Question

The electrostatic force between two point charges at a distance of separation \(d\) is \(F\). If one of the charge is moved away by a distance \(d/2\) then the force between them is

• A. \(\frac{2}{3} F\)
• B. \(\frac{9}{4} F\)
• C. \(\frac{4}{9} F\)
• D. \(\frac{3}{2} F\)
• E. \(\sqrt{2} F\)

Hint:

If the distance is increased by a factor of \(n\), the force decreases by a factor of \(n^2\). Here, distance becomes \(3/2\) times, so force becomes \((2/3)^2 = 4/9\) times.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem deals with Coulomb's Law, which describes the electrostatic force between two stationary point charges. The law states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.
Step 2: Key Formula or Approach:
Coulomb's Law is given by the formula:
\[ F = k \frac{q_1 q_2}{r^2} \] where \(F\) is the electrostatic force, \(q_1\) and \(q_2\) are the magnitudes of the charges, \(r\) is the distance between them, and \(k\) is Coulomb's constant.
Step 3: Detailed Explanation:
Let the two point charges be \(q_1\) and \(q_2\).
Initial situation:
The distance between the charges is \(d\).
The force \(F\) is given by:
\[ F = k \frac{q_1 q_2}{d^2} \quad \text{--- (1)} \] Final situation:
One of the charges is "moved away by a distance d/2". This means the initial distance \(d\) is increased by \(d/2\).
The new distance of separation, \(d'\), is:
\[ d' = d + \frac{d}{2} = \frac{2d + d}{2} = \frac{3d}{2} \] Now, we calculate the new force, \(F'\), using this new distance:
\[ F' = k \frac{q_1 q_2}{(d')^2} = k \frac{q_1 q_2}{\left(\frac{3d}{2}\right)^2} \] \[ F' = k \frac{q_1 q_2}{\frac{9d^2}{4}} \] \[ F' = \frac{4}{9} \left( k \frac{q_1 q_2}{d^2} \right) \quad \text{--- (2)} \] Now, substitute equation (1) into equation (2):
\[ F' = \frac{4}{9} F \] Step 4: Final Answer:
The new force between the charges is \(\frac{4}{9}F\), which corresponds to option (C). (Note: Options B and C are identical).