---
title: "In a region of space, source charges produce a static electric potential field.  Four identical protons, each having initial kinetic energy \\(K_0 = 100\\text{ eV}\\), travel along different paths between the initial and final electric potentials listed in the table, subject only to conservative electrostatic forces.  | Path | Initial Potential \\(V_i\\) | Final Potential \\(V_f\\) | Path Length | | :— | :— | :— | :— | | 1 | \\(+60\\text{ V}\\) | \\(+20\\text{ V}\\) | \\(0.10\\text{ m}\\) | | 2 | \\(+60\\text{ V}\\) | \\(+20\\text{ V}\\) | \\(0.35\\text{ m}\\) | | 3 | \\(+40\\text{ V}\\) | \\(-30\\text{ V}\\) | \\(0.20\\text{ m}\\) | | 4 | \\(+30\\text{ V}\\) | \\(+10\\text{ V}\\) | \\(0.15\\text{ m}\\) |  Which of the following correctly ranks the net change in kinetic energy \\(\\Delta K\\) of the protons along the four paths?"
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url: "https://nerd-notes.com/ubq/124635/"
date_modified: "2026-09-28T14:08:55+00:00"
---

# In a region of space, source charges produce a static electric potential field.

Four identical protons, each having initial kinetic energy \(K_0 = 100\text{ eV}\), travel along different paths between the initial and final electric potentials listed in the table, subject only to conservative electrostatic forces.

| Path | Initial Potential \(V_i\) | Final Potential \(V_f\) | Path Length |
| :— | :— | :— | :— |
| 1 | \(+60\text{ V}\) | \(+20\text{ V}\) | \(0.10\text{ m}\) |
| 2 | \(+60\text{ V}\) | \(+20\text{ V}\) | \(0.35\text{ m}\) |
| 3 | \(+40\text{ V}\) | \(-30\text{ V}\) | \(0.20\text{ m}\) |
| 4 | \(+30\text{ V}\) | \(+10\text{ V}\) | \(0.15\text{ m}\) |

Which of the following correctly ranks the net change in kinetic energy \(\Delta K\) of the protons along the four paths?

In a region of space, source charges produce a static electric potential field.

Four identical protons, each having initial kinetic energy \(K_0 = 100\text{ eV}\), travel along different paths between the initial and final electric potentials listed in the table, subject only to conservative electrostatic forces.

| Path | Initial Potential \(V_i\) | Final Potential \(V_f\) | Path Length |
| :--- | :--- | :--- | :--- |
| 1 | \(+60\text{ V}\) | \(+20\text{ V}\) | \(0.10\text{ m}\) |
| 2 | \(+60\text{ V}\) | \(+20\text{ V}\) | \(0.35\text{ m}\) |
| 3 | \(+40\text{ V}\) | \(-30\text{ V}\) | \(0.20\text{ m}\) |
| 4 | \(+30\text{ V}\) | \(+10\text{ V}\) | \(0.15\text{ m}\) |

Which of the following correctly ranks the net change in kinetic energy \(\Delta K\) of the protons along the four paths?

![A two-dimensional schematic showing five vertical parallel dashed lines representing equipotential lines, spaced horizontally across a rectangular region. The lines are labeled along their top ends from left to right as \(-30\text{ V}\), \(+10\text{ V}\), \(+20\text{ V}\), \(+40\text{ V}\), and \(+60\text{ V}\). Four paths with arrowheads indicating direction are shown. Path 1 is a straight horizontal solid arrow pointing to the left from the \(+60\text{ V}\) line to the \(+20\text{ V}\) line, labeled 'Path 1' above its center. Path 2 is a meandering, S-shaped curved solid line starting at the \(+60\text{ V}\) line and ending at the \(+20\text{ V}\) line, labeled 'Path 2' above its curve. Path 3 is a straight solid arrow pointing to the left from the \(+40\text{ V}\) line to the \(-30\text{ V}\) line, labeled 'Path 3' above its center. Path 4 is a smooth curved solid arc pointing to the left from an unlabeled position halfway between the \(+20\text{ V}\) and \(+40\text{ V}\) lines to the \(+10\text{ V}\) line, labeled 'Path 4' above its arc. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604535-9v1LbW.jpg)

- **A.** \(\Delta K_2 > \Delta K_3 > \Delta K_1 > \Delta K_4\)
- **B.** \(\Delta K_4 > \Delta K_1 = \Delta K_2 > \Delta K_3\)
- **C.** \(\Delta K_3 > \Delta K_2 > \Delta K_1 > \Delta K_4\)
- **D.** \(\Delta K_3 > \Delta K_1 = \Delta K_2 > \Delta K_4\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124635/*