Question

Two charges \( q_1 = 2 \, \mu \mathrm{C} \) and \( q_2 = -3 \, \mu \mathrm{C} \) are placed 10 cm apart in a vacuum. What is the magnitude and direction of the force between them?

• A. \( 5.4 \, \mathrm{N}, \, \text{Attractive} \)
• B. \( 5.4 \, \mathrm{N}, \, \text{Repulsive} \)
• C. \( 4.8 \, \mathrm{N}, \, \text{Attractive} \)
• D. \( 4.8 \, \mathrm{N}, \, \text{Repulsive} \)

Hint:

In Coulomb’s law, the force is attractive when the charges are of opposite signs and repulsive when the charges are of the same sign. Always convert the distance to meters in SI units for proper calculation.

Answer 1

The Correct Option is A

To find the force between two point charges, we apply **Coulomb's Law**, which states that the force \( F \) between two charges is given by the equation: \[ F = \frac{1}{4\pi\epsilon_0} \cdot \frac{|q_1 q_2|}{r^2}, \] where: \( q_1 \) and \( q_2 \) are the magnitudes of the charges, \( r \) is the distance between the charges, \( \frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \, \mathrm{N \, m^2 \, C^{-2}} \) is Coulomb's constant.
Step 1: Substitute the Given Values. We are given: \[ q_1 = 2 \, \mu \mathrm{C} = 2 \times 10^{-6} \, \mathrm{C}, \quad q_2 = -3 \, \mu \mathrm{C} = -3 \times 10^{-6} \, \mathrm{C}, \quad r = 10 \, \mathrm{cm} = 0.1 \, \mathrm{m}. \] Substitute these values into the formula: \[ F = 9 \times 10^9 \cdot \frac{|(2 \times 10^{-6})(-3 \times 10^{-6})|}{(0.1)^2}. \]
Step 2: Simplify the Calculation. First, calculate the numerator: \[ |q_1 q_2| = |(2 \times 10^{-6})(-3 \times 10^{-6})| = 6 \times 10^{-12}. \] Now, substitute into the formula: \[ F = 9 \times 10^9 \cdot \frac{6 \times 10^{-12}}{0.01}. \] Simplify: \[ F = 9 \cdot 6 \cdot 10^{-3} = 54 \times 10^{-3} = 5.4 \, \mathrm{N}. \]
Step 3: Determine the Direction. Since the charges \( q_1 \) and \( q_2 \) are opposite in sign (\( + \) and \( - \)), the force is **attractive**.
Conclusion: The magnitude of the force between the charges is \( 5.4 \, \mathrm{N} \), and the direction of the force is **attractive**. Thus, the correct answer is \( \mathbf{(1)} \).