Equal charges `q` are placed at the four corners `A,B,C,D`of a square of length `a`. The magnitude of the force on the charge at `B` will be

Options

A. `(3q^(2))/(4piepsilon_(0)a^(2))`
B. `(4q^(2))/(4piepsilon_(0)a^(2))`
C. `((1+2sqrt(2))/(2))(q^(2))/(4piepsilon_(0)a^(2))`
D. `(2+(1)/(sqrt(2)))(q^(2))/(4piepsilon_(0)a^(2))`

Explanation

To find the magnitude of the force on the charge at corner B of a square with equal charges placed at each corner, we can follow these steps:

### Step 1: Identify the charges and their positions
We have four equal charges \( q \) placed at the corners of a square ABCD, where the length of each side of the square is \( a \). The charges are located at:
- A (top left)
- B (top right)
- C (bottom right)
- D (bottom left)

### Step 2: Determine the forces acting on charge B
The charge at B will experience repulsive forces due to the charges at A, C, and D. We need to calculate the magnitude of these forces.

### Step 3: Calculate the force due to charge A on charge B
The distance between charges A and B is \( a \).
Using Coulomb's law, the force \( F_{AB} \) exerted by charge A on charge B is given by:
\[ F_{AB} = k \frac{q^2}{a^2} \]
where \( k \) is Coulomb's constant.

### Step 4: Calculate the force due to charge C on charge B
The distance between charges B and C is also \( a \).
The force \( F_{BC} \) exerted by charge C on charge B is:
\[ F_{BC} = k \frac{q^2}{a^2} \]

### Step 5: Calculate the force due to charge D on charge B
The distance between charges B and D is \( a \).
The force \( F_{BD} \) exerted by charge D on charge B is:
\[ F_{BD} = k \frac{q^2}{a^2} \]

### Step 6: Determine the directions of the forces
- The force \( F_{AB} \) acts to the left (towards A).
- The force \( F_{BC} \) acts to the right (towards C).
- The force \( F_{BD} \) acts downward (towards D).

### Step 7: Resolve the forces into components
Since \( F_{AB} \) and \( F_{BD} \) are perpendicular, we can use the Pythagorean theorem to find the resultant force \( F_{net} \) acting on charge B due to charges A and D.

### Step 8: Calculate the resultant force from A and D
The resultant force \( F_{AD} \) can be calculated as:
\[ F_{AD} = \sqrt{F_{AB}^2 + F_{BD}^2} = \sqrt{(k \frac{q^2}{a^2})^2 + (k \frac{q^2}{a^2})^2} = \sqrt{2} \cdot k \frac{q^2}{a^2} \]

### Step 9: Combine the forces
Now, we need to consider the force \( F_{BC} \) acting horizontally. The total force \( F_{net} \) on charge B is:
\[ F_{net} = F_{BC} + F_{AD} = k \frac{q^2}{a^2} + \sqrt{2} \cdot k \frac{q^2}{a^2} \]
\[ F_{net} = k \frac{q^2}{a^2} (1 + \sqrt{2}) \]

### Step 10: Final expression
Thus, the magnitude of the net force acting on charge B is:
\[ F_{net} = k \frac{q^2}{a^2} (1 + \sqrt{2}) \]