Conservation of Electric Energy
When a charge q moves from initial potential V_i to final potential V_f, its electric potential energy changes by ΔU_E=qΔV=q(V_f−V_i). If the electric force is the only force doing work, ΔK=−ΔU_E=−qΔV, so K+U_E is constant. The sign of q is essential: positive charges naturally speed up toward lower potential, while negative charges naturally speed up toward higher potential.
Topic sequence · 3 of 3
Learning objectives
Objective 1
Use ΔU_E=qΔV with signed charge and signed potential difference to predict energy changes.
Objective 2
Apply conservation of energy to charged-particle motion without unnecessarily solving position-dependent force and acceleration.
Objective 3
Translate among potential maps, field direction, electric potential energy, kinetic energy, speed, and energy diagrams.
- Topic 9.2 electric potential, potential difference, and field-potential relationships.
- Topic 9.1 electric potential energy as a system quantity.
- Mechanics Topic 3.4 conservation of energy and Topic 3.1 kinetic energy.
Definitions first
Representations you must read and create
Representation 1
Energy bar chart showing U_E decreasing by exactly the amount K increases when only electrostatic work occurs.
Representation 2
Potential map with initial/final locations and a charge sign, from which ΔV, ΔU_E, force direction, and speed trend are inferred.
Representation 3
K-versus-position and U_E-versus-position curves whose sum is constant for an isolated conservative model.
Representation 4
Table comparing positive and negative charges moving to higher/lower potential so sign reasoning is explicit rather than memorized.
Core relationships and calculus
Variables: ΔU_E J; q C; V volts
Use: electric potential-energy change between two locations
Variables: W J
Use: work done by the electric force
Variables: K J
Use: only when the electric force is the only force doing work on the particle
Variables: all terms J
Use: isolated conservative electric-motion model
Variables: m kg; v m/s
Use: nonrelativistic charged particle with only electrostatic work
Variables: e C
Use: particle-scale energy conversion
Energy conservation converts a spatially varying-force problem into an endpoint problem. Combining ΔV=−∫E⃗·dr⃗ with ΔU_E=qΔV yields W_electric=q∫E⃗·dr⃗=−ΔU_E. When only that conservative electric force transfers energy, the work-energy theorem gives ΔK=W_electric=−qΔV.
Start with the definition ΔV=ΔU_E/q, so ΔU_E=qΔV. The electric force is conservative, hence W_e=−ΔU_E. The work-energy theorem gives ΔK=W_net. If only the electric force does work, ΔK=W_e=−qΔV, or K_i+U_i=K_f+U_f. If another external force does work, add its energy transfer explicitly instead of forcing K+U_E to remain constant.
Use ΔK=−qΔV only when the electric force is the only force doing work in the selected model. The more general statement is total energy conservation with all transfers included. Keep the charge sign, use nonrelativistic kinetic energy only when appropriate, and do not assume a particle must move in the direction of decreasing V unless the charge is positive.
Physical examples
Example 1
A positive ion released from rest moves toward lower potential under the electric force, so ΔV<0, ΔU_E<0, and ΔK>0.
Example 2
An electron released in an electric field accelerates toward higher electric potential because q<0 makes qΔV negative when ΔV>0.
Example 3
A charge moved slowly against the electric force by an external agent can gain U_E without gaining kinetic energy because the external agent supplies the energy.
Example 4
A particle crossing an equipotential has ΔV=0, so electrostatic work over that displacement is zero even if the electric field is nonzero elsewhere.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Charge-sign reasoning: use qΔV before deciding whether U_E rises or falls.
Reasoning pattern 2
Speed without kinematics: if only electrostatic work acts, use ΔK=−qΔV rather than finding acceleration as a function of position.
Reasoning pattern 3
Representation translation: from equipotential spacing infer field magnitude, then use charge sign to infer force and energy trends.
Reasoning pattern 4
External-work check: if a particle is moved at constant speed by an external agent, ΔK=0 and the external work balances the electric-energy change.
Reasoning pattern 5
Electron-volt conversion: a charge magnitude e crossing |ΔV| volts changes energy magnitude by |ΔV| eV.
Deeper calculus and limiting-case check
Combine energy conservation with a known potential function V(x): ½mv²+qV(x)=E_total. Solving gives v(x)=√[(2/m)(E_total−qV(x))] where the radicand is nonnegative. This defines classically allowed regions and turning points for one-dimensional electrostatic motion without solving the differential equation of motion.
Model-selection workflow: decide what the problem is before choosing an equation
For Conservation of Electric Energy, the fastest reliable route is not equation hunting. Start by identifying the source configuration and reference, encode it with the potential/energy representation, choose the line integral or energy equation, and then test the reference-zero and electrostatic assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat electric energy conversion and charge-field system as physical quantities first and symbols second.
Choose a representation
Build Energy bar chart showing U_E decreasing by exactly the amount K increases when only electrostatic work occurs.. If the prompt supplies Potential map with initial/final locations and a charge sign, from which ΔV, ΔU_E, force direction, and speed trend are inferred., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is ΔU_E = qΔV = q(V_f−V_i). State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
Use ΔK=−qΔV only when the electric force is the only force doing work in the selected model. The more general statement is total energy conservation with all transfers included. Keep the charge sign, use nonrelativistic kinetic energy only when appropriate, and do not assume a particle must move in the direction of decreasing V unless the charge is positive. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | Energy bar chart showing U_E decreasing by exactly the amount K increases when only electrostatic work occurs. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | ΔU_E = qΔV = q(V_f−V_i) | electric potential-energy change between two locations. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | W_electric = −ΔU_E = −qΔV | work done by the electric force. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | Energy conservation converts a spatially varying-force problem into an endpoint problem. Combining ΔV=−∫E⃗·dr⃗ with ΔU_E=qΔV yields W_electric=q∫E⃗·dr⃗=−ΔU_E. When only that conservative electric force transfers energy, the work-energy theorem gives ΔK=W_electric=−qΔV. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
An electron starts from rest and moves through a potential increase of +250 V under electrostatic forces. With q=−e, ΔU_E=qΔV=(−1.602×10⁻¹⁹ C)(250 V)=−4.005×10⁻¹⁷ J. Therefore ΔK=+4.005×10⁻¹⁷ J, which is 250 eV. Using ½mv²=ΔK gives v≈9.38×10⁶ m/s in the nonrelativistic model. The key sign result is that a negative charge gains kinetic energy while moving to higher electric potential. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 9.3
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship ΔU_E = qΔV = q(V_f−V_i) should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Energy bar chart showing U_E decreasing by exactly the amount K increases when only electrostatic work occurs. and Potential map with initial/final locations and a charge sign, from which ΔV, ΔU_E, force direction, and speed trend are inferred.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Use a measured equipotential map from conductive paper or a low-voltage field tray. Choose several pairs of points, record ΔV, and calculate qΔV for a hypothetical positive and negative test charge. Build corresponding energy bar charts and compare the predicted force direction with the perpendicular electric-field direction inferred from the equipotentials.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Saying positive charges always move “downhill” and applying the same statement to electrons. Negative charges reverse the potential-direction intuition. | Return to the potential/energy representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Dropping the sign of q in ΔU_E=qΔV. | Return to the line integral or energy equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Using ΔK=−qΔV when an external force or other nonconservative interaction also transfers energy. | Return to the potential/energy representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Confusing volts with joules. Multiply potential difference by charge to obtain an energy change. | Return to the line integral or energy equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Assuming ΔV=0 means the electric field is zero everywhere along a path; an equipotential path can lie in a nonzero field. | Return to the potential/energy representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 9.2 supplies ΔV and the field/potential map; Topic 9.3 converts that spatial information into energy changes. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Mechanics Unit 3 provides the same conservative-energy bookkeeping structure used here. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 10 capacitors later stores electric energy in a field configuration; the same voltage-energy ideas reappear there. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write ΔU_E = qΔV = q(V_f−V_i) and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 9.3 mastery ladder
Interactive electric-energy conservation check
Enter signed charge, initial/final potentials, particle mass, and initial speed. The tool assumes only electrostatic work and reports whether the final state is energetically reachable.
Experimental connection
Use a measured equipotential map from conductive paper or a low-voltage field tray. Choose several pairs of points, record ΔV, and calculate qΔV for a hypothetical positive and negative test charge. Build corresponding energy bar charts and compare the predicted force direction with the perpendicular electric-field direction inferred from the equipotentials.
Connections across Physics C
Connection 1
Topic 9.2 supplies ΔV and the field/potential map; Topic 9.3 converts that spatial information into energy changes.
Connection 2
Mechanics Unit 3 provides the same conservative-energy bookkeeping structure used here.
Connection 3
Topic 10 capacitors later stores electric energy in a field configuration; the same voltage-energy ideas reappear there.
Connection 4
Electric circuits later use potential differences supplied by batteries, but circuit energy accounting must include sources and resistive transfers rather than treating K+U_E of a single charge as the whole system.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Saying positive charges always move “downhill” and applying the same statement to electrons. Negative charges reverse the potential-direction intuition.
Avoid this
Dropping the sign of q in ΔU_E=qΔV.
Avoid this
Using ΔK=−qΔV when an external force or other nonconservative interaction also transfers energy.
Avoid this
Confusing volts with joules. Multiply potential difference by charge to obtain an energy change.
Avoid this
Assuming ΔV=0 means the electric field is zero everywhere along a path; an equipotential path can lie in a nonzero field.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
Why do electrons accelerate toward higher potential?
Because q is negative. If ΔV is positive, qΔV is negative, so electric potential energy decreases and kinetic energy can increase.
When can I use ΔK=−qΔV?
When the electric force is the only force doing work on the particle in the selected model.
What is one electron volt?
One electron volt is the magnitude of energy change for a charge of magnitude e moving through one volt: 1 eV=1.602×10⁻¹⁹ J.
Does conservation of energy mean electric potential energy is constant?
No. U_E can change while kinetic or other energy changes by a compensating amount.
Can a charge move to higher potential and still speed up?
Yes. A negative charge can do exactly that under the electric force because its potential energy decreases when V increases.