Questions: Electric Potential Energy

Question 1

Two protons are initially held 0.5 nm apart and then released from rest. As they move apart, what happens to the total mechanical energy of the system?

• A. Total mechanical energy is conserved; potential energy decreases as kinetic energy increases
• B. Total mechanical energy increases because the electric field does positive work on the protons
• C. Total mechanical energy decreases because the protons shed their potential energy
• D. Kinetic energy increases and so does potential energy, so total energy increases

Answer: (A)

The electric force is conservative, so total mechanical energy (KE + PE) is conserved. Initially, both protons are at rest (KE = 0) with positive potential energy U = kq²/r > 0. As they repel and fly apart, r increases, U decreases, and by conservation of energy the decrease in U equals the increase in KE. Total KE + U remains constant. The work done by the electric force equals −ΔU = ΔKE — it converts PE to KE, but conserves the total.

Question 2

An electron and a proton are separated by distance r. Their electric potential energy U = −ke²/r is negative. The most physically meaningful interpretation is:

• A. Energy must be added to the system to separate them to infinity — the configuration is bound
• B. The system has 'anti-energy' that can be used to do work on other objects
• C. The electron has negative energy and the proton has positive energy; they sum to zero
• D. There was an error; potential energy cannot be negative

Answer: (A)

Negative potential energy means the current configuration has less energy than the reference state (U = 0 at infinite separation). To move from negative U to U = 0, energy must be added — the pair is bound. This parallels gravitational potential energy for two masses: U_grav = −Gm₁m₂/r is also negative, and we are gravitationally bound to Earth in the same sense. Negative U does not mean 'negative energy' belonging to individual particles — it belongs to the configuration.

Question 3

The electric potential energy of two charges is a property of each individual charge, not of the pair.

• T. True
• F. False

Answer: False

Electric potential energy belongs to the system — the pair of charges — not to either charge individually. It represents the work done to assemble the configuration against the interaction force. If you bring charge q₁ from infinity in isolation, no work is needed (no field to work against). Only when you bring q₂ into the field of q₁ does work appear, stored in the configuration as a whole. The formula U = kq₁q₂/r depends on both charges and their separation, not on one alone.

Question 4

If a positive charge is released near a fixed positive charge, it will move from a region of high electric potential energy to a region of lower electric potential energy, and the electric force does positive work on it.

• T. True
• F. False

Answer: True

The relationship is W_electric = −ΔU. If U decreases (ΔU < 0), then W_electric = −ΔU > 0: the force does positive work. For two like charges, U = kq₁q₂/r > 0 and decreases as r increases. The electric force points outward (repulsion), in the same direction as the displacement — so positive work is done, decreasing PE and increasing KE. This mirrors a ball rolling downhill: the force acts in the direction of motion, converting PE to KE.

Question 5

Explain why electric potential energy is a property of a system of charges rather than of an individual charge. Use the concept of work done in assembling the system.

Model answer: Potential energy represents stored work — specifically, the work done against the interaction force to assemble the configuration from a reference state (charges infinitely far apart). Bringing a single charge from infinity requires no work because no force acts on it in isolation. Work only appears when a second charge is brought into the field of the first — that work is stored in the spatial relationship between the pair. U = kq₁q₂/r depends on both charges and their separation, confirming it is a shared property of the system, not of either charge alone.

A single isolated charge has no electric potential energy — there is nothing to interact with. The moment a second charge is introduced, the configuration has potential energy. For three charges, U = U₁₂ + U₁₃ + U₂₃, where each term belongs to one pair. The total equals the work to assemble all three charges from infinity: bring q₁ (free), bring q₂ into q₁'s field (work = U₁₂), bring q₃ into the fields of both (work = U₁₃ + U₂₃). The energy resides in the configuration, not in any individual charge.