How would you change the distance between two positively charged particles to increase the electric potential energy by a factor of 4?
A. Increase the distance by a factor of 4.
B. Increase the distance by a factor of 16.
C. Reduce the distance by a factor of 4.
D. Reduce the distance by a factor of 16.
Answer (2)
The electric potential energy between two charged particles is inversely proportional to the distance between them: U = k r q 1 q 2 .
To increase the electric potential energy by a factor of 4, we set up the equation U ′ = 4 U , where U ′ is the new potential energy.
We solve for the new distance r ′ in terms of the original distance r , finding r ′ = 4 r .
Therefore, to increase the electric potential energy by a factor of 4, we must reduce the distance by a factor of 4. The answer is C: Reduce the distance by a factor of 4.
Explanation
Problem Analysis and Formula Let's analyze the problem. We have two positively charged particles, and we want to change the distance between them such that the electric potential energy increases by a factor of 4. The electric potential energy between two charged particles is given by the formula: U = k r q 1 q 2 where:
U is the electric potential energy,
k is Coulomb's constant,
q 1 and q 2 are the magnitudes of the charges of the two particles,
r is the distance between the particles.
Setting up the Equation We want to find the new distance r ′ such that the new potential energy U ′ is 4 times the original potential energy U . So, we have: U ′ = 4 U k r ′ q 1 q 2 = 4 k r q 1 q 2
Simplifying the Equation Now, we can solve for r ′ in terms of r . Divide both sides of the equation by k q 1 q 2 :
r ′ 1 = r 4
Solving for the New Distance Taking the reciprocal of both sides, we get: r ′ = 4 r
Conclusion This means the new distance r ′ is the original distance r divided by 4. In other words, we need to reduce the distance by a factor of 4 to increase the electric potential energy by a factor of 4.
Examples
Imagine you're adjusting the distance between two magnets. The closer they are, the stronger the force between them. Similarly, with charged particles, reducing the distance increases the electric potential energy, making the interaction stronger. This principle is crucial in designing electronic components, understanding chemical bonds, and even in medical imaging techniques like MRI, where precise control over electromagnetic fields is essential.
To increase the electric potential energy between two positively charged particles by a factor of 4, the distance between them must be reduced by a factor of 4. Therefore, the correct answer is C: Reduce the distance by a factor of 4.
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