A proton sits at the origin. Point A is 5 cm above the proton and point B is 1.5 cm to the right of the proton. The change in electrical potential energy of a 2.0 nC test charge moving from A to B is 3.15×10^-5 J. Which of the following statements is true about the potential energy at A and B?

• A. U_B > U_A
• B. U_B < U_A
• C. U_B = U_A
• D. Not enough information

Answer: (C)

The key idea is how potential energy in the field of a point charge depends on distance. For a positive source charge like a proton, the electric potential V goes as kQ/r, so it increases as you get closer to the charge (because 1/r grows as r shrinks). The potential energy of a test charge q in that field is U = qV, so for a positive test charge, moving closer to the proton raises its potential energy.

Here, A is 5 cm from the proton (r_A = 0.05 m) and B is 1.5 cm from the proton (r_B = 0.015 m). Since r_B < r_A, the potential at B is higher than at A, and with q = +2.0 nC (positive), the potential energy at B is greater than at A: U_B > U_A. The stated positive change in potential energy from A to B is consistent with this, because a positive ΔU means V_B > V_A when the test charge is positive.

So the true statement is that the potential energy at B is greater than at A. The numerical ΔU given would imply a very large potential difference, which doesn’t align with these distances for a proton’s field, but the directional conclusion remains that U_B > U_A.