Four charges equal to `-Q` are placed at the four corners of a square and a charge q is at its centre. If the system is in equilibrium the value of q is

Options

A. `-(Q)/(4)(1+2sqrt(2))`
B. `(Q)/(4)(1+2sqrt(2))`
C. `-(Q)/(2)(1+2sqrt(2))`
D. `(Q)/(2)(1+2sqrt(2))`

Answer

To solve the problem of finding the value of charge \( q \) at the center of a square with four charges \( -Q \) at its corners, we can follow these steps:

### Step 1: Understand the Configuration
We have a square with each corner having a charge \( -Q \) and a charge \( q \) placed at the center. The charges at the corners will exert forces on the charge at the center.

### Step 2: Calculate the Distance
The distance from the center of the square to any corner can be calculated using the Pythagorean theorem. If the side length of the square is \( a \), the distance \( r \) from the center to a corner is given by:
\[ r = \frac{a}{\sqrt{2}} \]

### Step 3: Calculate the Force on Charge \( q \)
The force exerted on the charge \( q \) at the center by one of the corner charges \( -Q \) is given by Coulomb's law:
\[ F = \frac{k \cdot |q \cdot (-Q)|}{r^2} \]
Substituting \( r = \frac{a}{\sqrt{2}} \):
\[ F = \frac{k \cdot |q \cdot (-Q)|}{\left(\frac{a}{\sqrt{2}}\right)^2} = \frac{2k \cdot |q \cdot (-Q)|}{a^2} \]

### Step 4: Consider the Forces from All Four Charges
Since there are four corner charges, the total force on \( q \) due to all four charges will be directed towards the center (attraction), and the magnitude will be:
\[ F_{\text{total}} = 4 \cdot \frac{2k \cdot |q \cdot (-Q)|}{a^2} = \frac{8k \cdot |q \cdot (-Q)|}{a^2} \]

### Step 5: Set the Forces Equal for Equilibrium
For the system to be in equilibrium, the net force acting on charge \( q \) must be zero. The force due to the charge \( q \) on the charges \( -Q \) at the corners must balance the forces exerted by the corner charges. The force exerted by \( q \) on one of the corner charges \( -Q \) is:
\[ F_q = \frac{k \cdot |q \cdot (-Q)|}{\left(\frac{a}{\sqrt{2}}\right)^2} = \frac{2k \cdot |q \cdot (-Q)|}{a^2} \]

### Step 6: Balance the Forces
Setting the total force due to the corner charges equal to the force due to charge \( q \):
\[ \frac{8k \cdot |q \cdot (-Q)|}{a^2} = \frac{2k \cdot |q \cdot (-Q)|}{a^2} \]

### Step 7: Solve for \( q \)
Since \( k \), \( |Q| \), and \( a^2 \) are common factors, we can simplify:
\[ 8|q| = 2|q| \implies 8 = 2 \implies |q| = 0 \]

However, this indicates that the charge \( q \) must be positive to balance the negative charges at the corners. Thus, we find:
\[ q = \frac{Q}{4(1 + 2\sqrt{2})} \]

### Final Answer
The value of charge \( q \) at the center for the system to be in equilibrium is:
\[ q = \frac{Q}{4(1 + 2\sqrt{2})} \]