The force between two charges 4C and -2C which are separated by a distance of 3km is

Options

A. `9 xx 10^(3)N`
B. `24 xx 10^(3)N`
C. `8 xx 10^(3)N`
D. `4 xx 10^(3)N`

Answer

To find the force between two charges using Coulomb's law, we can follow these steps:

### Step 1: Write down Coulomb's Law
Coulomb's law states that the force \( F \) between two point charges is given by the formula:
\[ F = \frac{1}{4 \pi \epsilon_0} \cdot \frac{q_1 q_2}{r^2} \]
where:
- \( F \) is the force between the charges,
- \( q_1 \) and \( q_2 \) are the magnitudes of the charges,
- \( r \) is the distance between the charges,
- \( \epsilon_0 \) is the permittivity of free space, approximately equal to \( 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \).

### Step 2: Substitute the values
Given:
- \( q_1 = 4 \, \text{C} \)
- \( q_2 = -2 \, \text{C} \)
- Distance \( r = 3 \, \text{km} = 3000 \, \text{m} \)

Now, substituting the values into the formula:
\[ F = \frac{1}{4 \pi \epsilon_0} \cdot \frac{(4)(-2)}{(3000)^2} \]

### Step 3: Calculate \( r^2 \)
Calculate the square of the distance:
\[ r^2 = (3000 \, \text{m})^2 = 9000000 \, \text{m}^2 = 9 \times 10^6 \, \text{m}^2 \]

### Step 4: Calculate the force
Now substitute \( r^2 \) back into the equation:
\[ F = \frac{1}{4 \pi \epsilon_0} \cdot \frac{(4)(-2)}{9 \times 10^6} \]

Using \( \frac{1}{4 \pi \epsilon_0} \approx 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \):
\[ F = 9 \times 10^9 \cdot \frac{-8}{9 \times 10^6} \]

### Step 5: Simplify the equation
Now simplify:
\[ F = 9 \times 10^9 \cdot \frac{-8}{9 \times 10^6} = -8 \times 10^3 \, \text{N} \]

### Step 6: State the final answer
The force between the two charges is:
\[ F = -8000 \, \text{N} \]
The negative sign indicates that the force is attractive (since one charge is positive and the other is negative).