Each of the following diagrams shows an isolated charge, plus capital 𝑄, at the center of a circle. In each case, a test charge, plus lowercase 𝑞, is moved from rest at A to rest at B by an external force along the path shown. In which of these cases is the electric potential at B higher than at A? a) I only. b) III only. c) I and II only. d) II and III only. e) I, II, and III.
Each of the diagrams is a circle with the isolated charge plus capital 𝑄 located at the center. Because the charge is isolated, all of the electric fields and potentials that are relevant to this problem are due to this charge. The other charge, plus lowercase 𝑞, is a test charge, which means that we’re interested in how this charge interacts with the fields and potentials due to the charge plus capital 𝑄.
There’re also two points of interest in this problem, point A, where the test charge starts, and point B, to where the test charge is moved. The question asks us, which of these diagrams represents a situation where the electric field is higher at point B than at point A?
Remember that the electric potential of interest is going to come from the isolated charge plus capital 𝑄. So let’s figure out what that potential is. Just like we can associate an electric field with a charge, we can also associate an electric potential. The electric potential is a series of scalars, that is, numbers, associated with the points in space. In such a way that they encode both the size and direction of the corresponding electric field at those points.
For an isolated charge like our charge plus 𝑄 in the problem, the electric potential at a point is given by. 𝑉, the electric potential, is equal to 𝑘, a constant approximately equal to nine times 10 to the ninth newtons meters squared per coulomb squared, times 𝑄, the charge, divided by 𝑟, the distance between the charge and the point of interest. We won’t discuss how exactly the electric potential encodes the electric field.
Note, however, that the expression for the magnitude of the electric field is very similar to the expression for the potential. The only difference being that the magnitude of the electric field has a factor of 𝑟 squared in the denominator, while the electric potential has a factor of just 𝑟.
Speaking of 𝑟 in the denominator, let’s make some observations about the electric potential of an isolated charge. Because the electric potential is inversely proportional to the distance away from the charge, as the distance increases, the electric potential decreases, and vice versa. On the other hand, if two points are the same distance away from the charge, then the electric potential is the same at both points.
Now let’s look at the diagrams. Each diagram has a circle. Since the circumference of a circle is defined as all of the points at distance 𝑟 from the center. In each diagram, all the points on the circumference of the circle are the same distance away from plus capital 𝑄 in the center. Remember that points that are the same distance away from an isolated charge have the same electric potential.
In diagram II, points A and B both lie on the circumference of the circle, which means they’re both the same distance away from the charge plus capital 𝑄 in the center of the circle. And so they have the same electric potential. This means that diagram number II does not represent a situation where the electric potential is higher at point B since the electric potential is the same at points A and B.
By the same logic, in diagram number III, points A and B both lie on the circumference of the circle, which means their electric potentials are the same. Because they’re the same distance from the charge plus capital 𝑄 in the center. Therefore, diagram number III is also not a situation where the electric potential is higher at point B.
This leaves only diagram I as a possible situation where the electric potential is higher at point B than at point A. And this is indeed the case. Point A lies on the circumference of the circle, while point B lies closer to the center. Point A then is farther from the charge than point B and so has a lower electric potential. Thus, point B indeed does have a higher electric potential than point A. And diagram I is one of these cases.
Looking at our results, we see that diagram I does represent a situation where the electric potential is higher at point B than at point A. And diagrams II and III do not. Thus, the correct answer is choice a, I only.
If we didn’t remember these two facts about the electric potential or the formula that we used to figure these facts out, we could’ve still guessed at the correct answer. We observed that an isolated charge has rotational symmetry, because it doesn’t matter what direction we turn the charge, it looks the same. Due to this symmetry, there should be no preference for particular orientations of the charge. In other words, quantities of interest should depend only on distance from the charge and not orientation about the charge.
Therefore, based on this observation about rotational symmetry, we would reason that points that are the same distance from the charge should have the same potential. We would then argue as we did before that points on the circumference of the circle have the same electric potential. This observation, just like before, would eliminate diagrams II and III since, in both diagrams, points A and B both lie on the circumference of the circle. Then, by eliminating answer choices that include diagrams II or III, we would be left with only answer choice a as the correct answer.