Question

Two point charges placed a distance d apart in vacuum exert a force of magnitude F on each other. One of the two charges is doubled. To keep the magnitude of force same the separation between the charges should be changed to

• A. 2d
• B. d/2
• C. \(\sqrt{2}\) d
• D. d/\(\sqrt{2}\)

Hint:

In problems involving ratios and proportions like Coulomb's law (\(F \propto \frac{q_1 q_2}{d^2}\)), you can set up a ratio. If \(F\) is to remain constant, then the ratio \(\frac{q_1 q_2}{d^2}\) must also be constant. So, \(\frac{q_1 q_2}{d_{initial}^2} = \frac{(2q_1) q_2}{d_{final}^2}\), which quickly leads to \(d_{final}^2 = 2d_{initial}^2\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem applies Coulomb's Law, which describes the electrostatic force between two point charges. The force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

Step 2: Key Formula or Approach:
Coulomb's Law states that the force \(F\) between two charges \(q_1\) and \(q_2\) separated by a distance \(d\) is: \[ F = k \frac{q_1 q_2}{d^2} \] where \(k\) is Coulomb's constant. We will use this relationship to compare the initial and final situations.

Step 3: Detailed Explanation:
Initial situation:
The charges are \(q_1\) and \(q_2\), the distance is \(d\), and the force is \(F\). \[ F = k \frac{q_1 q_2}{d^2} \cdots (1) \] Final situation:
One charge is doubled, let's say the new charges are \(q'_1 = 2q_1\) and \(q'_2 = q_2\). The new distance is \(d'\). The force \(F'\) must be the same as the initial force \(F\). \[ F' = k \frac{(2q_1) q_2}{(d')^2} \cdots (2) \] Equating the forces:
We are given that \(F' = F\). Therefore, we can set equation (1) equal to equation (2): \[ k \frac{q_1 q_2}{d^2} = k \frac{2q_1 q_2}{(d')^2} \] The terms \(k\), \(q_1\), and \(q_2\) cancel out from both sides: \[ \frac{1}{d^2} = \frac{2}{(d')^2} \] Now, we solve for the new distance \(d'\): \[ (d')^2 = 2d^2 \] \[ d' = \sqrt{2d^2} = \sqrt{2} d \]

Step 4: Final Answer:
To maintain the same force after doubling one of the charges, the new separation distance must be \(\sqrt{2} d\).