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 between two charges and separated by a distance is:
where 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 and , the distance is , and the force is .
Final situation:
One charge is doubled, let's say the new charges are and . The new distance is . The force must be the same as the initial force .
Equating the forces:
We are given that . Therefore, we can set equation (1) equal to equation (2):
The terms , , and cancel out from both sides:
Now, we solve for the new distance :
Step 4: Final Answer:
To maintain the same force after doubling one of the charges, the new separation distance must be .