A charge 'Q' is placed at each corner of a cube of side 'a'. The potential at the centre of the cube is

Options

A. `(8Q)/(pi epsi_(0)a)`
B. `(4Q)/(4pi epsi_(0)a)`
C. `(4Q)/(sqrt3 pi epsi_(0)a)`
D. `(2Q)/(pi epsi_(0)a)`

Explanation

To find the electric potential at the center of a cube with a charge \( Q \) placed at each corner, we can follow these steps:

### Step 1: Understand the configuration
A cube has 8 corners, and we have a charge \( Q \) at each corner. The goal is to find the total electric potential at the center of the cube due to these charges.

### Step 2: Formula for electric potential
The electric potential \( V \) due to a point charge \( Q \) at a distance \( r \) is given by the formula:
\[ V = \frac{KQ}{r} \]
where \( K \) is the Coulomb's constant, \( K = \frac{1}{4\pi \epsilon_0} \).

### Step 3: Calculate the distance from a corner to the center of the cube
The distance from a corner of the cube to the center can be calculated using the geometry of the cube. The length of the diagonal of the cube is given by:
\[ d = a\sqrt{3} \]
Thus, the distance from a corner to the center (which is half the diagonal) is:
\[ r = \frac{d}{2} = \frac{a\sqrt{3}}{2} \]

### Step 4: Calculate the potential due to one charge at the center
Using the formula for electric potential, the potential at the center due to one charge \( Q \) located at a corner is:
\[ V_{\text{one charge}} = \frac{KQ}{r} = \frac{KQ}{\frac{a\sqrt{3}}{2}} = \frac{2KQ}{a\sqrt{3}} \]

### Step 5: Calculate the total potential at the center
Since there are 8 charges, the total potential \( V_c \) at the center of the cube is the sum of the potentials due to all 8 charges:
\[ V_c = 8 \cdot V_{\text{one charge}} = 8 \cdot \frac{2KQ}{a\sqrt{3}} = \frac{16KQ}{a\sqrt{3}} \]

### Step 6: Substitute the value of \( K \)
Substituting \( K = \frac{1}{4\pi \epsilon_0} \) into the equation gives:
\[ V_c = \frac{16 \cdot \frac{1}{4\pi \epsilon_0} \cdot Q}{a\sqrt{3}} = \frac{4Q}{\pi \epsilon_0 a\sqrt{3}} \]

### Final Answer
Thus, the potential at the center of the cube is:
\[ V_c = \frac{4Q}{\pi \epsilon_0 a\sqrt{3}} \]

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