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Two massive, positively charged particles are initially held a fixed distance apart. When they are moved farther apart, the magnitude of their mutual gravitational force changes by a factor of n. Which of the following indicates the factor by which the magnitude of their mutual electrostatic force changes?
  1. $\dfrac{1}{n^2}$
  2. $\dfrac{1}{n}$
  3. $n$
  4. $n^2$
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OpenStax College Physics for AP® Courses Solution, Chapter 18, Problem 28 (Test Prep for AP® Courses) (2:26)

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This is College Physics Answers with Shaun Dychko. Two massive positively charged particles are initially held some distance apart and then they are moved further apart and the size of the gravity between them changes by a factor of n, which of the following is going to be the factor by which the size of their electrostatic force changes? So in the first case, the gravity will be the gravitational constant times the mass of the first particle times the mass of the second particle divided by the distance between them squared and then the second force of gravity after they move further apart it's going to be the same numerator because the masses are the same but divided by a new distance r 2 squared and we are told that this force is n times the first force of gravity and we can substitute in Gm 1m 2 over r 1 squared in place of Fg 1 and then we can solve for r 2 squared in terms of r 1 squared and we can multiply this by r 2 squared over n and then multiply this by r 2 squared over n and we are also dividing both sides by this numerator Gm 1m 2 and you end up with this here, which says r 2 squared equals r 1 squared over n so this tells us the factor by which the square of the distance changes and that's useful in this formula here, where we consider the magnitude of the electrostatic force between them. So in the first case, it will be kq 1q 2 over r 1 squared and in the second case, it will be the same numerator because it's the same charge divided by some new distance squared but we know that r 2 squared is r 1 squared over n— that's what we learned from considering the case of gravity— so when we substitute that in, we can multiply top and bottom by n and we get n times kq 1q 2 over r 1 squared and this is the first case electrostatic force and it's different from the second case by this factor n and the answer is (c). Now we know that n is going to be some number less than 1 by the way, it's a little bit confusing and you might be thinking well since things are going to decrease, I should have to divide by n but that's presuming that n is a number greater than 1 if you were to say that but n could be anything... it could be one-half and so on so there we go!