A charge Q is place at each of the opposite corners of a square. A charge q is placed at each of the other two corners. If the net electrical force on Q is zero, then `Q//q` equals:

Options

A. (a) `-1`
B. (b) `1`
C. (c) `-(1)/(sqrt2)`
D. (d) `-2sqrt2`

Explanation

To solve the problem, we need to analyze the forces acting on the charge \( Q \) placed at the corners of the square. Let's denote the side length of the square as \( L \).

### Step-by-Step Solution:

1. **Identify the Configuration**:
- We have a square with charges \( Q \) at two opposite corners (let's say A and C) and charges \( q \) at the other two corners (B and D).
- The configuration looks like this:
- A(Q) ------ B(q)
- | |
- D(q) ------ C(Q)

2. **Calculate Forces on Charge \( Q \)**:
- The force \( F_1 \) on charge \( Q \) at corner A due to charge \( q \) at corner B is given by Coulomb's law:
\[ F_1 = \frac{1}{4\pi \epsilon_0} \cdot \frac{Qq}{L^2} \]
- The force \( F_2 \) on charge \( Q \) at corner A due to charge \( q \) at corner D is also:
\[ F_2 = \frac{1}{4\pi \epsilon_0} \cdot \frac{Qq}{L^2} \]

3. **Direction of Forces**:
- The force \( F_1 \) acts horizontally towards charge \( q \) at B.
- The force \( F_2 \) acts vertically towards charge \( q \) at D.
- Since these forces are perpendicular to each other, we can use the Pythagorean theorem to find the resultant force \( F_r \) on charge \( Q \):
\[ F_r = \sqrt{F_1^2 + F_2^2} = \sqrt{2F_1^2} = F_1\sqrt{2} \]

4. **Net Force Condition**:
- For the net force on charge \( Q \) to be zero, the resultant force \( F_r \) must be balanced by the force due to the other charge \( Q \) at corner C (which is \( F_3 \)):
\[ F_r + F_3 = 0 \]
- The force \( F_3 \) on charge \( Q \) at corner A due to charge \( Q \) at corner C is:
\[ F_3 = \frac{1}{4\pi \epsilon_0} \cdot \frac{Q^2}{(L\sqrt{2})^2} = \frac{1}{4\pi \epsilon_0} \cdot \frac{Q^2}{2L^2} \]

5. **Setting Up the Equation**:
- Now we can set up the equation:
\[ F_1\sqrt{2} = \frac{Q^2}{8\pi \epsilon_0 L^2} \]
- Substituting \( F_1 \):
\[ \sqrt{2} \cdot \frac{Qq}{4\pi \epsilon_0 L^2} = \frac{Q^2}{8\pi \epsilon_0 L^2} \]

6. **Simplifying the Equation**:
- Canceling out common terms:
\[ \sqrt{2} \cdot q = \frac{Q}{2} \]
- Rearranging gives:
\[ \frac{Q}{q} = 2\sqrt{2} \]

### Final Result:
Thus, the ratio \( \frac{Q}{q} \) is:
\[ \frac{Q}{q} = 2\sqrt{2} \]