Electric Power
Connect current and voltage with power under a specified fixed-current or fixed-voltage constraint.
AP Physics 2 Topic 11.4 · College Board CED printed p. 66 / PDF p. 73 · See this topic in the course outline
Current tracks charge. Power tracks energy each second. Brightness is a useful qualitative clue about energy transfer.
For a supporting original reading, see Resistors in Circuits. Connect resistor voltage and current to energy transfer per second. Brightness comparisons require a stated bulb model.
One power law, two resistor forms
A charge amount Δq crossing a potential difference ΔV transfers energy of magnitude ΔqΔV. Divide by elapsed time to get , in watts (1 W = 1 J/s). An element absorbing energy has positive absorbed power when current enters its higher-potential terminal. A discharging ideal battery supplies the corresponding energy per second.
For an ohmic resistor substitute ΔV = IR or I = ΔV/R to obtain . Use the voltage across that resistor, not the battery voltage unless they are the same. Electrical energy converted in time Δt at constant power is .
At fixed current, larger R means larger power; at fixed voltage, larger R means smaller power. Those comparisons use different constraints, so “larger resistance means brighter” has no universal answer. A graph of P versus I at fixed R is an upward quadratic curve; at fixed voltage, P decreases as inverse R.
Brightness compares energy transfer, with a model stated
For a particular bulb, greater electrical power generally means greater brightness. For identical bulbs operating comparably, their power ranking predicts their brightness ranking. Two identical constant-R bulbs in series share the same current, so equal power and equal brightness follow. In parallel across the same ideal battery, each gets the full voltage and is brighter than each bulb in the corresponding series pair.
A real incandescent bulb’s resistance changes with filament temperature. Use constant-R bulb calculations only when the task specifies or approximates that model; power still provides the qualitative energy-transfer idea. Visible brightness need not be proportional to wattage across different bulb types or designs.
Energy per second and an energy check
A 15 Ω resistor has 0.40 A flowing through it for 50 s. Find its voltage, power, and converted energy. Treat R as fixed.
ΔV = IR = (0.40)(15) = 6.0 V.
Why this step? Ohm’s relation uses this element’s current and resistance.
P = IΔV = 2.4 W; I²R also gives (0.40)²(15) = 2.4 W.
Why this step? The general energy-transfer relation and its ohmic form must agree.
E = PΔt = (2.4)(50) = 120 J. Also Δq = IΔt = 20 C and ΔqΔV = 120 J.
Why this step? Charge bookkeeping independently confirms the energy calculation.
Answer and check
6.0 V, 2.4 W, and 120 J converted, primarily to thermal energy.
Make the model respond
Focused Physics 2 model
Two-resistor DC circuit
Predict first. Predict how total current changes when the same resistors are rewired in parallel.
Rₑq = R₁ + R₂; I₁ = I₂ = I; Vbattery = V₁ + V₂; P = I²R
On a narrow screen, scroll within a figure to read its labels. The table and explanation provide the same physics in text.
Series: I₁ = 0.2 A, I₂ = 0.2 A, and total current = 0.2 A.
Read the model data and comparisons
| Quantity | Value | Unit | Meaning or limit |
|---|---|---|---|
| Battery voltage | 12 | V | |
| Equivalent resistance | 60 | Ω | |
| Battery current | 0.2 | A | |
| R₁ current | 0.2 | A | |
| R₂ current | 0.2 | A | |
| R₁ voltage drop | 4 | V | |
| R₂ voltage drop | 8 | V | |
| R₁ power | 0.8 | W | |
| R₂ power | 1.6 | W | |
| Battery power | 2.4 | W | |
| Power residual Pbattery − P₁ − P₂ | -2.220 × 10^−16 | W |
In series, every resistor carries the same current, and the voltage drops sum to the battery voltage. In parallel, every branch has the full battery voltage, and the branch currents sum at the junction. In either topology, the ideal battery power equals the sum of resistor powers. The battery moves charge around a closed circuit; resistors transfer electric energy to thermal energy.
Assumptions and limits. Ideal constant-voltage battery, ideal wires, two ohmic resistors with fixed resistance and no temperature dependence. Steady DC state; conventional-current direction from the positive terminal. No internal battery resistance or transient behavior.
Try it from memory
Write your answer and reason before opening the worked answer. Your self-assessment is a study decision; it is not an automatically graded score.
Question 1
Double R once at fixed voltage and once at fixed current. How does power change? Why are the answers different?
Worked answer and reason
It halves at fixed voltage and doubles at fixed current. P = V²/R and P = I²R use different held-fixed quantities.
Constraints determine the functional dependence.
Question 2
Two identical constant-R bulbs are placed across the same ideal battery, first in series and then in parallel. Which pair arrangement gives each bulb more power?
Worked answer and reason
Parallel: each gets full battery voltage. In series each has half the battery voltage, so each has one fourth of its parallel power.
For identical constant resistance P ∝ V².
JavaScript enables saved drafts and self-assessment. You can always work on paper.
Catch the tempting mistake
- Current is used up as light is produced.
- Energy is transferred and converted; charge flow is conserved through an unbranched bulb path.
- Battery voltage can be used in every resistor’s P = V²/R.
- Only use the voltage across the element whose power you are finding.
Put it to work
Give the physical reason as well as the calculation. The worked solutions are on a separate page.
- Problem 7 · power derivation and energy ledger
- Problem 8 · brightness comparison with changing constraint
- Problem 17 · resistance experiment, uncertainty, and power
- Problem 18 · nonreducible bridge and independent equations
- Problem 19 · branching RC endpoints and empirical time scale
Scientific calculator workspace
Keep intermediate precision and report units. This calculator supports arithmetic and elementary functions.
Scientific calculator
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+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
For the teacher
- Demand a fixed-current or fixed-voltage statement in brightness comparisons.
- Ask students to check battery supplied power against sum of resistor powers.
Source objectives and scope
- 11.4.A — Describe the transfer of energy into, out of, or within an electric circuit, in terms of power.
Suggested skills are teaching guidance. The course applies all science practices across its objectives.
Science references for this teaching
- College Board AP Physics 2 Course and Exam Description, Unit 11 (PDF pages 68–81) — Official topic order, learning objectives, all essential-knowledge clauses and boundaries. Equation pages 68,71,73,75,77–80 and relevant continuation pages visually inspected; independent reviewed-source-equation overlay consulted.
- OpenStax University Physics Volume 2, 9.3 Resistivity and Resistance — Independent verification of real material temperature dependence and the fixed-temperature qualification for linear metallic response. Teaching and assessment prose are original.
- OpenStax University Physics Volume 2, 10.3 Kirchhoff’s Rules — Independent verification of conservation-based branch and loop analysis; no calculus is needed in these authored tasks.
- Corrected original reading: Current and Resistance — Follow charge-flow rate and conventional-current direction; keep electron drift opposite the arrows.Supporting context; original AP teaching and practice above remain independently authored.
- Corrected original reading: Introduction to Circuits — Practice node connectivity, switches and common symbols. Check that any short-circuit idealization retains a finite current-limiting resistance.Supporting context; original AP teaching and practice above remain independently authored.
- Corrected original reading: Resistors in Circuits — Connect resistor voltage and current to energy transfer per second. Brightness comparisons require a stated bulb model.Supporting context; original AP teaching and practice above remain independently authored.
- Corrected original reading: Kirchhoff’s Laws — Trace actual closed paths and keep signed voltage changes. Use independent equations rather than combining disconnected branches into one loop.Supporting context; original AP teaching and practice above remain independently authored.
- Corrected original reading: Capacitors in Circuits — Compare equal-charge series and equal-voltage parallel capacitor arrangements.Supporting context; original AP teaching and practice above remain independently authored.
- Corrected original reading: RC Circuits — Use initial and long-time states, τ and qualitative curves. Source exponential and logarithmic transient calculations extend beyond required AP Physics 2 modeling.Supporting context; original AP teaching and practice above remain independently authored.
Glossary and deeper reading · 10 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Electric current
Electric current is the net rate of charge flow through a specified cross section. It is measured in amperes, or coulombs per second, with direction defined by positive-charge motion. For example, the same steady current passes through each element of a simple unbranched series loop.
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Electric charge
Electric charge is a conserved property that determines electric interactions. It can be positive or negative and is measured in coulombs; ordinary net charges arise from an imbalance of charged particles. For example, transferring electrons between initially neutral objects leaves one negatively charged and the other positively charged.
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Potential difference
Potential difference is the change in electric potential between two locations. It represents electric potential-energy change per unit charge and is measured in volts. For example, the potential difference across a resistor describes the energy transferred per coulomb passing through it.
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Ohm’s law
Ohm’s law describes an element whose current is proportional to its potential difference under the stated conditions. Such an element has constant resistance over the modeled range, giving a straight-line current–voltage relationship through the origin. For example, a fixed-temperature ideal resistor follows this model, whereas a warming filament bulb need not.
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Mechanical work
Mechanical work is energy transferred by a force through displacement of its point of application. For a constant force on a particle, it depends on the force component along the particle’s displacement, and it can be positive, negative, or zero. For example, a magnetic force perpendicular to a particle’s displacement does zero work on that particle.
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Power
Power is the rate at which energy is transferred or converted. It is measured in watts, with one watt equal to one joule per second. For example, two motors can lift the same load through the same height but have different average powers if they take different times.
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Electrical resistance
Electrical resistance relates the potential difference across an element to the current through it. It is measured in ohms and depends on the element’s material, geometry, and conditions. For example, a longer uniform wire has greater resistance when cross-sectional area and material remain fixed.
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Series connection
A series connection places elements along one unbranched path. The same current passes through the series elements, while their potential differences can differ. For example, adding another resistor to a simple series loop raises the total resistance and reduces current from a fixed ideal source.
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Parallel connection
A parallel connection joins elements across the same two nodes. They therefore have the same potential difference, while their branch currents can differ. For example, two different resistors connected in parallel across a battery generally carry different currents.
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Temperature
Temperature is a state quantity that determines the direction of net energy transfer through thermal contact. For a classical ideal gas it is proportional to the average translational kinetic energy per particle, rather than to the total energy of the sample. For example, a small hot metal sample can have a higher temperature but less total internal energy than a large cooler sample.