A battery can show its full labelled voltage with almost no load, then deliver a lower voltage when asked for current. The missing voltage is a clue about the source, not a failure of charge conservation.
We connect that observation to charging, cell combinations, maximum power, equivalent sources and null measurements.
What you will learn to do
- Infer emf and internal resistance from a voltage–current relation.
- Keep terminal and chemical energy accounts consistent during delivery and absorption.
- Compare identical-cell arrangements for a stated load.
- Distinguish maximum load power from efficiency.
- Use equivalent-source and null-measurement methods within their assumptions.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Real sources, cell groups and useful power Learning goals: - Infer emf and internal resistance from a voltage–current relation. - Keep terminal and chemical energy accounts consistent during delivery and absorption. - Compare identical-cell arrangements for a stated load. - Distinguish maximum load power from efficiency. - Use equivalent-source and null-measurement methods within their assumptions. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BEFORE YOU READ
Bring back one idea
Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.
Why can a cell’s loaded terminal voltage differ from its emf?
Compare your reasoning
Current through its internal resistance produces an additional drop. With delivered current referenced positive, V = E − Ir; under absorption that same signed equation gives a voltage above E.
Revisit the explanation →01 / Real sources, cell groups and useful power
A battery label is an energy-per-charge claim
An ideal emf element raises electric potential by converting some other form of energy into electrical energy. Despite its historical name, electromotive force is not measured in newtons. Its unit is volt: joule of supplied energy per coulomb moved through the source.
A useful first model of a real cell combines an ideal emf E with a positive series resistance r. The emf describes conversion; the resistor describes internal dissipation. This model holds E and r constant over the stated interval. Temperature, chemistry and state of charge can change them in a real device, so do not treat a fit over a small range as a universal battery law.
Define I as current delivered outward from the positive terminal. The internal resistor uses up a drop Ir when I is positive. An open terminal gives I = 0 and measures E in this model. A finite-resistance voltmeter draws current and can read slightly less.
V is the positive-terminal potential minus the negative-terminal potential. The second formula additionally assumes a single passive load R across the source.
WORKED EXAMPLE
Separate the label from the loaded voltage
A model source has E = 18 V and r = 3 Ω. It supplies a 6 Ω resistor. Find current and terminal voltage.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Real sources, cell groups and useful power
Focus: A battery label is an energy-per-charge claim
Learning goals:
- Infer emf and internal resistance from a voltage–current relation.
- Keep terminal and chemical energy accounts consistent during delivery and absorption.
- Compare identical-cell arrangements for a stated load.
- Distinguish maximum load power from efficiency.
- Use equivalent-source and null-measurement methods within their assumptions.
Equations in this section (LaTeX):
V=E-Ir,\qquad I=\frac{E}{R+r}
Meaning: V is the positive-terminal potential minus the negative-terminal potential. The second formula additionally assumes a single passive load R across the source.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.02 / Real sources, cell groups and useful power
Use several readings to test the straight-line model
If E and r really are constant, terminal voltage plotted against delivered current is a straight line. Its vertical intercept is E and its slope is −r. Two exact observations determine a line; further observations test whether that line describes the source over the range of interest.
Specify which quantity is on which axis before reading a slope. Plotting I against V gives slope −1/r instead. Use amperes rather than milliamperes when you want the slope in ohms. Measurement uncertainty and drift affect fitted values, so report the model’s range as well as its parameters.
A source model inferred from a pair of ordinary load readings can predict another load reading. It does not license an actual short-circuit experiment. Here any short-circuit current is a mathematical limiting value of the assigned constant-parameter model.
WORKED EXAMPLE
Read resistance from the voltage lost per ampere
A source has terminal voltage 12 V while delivering 1.0 A and 9 V while delivering 2.5 A. Assume constant E and r. Find both.
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STOP & THINK
A model V–I line has intercept 9 V and voltage decreases by 0.8 V per extra ampere. Find terminal voltage at a delivered current of 2.5 A.
Answer in V.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Real sources, cell groups and useful power Focus: Use several readings to test the straight-line model Learning goals: - Infer emf and internal resistance from a voltage–current relation. - Keep terminal and chemical energy accounts consistent during delivery and absorption. - Compare identical-cell arrangements for a stated load. - Distinguish maximum load power from efficiency. - Use equivalent-source and null-measurement methods within their assumptions. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
03 / Real sources, cell groups and useful power
Charging changes the current sign, not the equation
A second source can drive current into a cell’s positive terminal. With our delivered-current reference, I is then negative. Keeping V = E − Ir gives V greater than E: the external source must overcome the emf and the internal resistive drop.
For an ideal rechargeable emf element, energy entering its positive terminal can increase chemical storage. The series resistor still heats because I²r is positive. Not every physical cell tolerates charging, and actual charging control is outside this model; we use the circuit as a specified reversible-source calculation.
Do not decide energy direction from the source symbol alone. Multiply its emf by the signed delivered current. Positive EI means delivery by the ideal emf; negative EI means absorption. The terminal power also includes the internal resistor’s loss.
WORKED EXAMPLE
Terminal voltage can exceed emf
An ideal rechargeable-source model has E = 10 V and r = 1.5 Ω. An external circuit drives 2 A into its positive terminal. Find terminal voltage, energy-storage rate and internal heating.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Real sources, cell groups and useful power Focus: Charging changes the current sign, not the equation Learning goals: - Infer emf and internal resistance from a voltage–current relation. - Keep terminal and chemical energy accounts consistent during delivery and absorption. - Compare identical-cell arrangements for a stated load. - Distinguish maximum load power from efficiency. - Use equivalent-source and null-measurement methods within their assumptions. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
04 / Real sources, cell groups and useful power
Combine both emf and internal resistance
Identical cells in series add their emfs when their polarities aid, and their internal resistances add too. Identical cells in parallel share terminal voltage and divide the delivered current; the equivalent emf stays that of one cell while the equivalent internal resistance decreases.
The word identical is doing real work. Unequal emf sources connected in parallel can drive a circulating current even with no external load. Solve that case using the network equations rather than averaging the labels. Ideal unequal voltage sources with zero series resistance cannot consistently impose two different voltages on the same terminals.
For N identical cells arranged into m equal parallel branches of n series cells each, N = mn. The equivalent emf is nE and the equivalent internal resistance is nr/m. The load current depends on the load as well as the arrangement. More series cells do not automatically give the greatest current for every external resistance.
All cells have the same emf, resistance and polarity; every parallel branch contains the same number of cells.
WORKED EXAMPLE
Series versus parallel for a stated load
Two identical cells each have E = 3 V and r = 1 Ω. Compare the current through a 2 Ω load when the cells are connected in aiding series and in parallel.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Real sources, cell groups and useful power
Focus: Combine both emf and internal resistance
Learning goals:
- Infer emf and internal resistance from a voltage–current relation.
- Keep terminal and chemical energy accounts consistent during delivery and absorption.
- Compare identical-cell arrangements for a stated load.
- Distinguish maximum load power from efficiency.
- Use equivalent-source and null-measurement methods within their assumptions.
Equations in this section (LaTeX):
E_{\rm eq}=nE,\qquad r_{\rm eq}=\frac{nr}{m}=\frac{n^2r}{N}
Meaning: All cells have the same emf, resistance and polarity; every parallel branch contains the same number of cells.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.05 / Real sources, cell groups and useful power
Maximum load power and maximum efficiency ask different questions
For a passive load R, the useful power is I²R and the source’s internal heating is I²r. A very small load lets a large current flow but receives little voltage. A very large load receives nearly the emf but takes little current. Between these limits, the load power has a maximum.
Differentiate the load-power function while holding E and r fixed. The maximum occurs at R = r for this simple positive-resistance DC model. Half the ideal source power is then lost internally. Maximum power transfer is therefore not a claim of maximum efficiency.
As R grows, the fraction of source power reaching the load approaches one, but the delivered power tends to zero. Always ask whether the task requests watts, an efficiency fraction, terminal voltage or current. Optimizing one need not optimize the others.
Efficiency here counts only the assigned internal resistor loss. It does not include every electrochemical or external conversion loss.
Power has a peak; efficiency keeps increasing
- Load power / maximum
- 0
- Efficiency fraction
- 0
WORKED EXAMPLE
Twelve watts at one-half efficiency
A model source has E = 12 V and r = 3 Ω. Find the load for maximum power and compare it with a 9 Ω load.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Real sources, cell groups and useful power
Focus: Maximum load power and maximum efficiency ask different questions
Learning goals:
- Infer emf and internal resistance from a voltage–current relation.
- Keep terminal and chemical energy accounts consistent during delivery and absorption.
- Compare identical-cell arrangements for a stated load.
- Distinguish maximum load power from efficiency.
- Use equivalent-source and null-measurement methods within their assumptions.
Equations in this section (LaTeX):
P_L=\frac{E^2R}{(R+r)^2},\qquad\eta=\frac{P_L}{EI}=\frac{R}{R+r}
Meaning: Efficiency here counts only the assigned internal resistor loss. It does not include every electrochemical or external conversion loss.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.06 / Real sources, cell groups and useful power
Replace a linear network only at the terminals you chose
A fixed linear resistive network can often be replaced, as seen from two chosen terminals, by an equivalent ideal voltage source in series with a resistance. The open-circuit terminal voltage is the equivalent emf. The equivalent resistance describes how terminal voltage changes as the load current changes.
For a network of resistors and independent ideal voltage sources, deactivate each ideal voltage source by replacing it with a short, then find the resistance seen from the chosen terminals. Keep all actual series resistances. Dependent sources require a test-source calculation rather than this simple shortcut; nonlinear devices require a stated operating model.
The equivalent preserves the relation between terminal voltage and load current. It does not preserve the current or heating of every internal component. If a question asks which of two original sources is absorbing energy, return to the original branches after finding the load voltage.
WORKED EXAMPLE
Two branches seen by one load
Two source branches in parallel have (E₁, r₁) = (12 V, 2 Ω) and (E₂, r₂) = (6 V, 3 Ω), with both positives toward the output. Find their equivalent source and the voltage across a 6 Ω load.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Real sources, cell groups and useful power Focus: Replace a linear network only at the terminals you chose Learning goals: - Infer emf and internal resistance from a voltage–current relation. - Keep terminal and chemical energy accounts consistent during delivery and absorption. - Compare identical-cell arrangements for a stated load. - Distinguish maximum load power from efficiency. - Use equivalent-source and null-measurement methods within their assumptions. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
07 / Real sources, cell groups and useful power
A null comparison avoids loading the tested cell
An ideal potentiometer comparison uses a uniform wire carrying a steady driver current. Its potential drop per unit length is a constant g. A test cell is compared with the drop along a chosen length using a detector branch. At balance, the detector current is zero, so the tested cell supplies no current through that branch.
The balance relation is E = gl for an unloaded test cell. Comparing two emfs with the same unchanged gradient gives their ratio from the two balance lengths. Reversing the test-cell polarity or choosing a driver gradient too small to reach its emf prevents a balance within the available wire; a missing null does not imply a zero emf.
To infer a cell’s internal resistance, compare its unloaded emf with its terminal voltage while a known resistor R draws current. The detector branch still carries zero current at its own balance, but the external load does not. Keep the driver gradient unchanged between readings, then combine the length ratio with V = ER/(R+r).
Uniform wire, stable driver gradient, negligible detector current at null and a constant-parameter cell over the comparison.
WORKED EXAMPLE
The shorter balance length reveals internal loss
A cell balances at 75 cm when unloaded and at 60 cm with a 4 Ω resistor across its terminals. The potentiometer gradient is unchanged. Find the cell’s internal resistance.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Real sources, cell groups and useful power
Focus: A null comparison avoids loading the tested cell
Learning goals:
- Infer emf and internal resistance from a voltage–current relation.
- Keep terminal and chemical energy accounts consistent during delivery and absorption.
- Compare identical-cell arrangements for a stated load.
- Distinguish maximum load power from efficiency.
- Use equivalent-source and null-measurement methods within their assumptions.
Equations in this section (LaTeX):
\frac{E}{V}=\frac{l_0}{l_R},\qquad r=R\left(\frac{l_0}{l_R}-1\right)
Meaning: Uniform wire, stable driver gradient, negligible detector current at null and a constant-parameter cell over the comparison.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.08 / Real sources, cell groups and useful power
Choose the source model before optimizing an answer
Begin by stating the terminals and whether the source parameters are fixed. Decide whether the source is alone, opposed by another source, or embedded in a network. Use current signs to decide delivery or absorption; use a graph or multiple readings to test whether a constant internal resistance is a useful approximation.
For cell arrangements, write the equivalent emf and resistance before computing current. For a maximum-power question, keep the variable being optimized explicit. For a measurement, ask how the instrument changes the circuit and whether a null removes that particular loading effect.
Check the result at open load, very small load, and zero internal resistance when those limits are meaningful. A model with an ideal voltage source and zero total loop resistance has no finite steady-current solution when its emf is nonzero; reporting an ordinary finite current would conceal the model’s contradiction.
WORKED EXAMPLE
Choose an arrangement with an integer number of cells
Eight identical cells have E = 1.5 V and r = 1 Ω each. They form equal parallel branches, each containing n series cells, to supply R = 2 Ω. Compare n = 1, 2, 4 and 8.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Real sources, cell groups and useful power Focus: Choose the source model before optimizing an answer Learning goals: - Infer emf and internal resistance from a voltage–current relation. - Keep terminal and chemical energy accounts consistent during delivery and absorption. - Compare identical-cell arrangements for a stated load. - Distinguish maximum load power from efficiency. - Use equivalent-source and null-measurement methods within their assumptions. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BRING THE IDEAS TOGETHER
What should stay with you
- V = E − Ir uses current delivered from the positive terminal; inward current has the opposite sign.
- A source model has a stated operating range.
- Maximum power occurs at matched positive resistances, with one-half efficiency in this model.
- An equivalent source preserves terminal behavior, not every internal current.
RETRIEVE, THEN EXPLAIN
Try it without the examples
Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.
SELF CHECK 1
A source has E = 8 V and r = 2 Ω. What load resistance maximizes its delivered load power?
Answer in Ω.
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SELF CHECK 2
Two unequal-emf branches with positive series resistances have no external load. Must all currents vanish?
- Yes, because the load is open
- No, a circulating current can flow
- Yes, because terminal voltage is shared
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SELF CHECK 3
A potentiometer detector reads zero while a resistor loads the tested cell. Which current must vanish?
- The external load current
- The detector-branch current
- Every current in the driver wire
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APPLY THE CHAPTER
Take the ideas into a problem
Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.
Find the emf that makes a branch quietAdvanced · Hints and guided reasoningInfer a source from a reversed branch currentIntermediate · Hints and guided reasoningFind the hidden open-circuit interceptIntermediate · Hints and guided reasoningChoose a physically available cell arrangementAdvanced · Hints and guided reasoningChoose efficiency instead of peak powerIntermediate · Hints and guided reasoningOpen terminals can hide an internal currentAdvanced · Hints and guided reasoningPredict a load through an equivalent sourceAdvanced · Hints and guided reasoningUse a null ratio to isolate internal resistanceAdvanced · Hints and guided reasoningThe load sees less than the emfIntermediate · Hints and guided reasoningCOME BACK TO THE IDEA
What can you explain later?
Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.
Next study session
At the load for maximum power, is this source model nearly lossless?
Compare your reasoning
No. When R = r, equal powers heat the load and the internal resistor. Efficiency is one half. A larger load improves this fraction while eventually reducing useful power toward zero.
Revisit the explanation →A few days later: change the situation
Does zero external load current imply zero current everywhere inside a network of sources?
Compare your reasoning
No. Unequal emfs with finite series resistances can drive circulating internal current. A terminal equivalent preserves load behavior, not the state of every internal branch.
Revisit the explanation →Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.
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