Four charges Q, q, Q and q are placed at the corners A, B, C and D of a square ABCD. If the resultant electric force on the charge at the corner C is zero, find the value of Q/q.

Answer

To solve the problem of finding the value of \( \frac{Q}{q} \) such that the resultant electric force on the charge at corner C of square ABCD is zero, we can follow these steps:

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

### Step 2: Identify Forces Acting on Charge at C
The charge at corner C experiences forces due to the charges at corners A, B, and D. We need to calculate these forces:
1. Force due to charge at A (F_AC)
2. Force due to charge at B (F_BC)
3. Force due to charge at D (F_DC)

### Step 3: Calculate the Force F_AC
The distance between A and C is the diagonal of the square, which is \( \sqrt{2}a \) (where \( a \) is the side length of the square). The force \( F_{AC} \) can be calculated using Coulomb's law:
\[ F_{AC} = k \frac{Q \cdot q}{(\sqrt{2}a)^2} = k \frac{Q \cdot q}{2a^2} \]
This force acts along the diagonal from A to C.

### Step 4: Calculate the Force F_BC
The distance between B and C is \( a \). The force \( F_{BC} \) is:
\[ F_{BC} = k \frac{q \cdot Q}{a^2} \]
This force acts horizontally towards B.

### Step 5: Calculate the Force F_DC
The distance between D and C is also \( a \). The force \( F_{DC} \) is:
\[ F_{DC} = k \frac{q \cdot Q}{a^2} \]
This force acts vertically towards D.

### Step 6: Set Up the Force Balance Equation
For the resultant force on charge C to be zero, the sum of the forces acting on it must equal zero. The forces F_BC and F_DC are both repulsive and act at right angles to each other. Thus, we can express the balance of forces as:
\[ F_{AC} = F_{BC} + F_{DC} \]

### Step 7: Substitute the Forces into the Equation
Substituting the expressions for the forces:
\[ k \frac{Q \cdot q}{2a^2} = k \frac{q \cdot Q}{a^2} + k \frac{q \cdot Q}{a^2} \]
This simplifies to:
\[ k \frac{Q \cdot q}{2a^2} = 2k \frac{q \cdot Q}{a^2} \]

### Step 8: Cancel Common Terms
We can cancel \( k \) and \( Q \) from both sides (assuming \( Q \neq 0 \) and \( k \neq 0 \)):
\[ \frac{q}{2a^2} = \frac{2q}{a^2} \]

### Step 9: Simplify the Equation
This leads to:
\[ Q = 4q \]

### Step 10: Find the Ratio \( \frac{Q}{q} \)
Thus, we find:
\[ \frac{Q}{q} = 4 \]

### Final Answer
The value of \( \frac{Q}{q} \) is \( 4 \).

---