The electric field at the centre of a square having equal charge q at each of the coner (side of square a )

Options

A. `(8kq)/(a^(2))`
B. `(2kq)/(a^(2))`
C. zero
D. `(4kq)/((a//sqrt2)^(2))`

Explanation

To find the electric field at the center of a square with equal charges \( q \) at each corner, we can follow these steps:

### Step 1: Understand the Configuration
We have a square with side length \( a \) and charges \( q \) placed at each of the four corners. The center of the square is equidistant from all four charges.

**Hint:** Visualize the square and label the corners with the charges.

### Step 2: Calculate the Distance from the Center to a Corner
The distance \( R \) from the center of the square to any corner can be calculated using the Pythagorean theorem. Since the center divides the square into two equal halves, we have:
\[ R = \frac{a}{\sqrt{2}} \]

**Hint:** Remember that the diagonal of the square can be used to find this distance.

### Step 3: Determine the Electric Field Due to One Charge
The electric field \( E \) due to a single charge \( q \) at a distance \( R \) is given by Coulomb's law:
\[ E = \frac{k \cdot q}{R^2} \]
where \( k \) is Coulomb's constant.

**Hint:** Make sure to substitute \( R \) from Step 2 into this formula.

### Step 4: Analyze the Direction of the Electric Fields
Each charge produces an electric field pointing away from the charge (if \( q \) is positive) or towards it (if \( q \) is negative). The electric fields due to opposite charges will have components that cancel each other out.

**Hint:** Draw vectors for the electric fields from each charge to visualize their directions.

### Step 5: Calculate the Net Electric Field
Since the electric fields from opposite corners cancel each other out, we can conclude that the net electric field at the center of the square is zero. Specifically:
- The electric fields due to charges at opposite corners will have equal magnitudes but opposite directions, leading to cancellation.

**Hint:** Consider symmetry; the configuration is symmetric, which simplifies the analysis.

### Conclusion
Thus, the electric field at the center of the square is:
\[ E_{\text{net}} = 0 \]

### Final Answer
The electric field at the center of the square is zero.

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