribution of Charge \( \frac{q}{2} \)
- Since the charge \( \frac{q}{2} \) is at the center of a face, only half of this charge contributes to the flux inside the cube.
- Therefore, the charge enclosed by the cube from this face charge is:
\[ Q_{enclosed, face} = \frac{q}{2} \times \frac{1}{2} = \frac{q}{4} \]

### Step 3: Calculate the Contribution of Charge \( 2q \)
- The charge \( 2q \) is located at a corner of the cube. A charge at a corner of a cube is shared by 8 adjacent cubes.
- Thus, only \( \frac{1}{8} \) of this charge contributes to the flux inside the cube:
\[ Q_{enclosed, corner} = 2q \times \frac{1}{8} = \frac{2q}{8} = \frac{q}{4} \]

### Step 4: Calculate the Total Enclosed Charge
- Now, we can find the total charge enclosed within the cube by adding the contributions from both charges:
\[ Q_{total} = Q_{enclosed, face} + Q_{enclosed, corner} = \frac{q}{4} + \frac{q}{4} = \frac{2q}{4} = \frac{q}{2} \]

### Step 5: Use Gauss's Law to Find the Total Electric Flux
- According to Gauss's Law, the total electric flux \( \Phi \) through a closed surface is given by:
\[ \Phi = \frac{Q_{enclosed}}{\epsilon_0} \]
- Substituting the total enclosed charge:
\[ \Phi = \frac{\frac{q}{2}}{\epsilon_0} = \frac{q}{2\epsilon_0} \]

### Final Answer
The total electric flux through the cube is:
\[ \Phi = \frac{q}{2\epsilon_0} \]

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