Which statement about the dependence of U12 on r12 is correct?

Which statement about the dependence of U12 on r12 is correct?

• A. U12 is independent of r12.
• B. U12 varies as 1/r12^2.
• C. U12 varies as 1/r12.
• D. U12 varies as r12.

Answer: (C)

The key idea is that the electrostatic potential energy between two point charges scales as 1 divided by their separation. Coulomb’s law gives the force as F ∝ 1/r^2, and the potential energy is the integral of the force with respect to distance, U ∝ ∫ F dr ∝ ∫ (1/r^2) dr ∝ 1/r. Therefore U12 depends on r12 as 1/r12. Doubling the separation halves the energy, and as the charges move infinitely far apart, the interaction energy goes to zero. The other options don’t fit this relationship: a distance-independent energy is not correct for Coulombic interaction; a 1/r^2 dependence applies to the force, not the potential energy; and a direct proportionality to r would imply energy grows with separation, which isn’t physical for this interaction.

The key idea is that the electrostatic potential energy between two point charges scales as 1 divided by their separation. Coulomb’s law gives the force as F ∝ 1/r^2, and the potential energy is the integral of the force with respect to distance, U ∝ ∫ F dr ∝ ∫ (1/r^2) dr ∝ 1/r. Therefore U12 depends on r12 as 1/r12.

Doubling the separation halves the energy, and as the charges move infinitely far apart, the interaction energy goes to zero. The other options don’t fit this relationship: a distance-independent energy is not correct for Coulombic interaction; a 1/r^2 dependence applies to the force, not the potential energy; and a direct proportionality to r would imply energy grows with separation, which isn’t physical for this interaction.