Ten electrons are equally spaced and fixed around a circle of radius `R`. Relative to `V = 0` at infinity, the electrostatic potential `V` and the electric field `E` at the centre `C` are

Options

A. `V != 0 and vecE != 0`
B. `V != 0 and vecE = 0`
C. `V = 0 and vecE = 0`
D. `V = 0 and vecE != 0`

Explanation

To solve the problem of finding the electrostatic potential \( V \) and the electric field \( E \) at the center \( C \) of a circle with 10 electrons fixed around it, we can follow these steps:

### Step 1: Understanding the Configuration
We have 10 electrons, each with a charge of \( -e \) (where \( e \) is the elementary charge, approximately \( 1.6 \times 10^{-19} \) C), equally spaced around a circle of radius \( R \).

### Step 2: Calculate the Electrostatic Potential \( V \) at the Center
The electrostatic potential \( V \) at a point due to a point charge is given by the formula:
\[ V = \frac{k \cdot q}{r} \]
where \( k \) is Coulomb's constant (\( k \approx 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)), \( q \) is the charge, and \( r \) is the distance from the charge to the point where we are calculating the potential.

Since all electrons are at the same distance \( R \) from the center \( C \), the total potential at the center due to all 10 electrons is:
\[ V = \sum_{i=1}^{10} \frac{k \cdot (-e)}{R} \]
\[ V = 10 \cdot \frac{k \cdot (-e)}{R} = \frac{-10ke}{R} \]

### Step 3: Calculate the Electric Field \( E \) at the Center
The electric field \( E \) due to a point charge is given by:
\[ E = \frac{k \cdot q}{r^2} \]
However, since the electrons are symmetrically placed around the center, the electric fields due to each electron will cancel each other out at the center. Therefore, the net electric field \( E \) at point \( C \) is:
\[ E = 0 \]

### Final Results
Thus, the electrostatic potential \( V \) at the center \( C \) is:
\[ V = \frac{-10ke}{R} \]
And the electric field \( E \) at the center \( C \) is:
\[ E = 0 \]