Four point charges `-Q, -q, 2q` and `2Q` are placed, one at each corner of the square. The relation between `Q` and `q` for which the potential at the centre of the square is zero is

Options

A. `Q=-q`
B. `Q=-1/q`
C. `Q=q`
D. `Q=1/q`

Answer

To find the relation between the charges \( Q \) and \( q \) such that the electric potential at the center of the square is zero, we can follow these steps:

### Step 1: Understand the Configuration
We have four charges placed at the corners of a square:
- Charge at corner A: \( -Q \)
- Charge at corner B: \( -q \)
- Charge at corner C: \( 2q \)
- Charge at corner D: \( 2Q \)

### Step 2: Determine the Distance to the Center
For a square with side length \( a \), the distance from each corner to the center (O) is given by:
\[ R = \frac{a}{\sqrt{2}} \]

### Step 3: Calculate the Potential at the Center
The electric potential \( V \) at a point due to a point charge is given by:
\[ V = \frac{kQ}{r} \]
where \( k \) is Coulomb's constant, \( Q \) is the charge, and \( r \) is the distance from the charge to the point.

The total potential at the center O due to all four charges is:
\[ V_O = V_A + V_B + V_C + V_D \]
Substituting the values:
\[ V_O = \frac{k(-Q)}{R} + \frac{k(-q)}{R} + \frac{k(2q)}{R} + \frac{k(2Q)}{R} \]
Factoring out \( \frac{k}{R} \):
\[ V_O = \frac{k}{R} \left( -Q - q + 2q + 2Q \right) \]
Simplifying the expression inside the parentheses:
\[ V_O = \frac{k}{R} \left( Q + q \right) \]

### Step 4: Set the Total Potential to Zero
For the potential at the center to be zero:
\[ \frac{k}{R} (Q + q) = 0 \]
Since \( k \) and \( R \) are not zero, we can set the expression inside the parentheses to zero:
\[ Q + q = 0 \]

### Step 5: Solve for the Relation Between \( Q \) and \( q \)
This gives us:
\[ Q = -q \]

### Conclusion
The relation between \( Q \) and \( q \) for which the potential at the center of the square is zero is:
\[ Q = -q \]