In a regular hexagon each corner is at a distance .r. from the centre. Identical charges of magnitude .Q. are placed at 5 corners. The field at the centre is `(K = (1)/(4pi in_(0)))`

Options

A. `KQ//r^(2)`
B. `(6KQ)/(r^(2))`
C. `(5KQ)/(r^(2))`
D. zero

Answer

To find the electric field at the center of a regular hexagon with identical charges placed at five of its corners, we can follow these steps:

### Step 1: Understand the Configuration
In a regular hexagon, all corners are equidistant from the center. Let’s denote the distance from the center to each corner as \( r \). We have identical charges \( Q \) placed at five corners of the hexagon.

### Step 2: Determine the Electric Field Contribution
The electric field \( E \) due to a point charge \( Q \) at a distance \( r \) is given by the formula:
\[ E = \frac{kQ}{r^2} \]
where \( k = \frac{1}{4\pi \epsilon_0} \).

### Step 3: Analyze the Directions of the Electric Fields
Since there are five charges, we need to consider the direction of the electric field produced by each charge at the center:
- The electric field due to each charge will point away from the charge (since they are positive charges).
- The symmetry of the hexagon means that some of these electric fields will cancel each other out.

### Step 4: Identify Canceling Pairs
- The electric fields from charges at opposite corners will cancel each other.
- For example, if we label the corners as 1, 2, 3, 4, 5, and 6 (where corner 6 has no charge), the pairs (1, 4) and (2, 5) will cancel each other out.

### Step 5: Calculate the Resultant Electric Field
The only charge that does not have a canceling counterpart is the charge at corner 3. The electric field due to this charge at the center will be:
\[ E_3 = \frac{kQ}{r^2} \]
This is the only contribution to the electric field at the center since the other fields cancel out.

### Step 6: Final Result
Thus, the total electric field at the center of the hexagon is simply:
\[ E_{total} = E_3 = \frac{kQ}{r^2} \]

### Summary
The electric field at the center of the hexagon, with charges at five corners, is given by:
\[ E = \frac{kQ}{r^2} \]