Two charges 2 C and 6 C are separated by a finite distance. If a charge of-4 C is added to each of them,the initial force of `12 xx 10^(3)` N will change to

Options

A. `4 xx 10^(3)N` repulsion
B. `4 xx 10^(2)N` repulssion
C. `6 xx 10^(3)N` attraction
D. `4 xx 10^(3)N` attraction

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the Initial Conditions
We have two charges:
- Charge \( q_1 = 2 \, \text{C} \)
- Charge \( q_2 = 6 \, \text{C} \)

The initial force \( F \) between them is given as:
\[ F = 12 \times 10^3 \, \text{N} \]

### Step 2: Use Coulomb's Law to Find the Distance
Coulomb's Law states that the force \( F \) between two point charges is given by:
\[ F = k \frac{q_1 q_2}{r^2} \]
where \( k \) is Coulomb's constant, approximately \( 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \).

We can rearrange this formula to find \( r^2 \):
\[ r^2 = k \frac{q_1 q_2}{F} \]

Substituting the known values:
\[ r^2 = 9 \times 10^9 \frac{(2)(6)}{12 \times 10^3} \]

### Step 3: Calculate \( r^2 \)
Calculating the numerator:
\[ 2 \times 6 = 12 \]

Now substituting back:
\[ r^2 = 9 \times 10^9 \frac{12}{12 \times 10^3} \]
\[ r^2 = 9 \times 10^9 \frac{1}{10^3} \]
\[ r^2 = 9 \times 10^6 \]

Taking the square root to find \( r \):
\[ r = \sqrt{9 \times 10^6} = 3 \times 10^3 \, \text{m} \]

### Step 4: Update the Charges
Now, we add a charge of \( -4 \, \text{C} \) to each charge:
- New \( q_1 = 2 - 4 = -2 \, \text{C} \)
- New \( q_2 = 6 - 4 = 2 \, \text{C} \)

### Step 5: Calculate the New Force
Now, we will calculate the new force between the charges using the same formula:
\[ F' = k \frac{q_1' q_2'}{r^2} \]
Substituting the new charges:
\[ F' = 9 \times 10^9 \frac{(-2)(2)}{(3 \times 10^3)^2} \]

### Step 6: Calculate the New Force
Calculating the numerator:
\[ (-2) \times 2 = -4 \]

Now substituting back:
\[ F' = 9 \times 10^9 \frac{-4}{(3 \times 10^3)^2} \]
Calculating \( (3 \times 10^3)^2 = 9 \times 10^6 \):
\[ F' = 9 \times 10^9 \frac{-4}{9 \times 10^6} \]

### Step 7: Simplify the Expression
The \( 9 \) in the numerator and denominator cancels out:
\[ F' = 10^3 \times -4 = -4 \times 10^3 \, \text{N} \]

### Step 8: Interpret the Result
The negative sign indicates that the force is attractive. Thus, the new force is:
\[ F' = 4 \times 10^3 \, \text{N} \, \text{(attractive)} \]

### Final Answer
The final answer is:
\[ F' = 4 \times 10^3 \, \text{N} \, \text{(attractive)} \]

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