Pictured are three point charges spaced equidistantly: All three charges have the same charge magnitude 0, but two charges are positive and one is negative, as shown in the figure. There are four points in the figure labeled A, B, C, and D. Let Ux denote the potential energy of the system when a negative test charge ~q is placed at some labeled point X ∈ {A, B, C, D}. Order the potential energies Ux from lowest to highest: Highest potential energy Lowest potential energy Answer Bank: UC UA UB UD
The Correct Answer and Explanation is:
To solve this problem, we must consider how electric potential energy behaves when a negative test charge (–q) is placed near other point charges.
Given:
- All three charges have magnitude Q.
- There are two positive charges (+Q) and one negative charge (–Q).
- The test charge is –q (a negative charge).
- The electric potential energy is given by: U=q⋅VU = q \cdot VU=q⋅V where VVV is the electric potential at the point, and q=−qq = -qq=−q here.
Key Concept:
- A positive potential (from nearby positive charges) times a negative test charge gives negative potential energy.
- A negative potential (from a nearby negative charge) times a negative test charge gives positive potential energy.
- Therefore, the test charge is most stable (lowest energy) when it is in regions of high electric potential created by positive charges.
Step-by-step:
- Point A: To the left of the first +Q. It is far from all charges, but closest to one +Q. The potential is relatively low and positive, so energy is slightly negative.
- Point B: Between the two +Q charges. This is a region of high positive potential because it is close to both positive charges. This makes UBU_BUB the most negative, or lowest potential energy.
- Point C: Between +Q and –Q. The positive and negative potentials cancel somewhat, giving a low net potential — so the energy is not very low.
- Point D: Closest to the –Q charge, and farther from the +Q charges. So the electric potential here is negative. Multiply that by –q, and you get a positive energy — so UDU_DUD is the highest.
Final Order from lowest to highest potential energy:
UB<UA<UC<UDU_B < U_A < U_C < U_DUB<UA<UC<UD
This matches the physical expectation: the negative test charge prefers to be near the positive charges, especially between them, and has the highest energy near the negative charge.
