Electromotive Force (EMF) and Batteries

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Core Idea

Electromotive force (EMF) ε is the work per unit charge a battery (or other source) does; EMF creates and maintains potential difference. Real batteries have internal resistance r; terminal voltage V = ε − Ir decreases with current.

Explainer

From your study of potential energy, you know that moving charges uphill in a potential field requires work. In a circuit, current flows from high to low potential through the external load, dissipating energy. But something must continuously pump charge back from low to high potential to sustain the current — that something is the electromotive force. Despite the name, EMF is not a force; it is energy per unit charge (measured in volts), representing the work done by the source (chemical, mechanical, thermal) per coulomb moved through it. A 12 V car battery does 12 joules of work for every coulomb it drives around the circuit.

The idealized battery maintains a fixed potential difference ε across its terminals regardless of the current drawn. Real batteries do not behave this way. Every real battery has internal resistance r — resistance inside the battery itself due to the ionic solution and electrodes. When current I flows, there is a voltage drop Ir within the battery, so the actual terminal voltage is V = ε − Ir. Draw more current and the terminal voltage sags. This is why a nearly dead battery reads 12 V when disconnected but might only sustain 9 V when trying to start a car engine. The EMF is still 12 V; the internal resistance has increased, causing a larger voltage drop at the high current demanded.

You already know from Ohm's law that V = IR for a resistor. Combining that with the battery model gives you the complete single-loop circuit: ε = I(R + r). The total driving EMF equals the total resistive voltage drop across both the external load R and the internal resistance r. Rearranging, I = ε/(R + r). Notice the limiting cases: if r → 0 (ideal battery), I = ε/R as you would naively expect; if R → 0 (short circuit), I = ε/r, which can be dangerously large since r is small.

The power perspective ties it together. The battery delivers power P = εI. Of this, P_load = I²R goes to the external load doing useful work, and P_lost = I²r is wasted as heat inside the battery. Maximum power is delivered to the load when R = r — a result called the maximum power transfer theorem that will reappear in circuit analysis. The EMF concept is the bridge between the energy-source view (chemistry pumping charge) and the circuit-analysis view (voltage sources driving currents through resistances); mastering it is essential before you move to multi-loop circuits with Kirchhoff's laws.

Practice Questions 5 questions

Prerequisite Chain

Counting to 10Counting to 20Understanding ZeroThe Number ZeroCounting to FiveOne-to-One CorrespondenceCombining Small Groups Within 5Addition Within 10Addition Within 20Two-Digit Addition Without RegroupingTwo-Digit Addition with RegroupingAddition Within 100Repeated Addition as MultiplicationMultiplication Facts Within 100Division as Equal SharingDivision as Grouping (Measurement Division)Division: Grouping (Repeated Subtraction) ModelDivision: Fair Sharing ModelDivision as Equal SharingDivision as GroupingBasic Division FactsDivision Facts Within 100Two-Digit by One-Digit DivisionDivision with RemaindersRemainders and Quotients in DivisionDivision Word ProblemsIntroduction to Long DivisionFactors and MultiplesPrime and Composite NumbersEquivalent FractionsRelating Fractions and DecimalsDecimal Place ValueReading and Writing DecimalsComparing and Ordering DecimalsAdding and Subtracting DecimalsMultiplying DecimalsDividing DecimalsDividing FractionsMixed Number ArithmeticOrder of OperationsInteger Order of OperationsVariable ExpressionsCombining Like TermsOne-Step EquationsTwo-Step EquationsSolving Multi-Step EquationsEquations with Variables on Both SidesAngle Pairs: Complementary, Supplementary, and VerticalParallel Lines and TransversalsCorresponding AnglesAlternate Interior AnglesTriangle Angle Sum TheoremExterior Angle TheoremTriangle Inequality TheoremSimilar Triangles: AA SimilaritySimilar Triangles: SSS and SAS SimilarityProportions in Similar TrianglesRight Triangle Trigonometry IntroductionTrigonometric Ratios ReviewRadian MeasureConverting Between Degrees and RadiansThe Unit CircleGraphing Sine and CosineGraphing Tangent and Reciprocal Trigonometric FunctionsDerivatives of Trigonometric FunctionsAntiderivativesIterated Integrals and Fubini's TheoremDouble Integrals in Cartesian CoordinatesDouble Integrals over Rectangular RegionsDouble Integrals in Polar CoordinatesDouble Integrals: Definition and SetupIterated Integrals and Fubini's TheoremDouble Integrals over Rectangular RegionsDouble Integrals over General RegionsApplications of Double Integrals: Area, Mass, and MomentsTriple Integrals in Cartesian CoordinatesTriple Integrals in Cylindrical and Spherical CoordinatesChange of Variables and the Jacobian DeterminantApplications of Triple Integrals: Volume and MassVector Fields and Their RepresentationsLine Integrals of Vector FieldsGreen's TheoremSurface Integrals and Flux of Vector FieldsSurface Integrals and Flux of Vector FieldsDivergence Theorem: Flux and OutflowDivergence TheoremElectric FluxGauss's LawConductors in Electrostatic EquilibriumCapacitance and CapacitorsDielectricsDielectric Constant and Relative PermittivityElectric Field Inside Dielectric MaterialsDielectric Materials and PolarizationDielectric Susceptibility and PermittivityEnergy Density in Electric FieldsElectric Current and Current DensityElectrical Resistance and ResistivityOhm's Law and Circuit ElementsElectromotive Force (EMF) and Batteries

Longest path: 100 steps · 485 total prerequisite topics

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