How does the electric force between two charged particles change if the distance between them is reduced by a factor of 2?
A. It is reduced by a factor of 4.
B. It is increased by a factor of 2.
C. It is reduced by a factor of 2.
D. It is increased by a factor of 4.
Answer (1)
The electric force is inversely proportional to the square of the distance between charged particles.
When the distance is reduced by a factor of 2, the new distance becomes 2 r .
The new electric force F ′ is calculated as F ′ = k ( 2 r ) 2 q 1 q 2 = 4 k r 2 q 1 q 2 .
The electric force is increased by a factor of 4, so the answer is D .
Explanation
Problem Analysis Let's analyze the problem. The electric force between two charged particles is described by Coulomb's law, which states that the force is inversely proportional to the square of the distance between the particles. We need to determine how the force changes when the distance is reduced by a factor of 2.
Coulomb's Law Let F be the initial electric force and r be the initial distance between the charged particles. According to Coulomb's law, we have: F = k r 2 q 1 q 2 where k is Coulomb's constant, and q 1 and q 2 are the charges of the particles.
New Distance Now, let's reduce the distance by a factor of 2. The new distance r ′ is given by: r ′ = 2 r We want to find the new electric force F ′ in terms of the original force F .
New Electric Force The new electric force F ′ is given by: F ′ = k ( r ′ ) 2 q 1 q 2 = k ( 2 r ) 2 q 1 q 2 = k 4 r 2 q 1 q 2 = 4 k r 2 q 1 q 2 Since F = k r 2 q 1 q 2 , we can substitute this into the equation for F ′ :
F ′ = 4 F
Conclusion Therefore, the new electric force F ′ is 4 times the original electric force F . This means the electric force is increased by a factor of 4.
Examples
Imagine you have two magnets. When you bring them closer, the force between them increases. If you halve the distance between them, the magnetic force (which behaves similarly to the electric force) becomes four times stronger. This principle is crucial in many applications, from designing efficient electric motors to understanding how atoms interact to form molecules. Understanding how forces change with distance helps engineers and scientists create and improve technologies.
