Two point charges +2C and +6C repel each other with a force of 12N. If a charge q is given the each of the these charges then they attract with 4N. Then value q is

Options

A. A) `+4C`
B. B) `-2C`
C. C) `-4C`
D. D) `+2C`

Answer

To solve the problem step by step, let's break it down into manageable parts.

### Step 1: Understand the Given Information
We have two point charges:
- \( Q_1 = +2C \)
- \( Q_2 = +6C \)

Initially, they repel each other with a force of \( F_1 = 12N \).

### Step 2: Apply Coulomb's Law for the Initial Situation
According to Coulomb's Law, the force between two point charges is given by:
\[ F = k \frac{|Q_1 Q_2|}{r^2} \]
Where:
- \( k \) is Coulomb's constant,
- \( r \) is the distance between the charges.

Substituting the known values:
\[ 12 = k \frac{(2)(6)}{r^2} \]
This simplifies to:
\[ 12 = k \frac{12}{r^2} \]
From this, we can derive:
\[ k \frac{1}{r^2} = 1 \quad \text{(Equation 1)} \]

### Step 3: Modify Charges and Analyze the New Situation
Now, we add a charge \( q \) to each of the charges:
- New \( Q_1 = 2 + q \)
- New \( Q_2 = 6 + q \)

In this case, they attract each other with a force of \( F_2 = 4N \).

### Step 4: Apply Coulomb's Law for the New Situation
Using Coulomb's Law again for the new charges:
\[ 4 = k \frac{|(2 + q)(6 + q)|}{r^2} \]
Substituting \( k \frac{1}{r^2} = 1 \) from Equation 1:
\[ 4 = (2 + q)(6 + q) \]

### Step 5: Expand and Rearrange the Equation
Expanding the right-hand side:
\[ 4 = 12 + 2q + 6q + q^2 \]
This simplifies to:
\[ 4 = 12 + 8q + q^2 \]
Rearranging gives:
\[ q^2 + 8q + 12 - 4 = 0 \]
Thus:
\[ q^2 + 8q + 8 = 0 \]

### Step 6: Solve the Quadratic Equation
We can solve this quadratic equation using the quadratic formula:
\[ q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 1, b = 8, c = 8 \):
\[ q = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1} \]
Calculating the discriminant:
\[ q = \frac{-8 \pm \sqrt{64 - 32}}{2} = \frac{-8 \pm \sqrt{32}}{2} = \frac{-8 \pm 4\sqrt{2}}{2} \]
This simplifies to:
\[ q = -4 \pm 2\sqrt{2} \]

### Step 7: Determine the Value of \( q \)
Since we are looking for a charge \( q \), we consider the negative solution:
\[ q = -4 \]

### Conclusion
The value of charge \( q \) is \( -4C \).

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