Electric Potential Energy & Work — Relate electric potential energy to work and potential difference - Physics 2

SECTION 1

Historical Context & Motivation

The modern understanding of electric potential energy did not emerge in a single moment of insight; rather, it developed across nearly two centuries of investigation into the nature of electricity. In the eighteenth century, experimentalists such as Benjamin Franklin observed that charged objects could exert forces over a distance, hinting at an underlying mechanism for energy storage. The formalization of these observations into a coherent mathematical framework required contributions from Coulomb, Volta, and eventually the theoretical syntheses of Faraday and Maxwell. Understanding this historical trajectory illuminates why physicists introduced the concept of potential energy in the electric context — it was the key to connecting mechanical work with electrostatic phenomena and paving the way toward practical devices like capacitors, batteries, and eventually entire power grids.

1785 • Coulomb's Inverse-Square Law

Charles-Augustin de Coulomb published his precise measurements of the force between charged spheres, establishing that the electrostatic force is proportional to the product of the charges and inversely proportional to the square of their separation — a result structurally analogous to Newton's gravitational law.

1800 • Volta's Pile

Alessandro Volta constructed the first electrochemical battery, demonstrating that a sustained potential difference could drive charges through a circuit. This invention gave concrete meaning to the idea that work is performed when charges move between points of differing electric potential.

1831 • Faraday's Field Concept

Michael Faraday introduced the concept of electric and magnetic fields, shifting the discourse from action-at-a-distance to a local field-based picture. His notion of lines of force provided the intuitive foundation for defining potential energy as a property of a charge's position within a field.

1865 • Maxwell's Electromagnetic Theory

James Clerk Maxwell unified electricity and magnetism into a single theoretical framework, rigorously connecting the scalar potential to the work done by electric fields and establishing the conservation of energy across all electromagnetic phenomena.

The central question motivating this lesson is deceptively simple: How much energy does it take to move a charge from one point to another in an electric field, and how do we quantify that energy? Answering this question requires linking three tightly interwoven concepts — the work done by or against the electric field, the electric potential energy stored in the configuration, and the electric potential difference (voltage) between two locations. Mastering their relationship is essential before tackling capacitors, circuits, and more advanced electrodynamics.

SECTION 2

Core Principles & Definitions

Before diving into equations, it is important to establish the conceptual pillars that support the relationship between work, potential energy, and potential difference. The electrostatic force is a conservative force, meaning the work it does on a charge depends only on the initial and final positions, not on the path taken between them. This path-independence is what allows us to define a scalar potential energy function U that depends solely on configuration. All three quantities — work, potential energy, and voltage — are intimately related through energy conservation, and grasping their definitions is the first step toward fluency with electrostatic energy arguments.

Electric Potential Energy (U)

The energy stored in a system of charges by virtue of their positions relative to one another. For a charge q in an external electric field, U represents the capacity of the configuration to do work. Like gravitational PE, it is defined relative to a chosen reference point (often at infinity, where U = 0).

Work Done by the Electric Field (W)

The energy transferred to or from a charge as it moves through an electric field. When the field does positive work, the charge's kinetic energy increases and the system's potential energy decreases. The work–energy theorem still applies: W = −ΔU for the work done by the conservative electric force.

Electric Potential Difference (ΔV)

The potential energy change per unit charge between two points: ΔV = ΔU / q. Measured in volts (1 V = 1 J / C), voltage is a scalar field property independent of the test charge. Moving a positive charge from low to high potential requires external work.

Conservative Nature of the Electrostatic Force

Because ∮ E · dl = 0 for a static field, the work done by the electric force around any closed loop is zero. This path-independence guarantees that U is a well-defined state function, making energy-based problem solving both valid and powerful.

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KEY TAKEAWAY

Think of electric potential energy like water stored behind a dam. The height of the water (analogous to voltage) determines how much energy is available per unit volume (per unit charge). Releasing the water (allowing charges to move) converts stored potential energy into kinetic energy, and the total work the water can do depends on both the height difference and the volume released — just as W = qΔV depends on both the charge and the potential difference.

SECTION 3

Visual Explanation — Charges Moving Through a Potential Difference

A positive test charge +q placed in a uniform electric field (amber arrows) moves from the high-potential plate (red, left) to the low-potential plate (cyan, right). The electric field performs positive work on the charge, decreasing the system's potential energy and increasing the charge's kinetic energy. The dashed green path shows the displacement direction.

The diagram above captures the essential physics of work and potential energy in an electrostatic context. A uniform electric field E points from left to right (amber arrows), produced by parallel plates at different potentials. The left plate is held at a higher potential V₊ (shown in red), while the right plate is at a lower potential V₋ (shown in cyan). When a positive test charge +q is released, the electric force qE accelerates it in the direction of the field — from high to low potential. During this displacement, the electric field does positive work on the charge, and the system's electric potential energy decreases by an amount equal to that work. Energy conservation then dictates that the charge's kinetic energy increases by the same amount. If instead a negative charge were placed in the same field, it would be pushed from right to left — from low to high potential — because the force on it is opposite to E. In that case, moving from low to high potential still satisfies W = qΔV, but now q is negative and ΔV is positive, yielding negative work by the field (the charge decelerates unless an external agent pushes it).

SECTION 4

Mathematical Framework

The relationship between work, potential energy, and potential difference is grounded in the work–energy theorem applied to conservative forces. Because the electrostatic force is conservative, we can write all the central results in terms of a scalar potential energy function U(r) and its per-unit-charge counterpart V(r). The following equations form a tightly linked chain: each can be derived from the others, and together they provide a complete toolkit for solving electrostatic energy problems.

WORK–POTENTIAL ENERGY RELATION

W_E = −ΔU = −(U_f − U_i) = U_i − U_f

W_E = work done by the electric force; ΔU = U_f − U_i = change in electric potential energy. Positive W_E means the field transfers energy to the charge (U decreases).

ELECTRIC POTENTIAL DIFFERENCE

ΔV = V_f − V_i = ΔU / q ⟹ ΔU = qΔV

ΔV = potential difference (volts, V); q = charge (coulombs, C). This definition factors out the test charge so that V is a property of the field alone. Note: 1 V = 1 J/C.

WORK IN TERMS OF VOLTAGE

W_E = −qΔV = −q(V_f − V_i) = q(V_i − V_f)

Combining the two preceding results yields the most commonly used form. For a positive charge moving from high to low potential, V_i > V_f, so W_E > 0 — the field does positive work.

INTEGRAL FORM (GENERAL FIELDS)

W_E = q ∫_A^B E · dl = −q(V_B − V_A)

For non-uniform fields, the work is computed via a line integral of the electric field E along any path from point A to point B. The path-independence of this integral for electrostatic fields is what makes V a well-defined scalar function.

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Sign Convention Warning

Be meticulous with signs. The work done by the electric field is W_E = −qΔV. The work done by an external agent (moving the charge quasi-statically) is W_ext = +qΔV = +ΔU. Confusing these two is the single most common error in potential energy problems.

SECTION 5

Energy Diagrams & Sign Analysis

One of the most effective ways to develop physical intuition for potential energy and work is through energy bar charts and sign-analysis tables. These tools force you to track where energy comes from and where it goes, preventing sign errors and reinforcing the conservation principle. The diagram below illustrates how kinetic energy, electric potential energy, and work are related for a positive charge moving through a potential difference in two scenarios — one where the field does the work, and one where an external agent does.

Energy bar charts comparing two scenarios. Scenario A (left): A free positive charge accelerates from high to low potential; the field does positive work, converting potential energy (red) into kinetic energy (green). Scenario B (right): An external agent pushes a positive charge from low to high potential at constant speed; the agent's work is stored as increased potential energy.

Scenario	Sign of $W_E$	Sign of ΔU	Physical Interpretation
+q moves high V → low V (with field)	W_E > 0	ΔU < 0	Field accelerates charge; PE converts to KE
+q moves low V → high V (against field)	W_E < 0	ΔU > 0	External agent must do positive work; energy stored as PE
−q moves high V → low V (against field)	W_E < 0	ΔU > 0	Negative charge pushed toward low V requires external work
−q moves low V → high V (with field)	W_E > 0	ΔU < 0	Field pushes negative charge toward high V; PE decreases

Sign analysis for different charge and potential configurations

Notice the symmetry: a positive charge "falls" from high to low potential, while a negative charge "falls" from low to high potential. In both cases, the electric force does positive work and the potential energy decreases. The sign analysis table above is an invaluable reference for checking your work on exam problems. When in doubt, remember that the electric force always acts to reduce the system's potential energy, just as gravity always pulls masses toward lower gravitational PE.

SECTION 6

Worked Example — Proton Accelerated Through a Potential Difference

A proton (q = +1.60 × 10⁻¹⁹ C, m = 1.67 × 10⁻²⁷ kg) is released from rest near a positive plate and accelerates through a potential difference of 500 V toward a negative plate. Find (a) the work done by the electric field on the proton, (b) the change in the proton's potential energy, and (c) the proton's final speed.

Proton Through 500 V

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Step 1 — Identify Given Values and Setup

The proton has charge q = +1.60 × 10⁻¹⁹ C and mass m = 1.67 × 10⁻²⁷ kg. It starts from rest (v_i = 0) near the positive plate (high potential) and moves to the negative plate (low potential). Define ΔV = V_f − V_i = −500 V (since V_f < V_i).
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Step 2 — Calculate Work Done by the Electric Field

Using W_E = −qΔV:

W_E = −(1.60 × 10⁻¹⁹ C)(−500 V) = +8.00 × 10⁻¹⁷ J
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Step 3 — Determine the Change in Potential Energy

Since W_E = −ΔU, we have ΔU = −W_E:

ΔU = −8.00 × 10⁻¹⁷ J = −8.00 × 10⁻¹⁷ J. The potential energy decreases, as expected for a positive charge moving with the field.
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Step 4 — Find the Final Speed Using Energy Conservation

By the work–energy theorem, W_E = ΔKE = ½mv_f² − 0. Solve for v_f: v_f = √(2W_E / m) = √(2 × 8.00 × 10⁻¹⁷ / 1.67 × 10⁻²⁷)

v_f = √(9.58 × 10¹⁰) ≈ 3.10 × 10⁵ m/s — about 0.1% the speed of light, so the non-relativistic treatment is valid.

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Electron-Volt Shortcut

For particles with charges that are integer multiples of the elementary charge e, the electron-volt (eV) provides a natural energy unit. 1 eV = 1.60 × 10⁻¹⁹ J is the kinetic energy gained by a charge of magnitude e falling through a 1 V potential difference. In this problem, the proton gained 500 eV of kinetic energy — a single multiplication instead of the multi-step SI calculation.

SECTION 7

Gravitational vs. Electric Potential Energy — A Parallel Analysis

The structural analogy between gravitational and electric potential energy is both powerful and instructive. Understanding where the analogy holds — and where it breaks down — deepens your grasp of both systems. In gravitational physics, there is only one type of "charge" (mass), and it is always positive, so gravity is always attractive. Electricity, by contrast, admits both positive and negative charges, enabling both attractive and repulsive interactions. This key difference has significant implications for the behavior of potential energy.

Feature	Gravitational PE	Electric PE
Force law	F = Gm₁m₂ / r² (always attractive)	F = kq₁q₂ / r² (attractive or repulsive)
PE for two-particle system	U = −Gm₁m₂ / r (always negative)	U = kq₁q₂ / r (sign depends on charges)
PE near surface (uniform field)	U = mgh	U = qEd (uniform field, separation d)
Potential (per unit charge/mass)	V = gh (gravitational potential near surface)	V = U/q (electric potential)
Work by field	W = −ΔU = mg( $h_i$ − $h_f$ )	W = −ΔU = q( $V_i$ − $V_f$ )
Sign of charge	Mass always positive	Charge can be + or −; must track signs carefully

Structural comparison: gravitational vs. electric potential energy

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KEY TAKEAWAY

Gravitational PE is the scaffolding on which electric PE is built. Replacing mass with charge, g with E, and height with voltage yields the electric analogs. But the electric case is richer because charges have two signs: like charges have positive PE (repulsive, like a compressed spring), while opposite charges have negative PE (bound, like gravitational pairs). This distinction is crucial when analyzing bound states in atoms or repulsion in nuclear physics.

SECTION 8

Connection to Advanced Theory — From Voltage to Capacitance and Beyond

The ideas developed in this lesson form the bedrock for several more advanced topics in electromagnetism. When you understand that potential energy is stored in the electric field configuration, you are naturally led to ask how much energy can be stored in a given geometry — the domain of capacitance and energy density. Similarly, when charges move continuously through a potential difference, the rate of energy transfer is electric power, P = IV, which underpins all of circuit theory. Looking further ahead, the concept of electric potential connects to the Laplace and Poisson equations in advanced E&M, where solving for V(r) in complex boundary conditions becomes a central mathematical technique.

This Lesson's Concept	Advanced Extension	Key New Idea
W = qΔV (work on a single charge)	Energy stored in a capacitor: U = ½CV²	Energy is stored in the field, not on the plates
ΔV = −∫E · dl	E = −∇V (gradient relationship)	The field points in the direction of steepest potential decrease
V = kq/r (point charge potential)	Poisson's equation: ∇²V = −ρ/ε₀	Charge distribution determines V everywhere
W = qΔV for charges in steady flow	Power: P = IV = I²R = V²/R	Rate of energy transfer in circuits

How this lesson's concepts extend into advanced electromagnetism and circuits

In your upcoming study of capacitors, you will find that the energy stored can be expressed as U = ½QV = ½CV² = Q²/(2C), all of which are direct consequences of the W = qΔV relationship applied incrementally as charge is transferred. Similarly, when you encounter Kirchhoff's voltage law (the sum of voltage drops around a closed loop equals zero), you will recognize it as a statement of energy conservation — precisely the same principle that makes potential energy and work equivalent bookkeeping tools.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A positive charge is moved from point A to point B in a uniform electric field, and the electric potential at B is higher than at A (V_B > V_A). Is the work done by the electric field on the charge positive, negative, or zero? Does the system's electric potential energy increase or decrease?

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The work done by the electric field is negative. Using W_E = −qΔV = −q(V_B − V_A), since q > 0 and ΔV > 0, the product is negative. Physically, moving a positive charge to a higher potential is like pushing a ball uphill — the field opposes the motion, doing negative work. The system's electric potential energy increases because ΔU = qΔV > 0, meaning an external agent must supply the energy.

Question 2: BASIC CALCULATION

An electron (q = −1.60 × 10⁻¹⁹ C) is accelerated from rest through a potential difference of 200 V. How much kinetic energy does it gain? Express your answer in both joules and electron-volts.

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The work done by the electric field equals the kinetic energy gained: W_E = −qΔV. An electron accelerates from low potential to high potential (ΔV = +200 V), so W_E = −(−1.60 × 10⁻¹⁹)(200) = +3.20 × 10⁻¹⁷ J. In electron-volts, KE = 200 eV, since each volt of potential difference gives one electron-volt of energy to a charge of magnitude e. The answer is 3.20 × 10⁻¹⁷ J = 200 eV.

Question 3: INTERMEDIATE

Two parallel plates are separated by 2.0 cm and connected to a 120 V battery. A doubly-charged ion (q = +3.20 × 10⁻¹⁹ C, m = 6.64 × 10⁻²⁶ kg) is released from rest at the positive plate. (a) What is the electric field between the plates? (b) What is the ion's speed when it reaches the negative plate?

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(a) For uniform parallel plates, E = ΔV/d = 120 V / 0.020 m = 6000 V/m. (b) The work done by the field equals the kinetic energy gained: W = qΔV = (3.20 × 10⁻¹⁹)(120) = 3.84 × 10⁻¹⁷ J. Setting ½mv² = W and solving: v = √(2 × 3.84 × 10⁻¹⁷ / 6.64 × 10⁻²⁶) = √(1.157 × 10⁹) ≈ 3.40 × 10⁴ m/s. Note that the answer depends on the charge and mass of the ion, not on the plate separation (which affects E and d independently but their product Ed = ΔV is fixed by the battery).

Question 4: APPLIED

In a Van de Graaff accelerator, protons are accelerated from rest through a potential difference of 2.00 × 10⁶ V. (a) What is the final kinetic energy in MeV? (b) What is the final speed? (c) Is a relativistic correction needed? (Use m_p = 1.67 × 10⁻²⁷ kg, c = 3.00 × 10⁸ m/s.)

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(a) KE = qΔV = (1 e)(2.00 × 10⁶ V) = 2.00 MeV = 2.00 MeV = 3.20 × 10⁻¹³ J. (b) v = √(2KE/m) = √(2 × 3.20 × 10⁻¹³ / 1.67 × 10⁻²⁷) = √(3.83 × 10¹⁴) ≈ 1.96 × 10⁷ m/s. (c) The ratio v/c ≈ 0.065, or about 6.5% the speed of light. Since the Lorentz factor γ = 1/√(1 − v²/c²) ≈ 1.002, the non-relativistic kinetic energy overestimates the true value by only about 0.2%, so no significant relativistic correction is needed. Relativistic effects typically become important above v ≈ 0.1c.

Question 5: CRITICAL THINKING

Two point charges, Q₁ = +4.0 μC and Q₂ = −2.0 μC, are initially separated by 0.30 m. An external agent slowly pushes them together until they are 0.10 m apart. (a) Find the initial and final electric potential energies of the system. (b) Calculate the work done by the external agent and the work done by the electric field. (c) Explain physically why the external agent does negative work in this scenario.

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(a) U = kQ₁Q₂/r. Initially: U_i = (8.99 × 10⁹)(4.0 × 10⁻⁶)(−2.0 × 10⁻⁶)/0.30 = −0.240 J. Finally: U_f = (8.99 × 10⁹)(4.0 × 10⁻⁶)(−2.0 × 10⁻⁶)/0.10 = −0.719 J. (b) W_ext = ΔU = U_f − U_i = −0.719 − (−0.240) = −0.479 J. W_E = −ΔU = +0.479 J. (c) Because Q₁ and Q₂ are opposite in sign, they attract each other. The electric force already pulls them together, so the agent must actually resist their approach (pushing back) to maintain the quasi-static condition. The agent's force is opposite to the displacement, yielding negative work. The energy released by the attractive electric force (positive W_E) is absorbed by the agent — physically, the agent acts like a brake, preventing runaway acceleration.

SUMMARY

Lesson Summary

Summary

This lesson established the deep connection between three core electrostatic quantities. Electric potential energy U is the energy stored in a charge configuration by virtue of the charges' positions. The work done by the electric field equals the negative of the change in potential energy: $W_E$ = −ΔU. Dividing by the test charge yields the electric potential difference ΔV = ΔU / q, a scalar field property measured in volts. These three results combine into the master relation $W_E$ = −qΔV, which governs every problem involving charges moving through potential differences.

The conservative nature of the electrostatic force ensures that work is path-independent, making U a well-defined state function. Positive charges naturally "fall" from high potential to low potential; negative charges fall from low to high. The gravitational analogy (mgh ↔ qEd, g ↔ E, height ↔ voltage) is a powerful guide, but the electric case is richer due to the existence of two charge signs. These foundational ideas extend directly into capacitor energy storage (U = ½CV²), circuit power (P = IV), and the gradient relationship E = −∇V that forms the backbone of advanced electrostatics.