Question

Two identical charges $ q $ are placed 1 m apart. The electrostatic force between them is $ 9 \times 10^{-9} \, \text{N} $. What is the magnitude of each charge? (Take $ k = 9 \times 10^9 \, \text{Nm}^2/\text{C}^2 $)

• A. \( 1 \times 10^{-9} \, \text{C} \)
• B. \( 3 \times 10^{-9} \, \text{C} \)
• C. \( 1 \times 10^{-9} \, \text{C} \)
• D. \( 3 \times 10^{-9} \, \text{C} \)

Hint:

To find the magnitude of identical charges when the force and separation are known, use Coulomb’s law: \[ F = \frac{k q^2}{r^2} \Rightarrow q = \sqrt{\frac{F r^2}{k}} \] Always isolate \( q^2 \) and then take the square root.

Answer 1

The Correct Option is A

Coulomb's Law is used to determine the magnitude of each charge. The law is stated as:

\[ F = \frac{k \cdot q_1 \cdot q_2}{r^2} \]

Where:

• \( F \) represents the electrostatic force between the charges.
• \( k \) is Coulomb's constant, equal to \( 9 \times 10^9 \, \text{Nm}^2/\text{C}^2 \).
• \( q_1 \) and \( q_2 \) denote the magnitudes of the charges.
• \( r \) is the distance between the charges, given as \( 1 \) m.

Given that the charges are identical (\( q_1 = q_2 = q \)) and the force \( F \) is \( 9 \times 10^{-9} \, \text{N} \), the formula becomes:

\[ 9 \times 10^{-9} = \frac{9 \times 10^9 \cdot q \cdot q}{1^2} \]

This simplifies to:

\[ 9 \times 10^{-9} = 9 \times 10^9 \cdot q^2 \]

Dividing both sides by \( 9 \times 10^9 \) yields:

\[ q^2 = \frac{9 \times 10^{-9}}{9 \times 10^9} \]

\[ q^2 = 1 \times 10^{-18} \]

Taking the square root of both sides gives:

\[ q = \sqrt{1 \times 10^{-18}} = 1 \times 10^{-9} \, \text{C} \]

Therefore, the magnitude of each charge is \( 1 \times 10^{-9} \, \text{C} \).