Eight point charge of charge `q` each are placed on the eight corners of a cube of side a. A solid neutral metallic sphere of radius ` a//3` is placed with its centre at the centre of the cube . As a result charges are induced on the sphere , which form certain pattern on its surface . What is the potential at the center of the sphere .

Options

A. `8/sqrt(3) (kq)/a`
B. `16/(sqrt3) (kq)/a`
C. `(-8)/sqrt(3) (kq)/a`
D. zero

Explanation

To find the potential at the center of the sphere due to the eight point charges placed at the corners of the cube, we can follow these steps:

### Step 1: Understand the Configuration
We have a cube with side length \( a \) and eight point charges \( q \) located at each corner of the cube. A neutral metallic sphere with radius \( \frac{a}{3} \) is placed at the center of the cube.

### Step 2: Determine the Distance from the Charges to the Center of the Sphere
The distance from the center of the cube (and the sphere) to any corner of the cube can be calculated using the geometry of the cube. The distance \( r \) from the center of the cube to a corner is given by:
\[ r = \frac{a}{2} \sqrt{3} \]
This is derived from the diagonal distance in a cube.

### Step 3: Calculate the Potential Due to One Charge at the Center of the Sphere
The potential \( V \) due to a single point charge \( q \) at a distance \( r \) is given by:
\[ V = \frac{kq}{r} \]
where \( k \) is Coulomb's constant.

### Step 4: Substitute the Distance into the Potential Formula
Substituting \( r = \frac{a}{2} \sqrt{3} \) into the potential formula, we get:
\[ V = \frac{kq}{\frac{a}{2} \sqrt{3}} = \frac{2kq}{a \sqrt{3}} \]

### Step 5: Calculate the Total Potential at the Center of the Sphere
Since there are 8 charges, the total potential \( V_0 \) at the center of the sphere is:
\[ V_0 = 8 \times \frac{2kq}{a \sqrt{3}} = \frac{16kq}{a \sqrt{3}} \]

### Step 6: Conclusion
Thus, the potential at the center of the sphere due to the eight point charges is:
\[ V_0 = \frac{16kq}{a \sqrt{3}} \]