Compare maximum load power with efficient transfer
Electromotive force is energy supplied per unit charge, measured in volts despite its name. It is not a mechanical force. The supplied source has fixed emf 9 V and series internal resistance r. With load R, current is 9/(r+R), and terminal voltage is 9−Ir=IR. The open-circuit limit approaches the emf because current and internal drop approach zero.
Learning outcome: Hold r=1 Ω and compare supported loads. Plot load power and separately retain efficiency. Explain why a higher terminal voltage can coexist with lower delivered power and name an independent observation needed before treating r as constant.
Starting recap
Use charge continuity, energy per charge, consistent SI units and an explicit circuit topology. Exponential responses are supplied algebraic models; initial state, source constraint and measurement loading must be declared.
All lessons are open directly. Your saved experiments remain when you leave and return.
Try
At r=1 Ω and R=4 Ω, current is 1.8 A, terminal voltage 7.2 V, load power 12.96 W and internal power 3.24 W. Their sum is source power 16.2 W, with efficiency .8. The theoretical maximum load power for this r is 20.25 W at R=1 Ω, where efficiency is .5.
Open the experimentPredict
Matching R=r maximizes the stated load power at efficiency:
Experiment
Hold r=1 Ω and compare supported loads. Plot load power and separately retain efficiency. Explain why a higher terminal voltage can coexist with lower delivered power and name an independent observation needed before treating r as constant.
Distinguish load power from source efficiency
Worked teaching model · synthetic values · Fictional fixed emf 9 V source with series internal resistance and an ohmic load. No chemical depletion, temperature change or current limit modeled. Algebraic efficiency/power comparison, not a battery operating recommendation.
Reset keeps your prediction, captures and practice history.
Load power is not the same objective as efficiency
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: fixed emf and internal resistance (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Load power is not the same objective as efficiency
fixed emf and internal resistance
| load resistance (Ω) | load power (W) |
|---|---|
| 0.25 | 12.96 |
| 0.49375 | 17.9241 |
| 0.7375 | 19.7878 |
| 0.98125 | 20.2482 |
| 1.225 | 20.0429 |
| 1.46875 | 19.5199 |
| 1.7125 | 18.8528 |
| 1.95625 | 18.1312 |
| 2.2 | 17.4023 |
| 2.44375 | 16.6909 |
| 2.6875 | 16.0092 |
| 2.93125 | 15.363 |
| 3.175 | 14.7542 |
| 3.41875 | 14.1825 |
| 3.6625 | 13.6466 |
| 3.90625 | 13.1445 |
| 4.15 | 12.6741 |
| 4.39375 | 12.2332 |
| 4.6375 | 11.8194 |
| 4.88125 | 11.4308 |
| 5.125 | 11.0654 |
| 5.36875 | 10.7214 |
| 5.6125 | 10.397 |
| 5.85625 | 10.0909 |
| 6.1 | 9.80163 |
| 6.34375 | 9.52786 |
| 6.5875 | 9.26847 |
| 6.83125 | 9.02242 |
| 7.075 | 8.78874 |
| 7.31875 | 8.56655 |
| 7.5625 | 8.35505 |
| 7.80625 | 8.15353 |
| 8.05 | 7.9613 |
| 8.29375 | 7.77775 |
| 8.5375 | 7.60233 |
| 8.78125 | 7.43451 |
| 9.025 | 7.27384 |
| 9.26875 | 7.11985 |
| 9.5125 | 6.97217 |
| 9.75625 | 6.8304 |
| 10 | 6.69421 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Current | 1.8 | A |
| Terminal voltage | 7.2 | V |
| Source chemical power in ideal emf model | 16.2 | W |
| Load power | 12.96 | W |
| Internal thermal power | 3.24 | W |
| Load efficiency | 0.8 | ratio |
| Theoretical load resistance for maximum power | 1 | Ω |
| Theoretical maximum load power | 20.25 | W |
| Power residual | -1.77636e-15 | W |
Maximum load power occurs at matched resistances, where only half the source power reaches the load.
Current E/(r+R) gives Vterminal=E−Ir and Pload=E²R/(r+R)². The inequality (R+r)²≥4Rr proves Pload≤E²/(4r) without calculus; equality requires R=r. Efficiency R/(R+r) rises toward one as load resistance grows, while power eventually falls toward zero. Neither ideal objective establishes a real safe operating point.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Maximum load power occurs at matched resistances, where only half the source power reaches the load.
Current E/(r+R) gives Vterminal=E−Ir and Pload=E²R/(r+R)². The inequality (R+r)²≥4Rr proves Pload≤E²/(4r) without calculus; equality requires R=r. Efficiency R/(R+r) rises toward one as load resistance grows, while power eventually falls toward zero. Neither ideal objective establishes a real safe operating point.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
Understand
Apply
Public practice checks a specific claim. It does not establish independent mastery.
Which independent observation could reject this model rather than merely refit its parameters?
Continue
Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.
Skipping retains experiments and awards no mastery.
Save a science backup or import one
Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.
A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.
Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.
Complete file sets and course backups
A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.
The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.
0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.
Science storage: loading. No account or remote computation is required.