EMF Internal Resistance & Cells EMF Internal Resistance & Cells A real battery is not a perfect supplier of voltage. It is a chemical energy source (with a fixed electromotive force, EMF) in series with a small hidden resistance inside it. When no current flows, the battery’s terminal voltage equals its EMF. But the moment you draw current, part of the EMF gets “spent” inside the battery across its internal resistance, so the terminal voltage drops. This is why a phone that shows 100% off-load can suddenly dip under a heavy app load: the battery sags due to internal resistance. Everything practical in circuits—how bright a bulb glows, how hot a wire gets, how much power a device actually receives—depends on this simple but crucial idea: terminal voltage equals EMF minus the internal drop. Building on this, we group multiple cells in series or parallel to reach the voltage or current we need. Series raises the EMF (and internal resistance), parallel keeps the EMF the same but reduces the internal resistance. With these tools, we can design battery packs for torches, toys, cars, and medical devices, and also predict efficiency, heating, and safety limits like short-circuit currents. Mastering this chapter means reading circuits from the battery’s point of view: track the current, account for the internal resistance, and you will get the right terminal voltage, power distribution, and performance—every time. Analogy: A water pump (EMF) pushes water through a hidden narrow pipe (internal resistance). With the tap closed (no current), gauge pressure at the outlet equals the pump’s rating. Open the tap wide (large current), pressure at the outlet drops because some pressure is lost inside the narrow pipe. remember Work done per unit charge by the chemical source. It is the ideal source voltage of a cell when no current flows; symbol ; unit V . EMF (Electromotive Force) The effective resistance inside a cell due to electrolyte and electrodes. Symbol r ; causes a voltage drop Ir inside the cell when current I flows. Internal Resistance Terminal Voltage The actual voltage across the external terminals of the cell. For a discharging cell, V = - Ir . The external resistance R connected to the cell. Receives power P load = I 2 R = VI . Load Resistance External resistance R 0 . Current tends to I / r and terminal voltage V 0 , causing severe heating inside the cell. Short Circuit EMF and terminal voltage are equal only when current is zero (open circuit). As soon as current flows, the internal resistance r causes a drop Ir inside the cell, so the terminal voltage falls to - Ir . This is why high-drain devices cause more voltage sag than low-drain ones. The size of sag depends on both I and r : fresh, high-quality batteries have smaller r , so they sag less under load; old or cold batteries have larger r , so they sag more and waste more power as internal heat. Terminal voltage (discharge) When the cell supplies current I to an external load, the terminal voltage drops below EMF by Ir . V = - I r By Ohm’s law and series addition: external R plus internal r . Terminal voltage is the drop across external R only; use I(R + r) = . V = - I r Quasi-steady DC conditions; temperature effects ignored. Internal resistance r modeled in series with ideal source of EMF . Current I flows from positive to negative terminal through the external load (discharge). Boundary checks: If the circuit is open ( I=0 ), V= (no drop inside). If the terminals are shorted ( R=0 ), current becomes I = /r ideally, giving V=0 across the terminals but a large internal drop Ir= and dangerous heating. Real cells may have additional limits (electrochemistry, fuses). Open-circuit voltage ε/r Short-circuit current control dependent Current I Terminal voltage vs current: intercept gives EMF, magnitude of slope gives internal resistance. A straight line V = ε − I r with intercept ε at I = 0 and zero at I = ε/r; slope −r. custom Terminal voltage V 2D PLOT Terminal voltage vs current (V = ε − I·r) V = emf - I r emf EMF ε Internal resistance The V – I straight line is experimentally useful. Plot terminal voltage versus current for various loads R , and fit a straight line. The intercept at I=0 gives , and the magnitude of slope gives r . This method avoids directly shorting the cell and gives both parameters from safe measurements. Discharge: current leaves positive terminal into the circuit; V = - Ir . Charge: an external source pushes current into the positive terminal; then across the cell, drop is + Ir . KVL sense: Going from negative to positive plate inside the cell raises potential by for the ideal source, but the internal resistor causes a drop Ir along current. Sign conventions (discharge vs charge) Boundary conditions: Open circuit ( I=0 ) ⇒ V= . Short circuit ( R=0 ) ⇒ I= /r , V=0 , internal heating I 2r is maximum. Very large R ⇒ small current, negligible sag, V . tip Power splits between the load and the internal resistance. The source’s chemical power is P chem = I . Of this, P int = I 2 r is wasted as heat inside the cell, and P load = IV = I( - Ir) is delivered to the external circuit. The fraction delivered to the load (efficiency) is = V/ = R/(R + r) . Efficiency approaches 100% only when r R . Chemical power equals load power plus internal heating. Power split and efficiency This formula applies when a battery or electrochemical cell is discharging current (I) through an external load resistance, accounting for i Single source with EMF and internal resistance r feeding a variable load R . Steady DC; temperature constant; linear elements. P load ; is maximum when ; R = r Express load power in terms of R . Differentiate with respect to R . Critical point at R=r ; it is a maximum. Maximum power transfer at R = r At R=r , the power delivered to the load is maximized, but the efficiency is only = R/(R + r) = 1/2 . This result is important for matching in communication circuits but is not energy-efficient for batteries. For long battery life, we prefer R r so that most of the chemical power goes to the load, not into internal heating. EMF and terminal voltage are always the same. They are equal only at zero current (open circuit). Under load, terminal voltage is - Ir and is strictly less than EMF for discharge. Power from the chemical source is I , while power received by the load is IV with V= - Ir . The difference I 2r is internal heating. Battery’s power output is always VI where V is EMF. Cells in Series and Parallel We combine cells to obtain desired voltage and current. Series combination adds EMFs and internal resistances, useful for higher voltage. Parallel combination keeps EMF the same (for identical cells) but reduces internal resistance, useful for higher current with less sag. Mixed grouping (series-parallel) tunes both. Real packs must also consider cell matching: cells should be identical and of similar age/state to prevent unhealthy internal currents. Series combination In series, the same current flows through all cells; EMFs and internal resistances add. This analysis applies when multiple batteries or internal components are connected in series, or when calculating the performance of a singl Parallel combination (general) For unequal cells in parallel, equivalent EMF is a resistance-weighted average; internal resistances combine like parallel resistors. This rule applies when multiple galvanic cells (batteries) are connected end-to-end, such that the positive terminal of one cell is connecte Terminals tied together so both cells have the same terminal voltage V . Currents in branches are I 1 = ( 1 - V)/r 1 , I 2 = ( 2 - V)/r 2 (passive sign). External load draws current I = I 1 + I 2 ; equivalent model is a single source ( eq , r eq ) with V = eq - Ir eq . eq = 1 /r 1 + 2 /r 2 1/r 1 + 1/r 2 , r eq = 1 1/r 1 + 1/r 2 Rearrange to isolate the common voltage V . Identify the form I = (1/r eq )( eq - V) , hence 1/r eq = 1/r 1 + 1/r 2 . Equivalent EMF of two parallel cells with ( 1 , r 1 ) and ( 2 , r 2 ) For identical cells: in series, eq = n , r eq = nr ; in parallel, eq = , r eq = r/n . For a mixed pack with m cells in series per row and k such rows in parallel (all identical), eq = m , r eq = (mr)/k . This pattern guides battery-pack design: choose m for voltage, k for current capability. Grouping Equivalent EMF Equivalent r Use When Pros/Cons Series (n cells) eq = n r eq = nr Need higher voltage Pros: raises voltage; Cons: raises internal resistance, more sag at high current Parallel (n cells) eq = r eq = r/n Need more current with less sag Pros: lower internal resistance; Cons: careful cell matching needed Mixed (m×k) eq = m r eq = (mr)/k Need both voltage and current Pros: tunable; Cons: increased complexity Reversed cell in series: If one cell is flipped, its EMF subtracts. For identical cells, net EMF becomes (n - 2) , but internal resistances always add: nr . Many students forget to subtract the EMF while still adding r . neet-alert Internal currents can flow even without an external load if unequal cells are paralleled: the higher-EMF cell drives current into the lower-EMF one. This wastes power and can damage cells. Hence, parallel connection requires identical cells with similar state-of-charge. In exams, if unequal parallel cells are given, use the weighted-average EMF and parallel r formula and watch the sign of branch currents carefully. Current, terminal voltage, power in load, efficiency. A 12 V cell with internal resistance 1.0 Ω supplies a 5.0 Ω resistor. EMF = 12 , V Internal resistance r = 1.0 , Load R = 5.0 , A, V, W, dimensionless Use I = /(R + r) and V = - Ir , then P load = I 2 R , = V/ . easy Interpretation: Two-fifths of the total resistance is internal, so 17% of chemical power is lost as heat in the cell. The 10 V terminal voltage explains why a “12 V” battery may show less under load. Efficiency improves if we reduce r or increase R . Current and load power for (A) and (B); which gives higher power? Four identical cells each of = 1.5 , V and r = 0.50 , are used to power a 3.0 , load. Compare two configurations: (A) all in series; (B) two-in-series per row, and two such rows in parallel. Compute ( eq , r eq ) for each configuration, then I = eq /(R + r eq ) and P = I 2 R . medium A, W Each cell: = 1.5 , V , r = 0.50 , Load: R = 3.0 , Why series wins here: the load is moderate, so higher EMF outweighs the penalty of larger internal resistance. If the load were very small (demanding large current), the parallel-heavy arrangement could reduce sag more effectively. Always compare R with r eq . 1 = 2.0 , V , r 1 = 1.0 , 2 = 1.5 , V , r 2 = 0.50 , Load for part (b): R = 2.0 , V, A Use the parallel formulas or branch relations I i = ( i - V)/r i . hard Negative means current enters cell 2 (it gets charged). (a) Open-circuit terminal voltage V and I 1 , I 2 . (b) With load, V , I , and branch currents. Two cells are connected in parallel: ( 1 =2.0 , V , r 1 =1.0 , ) and ( 2 =1.5 , V , r 2 =0.50 , ) . (a) With no external load, find the common terminal voltage and branch currents. (b) If a 2.0 , load is connected across the combination, find the terminal voltage and total current. Key lesson: unequal parallel cells share current unequally; the higher-EMF cell supplies more. With no load, their terminal voltage is the resistance-weighted average. With a load, treat the pair as an equivalent source to find I and V , then back-calculate individual branch currents. Battery modeled as EMF in series with internal resistance r and external load R in series. Schematic: battery symbol with a small resistor r inside the battery box, then an external resistor R. Model of a real cell: an ideal EMF source in series with an internal resistor r feeding an external load R. How to use the visualizer: Start with small r and vary R to see that V stays near (high efficiency). Increase r and note how V sags more for the same current. Try R 0 to glimpse the dangerously large short-circuit current /r and zero terminal voltage. Connect a variable resistor (rheostat) as load to the cell. Measure current I with an ammeter in series and terminal voltage V with a high-resistance voltmeter in parallel with the load. Record pairs (I, V) for several R settings, avoiding very large currents. Plot V on y-axis vs I on x-axis. Fit a straight line. Intercept at I=0 is . Slope dV/dI is -r . Experiment: Find ε and r from V–I data This V–I plot method is robust and does not require opening the circuit. Since the voltmeter has large resistance, it samples terminal voltage without significantly altering the current. Avoid regions where the cell warms up noticeably; temperature dependence can change r and distort your slope. Voltmeter reading vs EMF: A high-resistance voltmeter across the cell with no external load reads EMF. But across a working load, it reads terminal voltage V = - Ir , not EMF. Many errors come from mixing up these two situations. neet-alert Move inside the ideal source from − to + plate: potential rises by + . Move through internal resistor along current: potential drops by Ir . Around a loop, algebraic sum of potential changes is zero: + - Ir - IR = 0 for discharge. KVL sign rules through a cell Charging a cell flips the internal drop sign relative to the external supply. If a DC source of voltage V ext charges the cell with current I (entering the positive terminal), then the drop across the cell terminals equals V ext = + Ir . The extra Ir term is the voltage required to push current through the internal resistance while also overcoming the EMF. External charger voltage must exceed EMF by Ir to drive current into the cell. Charging relation This formula calculates the terminal voltage of a single battery or cell when it is connected to an external circuit, regardless of whether Joule heating and safety: Internal heat I 2 r rises rapidly with current. A short circuit produces I /r and P int = I 2 r = I , which can overheat the cell, damage chemistry, and start fires. Fuses and protection circuits limit current. In daily life, thin wires and old batteries (larger r ) warm up more for the same current. remember Old batteries show large voltage drop under load because their internal resistance has increased. Off-load voltage may look fine, but the moment you draw current, terminal voltage collapses. Source Typical EMF Typical Internal r Notes AA alkaline cell 1.5 V 0.1–0.3 Ω (fresh), higher when old Good for moderate currents Lead–acid car battery 12 V (6 × 2 V) ≈ 0.01–0.02 Ω Very low r for high starter currents Phone Li-ion cell 3.7 V nominal ≈ 0.05–0.1 Ω (varies) Low r but rises with age and cold Lab DC supply (regulated) Variable Very small (effective) Active regulation mimics near-ideal source Potentiometer link: A potentiometer compares EMFs without drawing current, so it measures the true EMF, not the terminal voltage under load. It can also measure internal resistance by noting terminal voltage at different currents and extrapolating to I=0 . This will appear later as a dedicated experiment. SPRS rule for identical cells: Series → Potential (EMF) Sums, Resistance Sums; Parallel → EMF Same, Resistance Splits (r/n). Memory hook for quick NEET calculations. neet-alert Do not apply V = + Ir during discharge. That form is for charging (current forced into the positive terminal). In discharge, terminal voltage is V = - Ir . Edge cases in combinations: For a series string where one cell fails open, the circuit is broken and I=0 despite other healthy cells. If one cell becomes shorted internally ( r 0 and 0 effectively), it drags down the string’s voltage and wastes power. In parallel packs, a weak cell with higher r naturally shares less current, but a cell with much lower EMF can get charged unintentionally by neighbors. Mixed grouping in practice: Suppose you need about 9 V at moderate current using 1.5 V cells. Choose m=6 in series to get 9 V EMF. If the load current causes too much sag, add k identical series-rows in parallel to reduce r eq = (mr)/k . Balance between voltage target (by m ) and sag control (by k ). Measurement caution: Internal resistance is not strictly constant; it can depend on temperature and state-of-charge. Short bursts of current may show a different effective r than sustained loads. In exam problems, assume r is constant unless stated otherwise. Practical design tip: To maximize efficiency for a given load R , minimize the ratio r/R . This can be done by choosing a source with small internal resistance (e.g., a car battery for high current tasks) or by paralleling identical cells to reduce r eq when higher current is needed at modest voltage. Reverse-polarity hazards: Connecting a cell backward in a series pack creates opposing EMF and large internal power loss. The reversed cell can get driven into reverse charging, which is unsafe for many chemistries. In calculations, always sum EMFs algebraically with sign but sum internal resistances arithmetically. Troubleshooting with V–I: If a device intermittently resets under load, measure its supply terminal voltage while it draws peak current. A large drop indicates either high internal resistance of the source or high resistance in wiring/contacts. Replace the source or reduce series resistance to stabilize the device. Energy accounting over time: Over an interval t , chemical energy supplied is E chem = (I ) t , external energy received is E load = (IV) t , and internal heat is E int = I 2 r t . If current varies with time, integrate: E = I(t) , ,dt , etc. For NEET, steady cases dominate, but keep the split in mind. Multiple sources in general loops: Use Kirchhoff’s Voltage Law (KVL). Assign current directions, write KVL around loops with sign conventions for EMF and resistor drops, and solve simultaneous equations. Internal resistances are treated like any other series resistance that happens to sit inside the source symbol. Real meters: An ideal ammeter has negligible internal resistance, so it measures true current without altering it much. An ideal voltmeter has very large resistance, so it measures terminal voltage without drawing significant current. In practice, meter imperfections can slightly change readings, but exam problems often take meters as ideal unless specified. Temperature effects: Most cells exhibit higher internal resistance when cold, so cars struggle to start on winter mornings. Heating raises reaction rates and can lower r temporarily, but overheating is dangerous. Exam numericals usually ignore temperature dependence unless data are provided. Charging window: For safe charging, external voltage must exceed EMF by Ir , but should remain within the chemistry’s limits. Slow charging corresponds to small I , so V ext only slightly exceeds ; fast charging needs a larger margin but produces more internal heat I 2 r . Power rating vs grouping: If a device needs 9 V at 0.5 A (4.5 W), a 9 V series pack may provide the voltage but could have too high r eq to sustain 0.5 A without heavy sag. Adding parallel rows lowers r eq , reducing sag and heat, ensuring the device receives near-rated power. Work backward from measurements: If a 12 V battery under a 6 Ω load shows 10.8 V at the terminals, then current is I=10.8/6 = 1.8 , A and internal resistance is r = ( - V)/I = (12 - 10.8)/1.8 = 0.67 , . This quick method is often tested. Battery aging signature: Over months, open-circuit voltage of a rechargeable cell may remain near nominal, but under the same load the terminal voltage dips more and recovers more slowly. This indicates rise of internal resistance rather than a big change in EMF. Replacement time. Multiple-load sharing: If two resistors are in parallel across a cell, they share the same terminal voltage V = - Ir . The current I is the sum of branch currents set by V . Increasing total load (decreasing equivalent R ) increases current and deepens the sag, reducing V for every branch. Quick estimate trick: For identical cells, if you need roughly double the voltage of one cell but the same current capacity, use two in series (voltage ×2, internal resistance ×2). If you need twice the current capacity at the same voltage, use two in parallel (same EMF, internal resistance halved). Sign check drill: While applying KVL, write rises as positive and drops as negative consistently. For a discharging cell loop: + - Ir - IR = 0 . For a charging scenario: - - Ir + V ext = 0 . Consistency removes most sign errors. Internal resistance in measurements: If a student uses a low-resistance voltmeter, it draws current and under-reads the true open-circuit EMF. The fix is to use a high-resistance voltmeter or a potentiometer that draws no current. NEET questions may exploit this subtlety. Loops with mixed sources: When two sources oppose each other in a loop, the current depends on the algebraic difference of EMFs and the sum of all internal and external resistances. Identify polarity properly before writing KVL to avoid sign mistakes. Measurement with a series ammeter and parallel voltmeter is standard for characterizing a source. Keep lead resistances short and contacts clean; otherwise, added series resistance appears as part of r and inflates the estimate. Rule of thumb: If a source’s internal resistance is more than about 10% of the load resistance, expect noticeable efficiency loss and heating. Either reduce the load current requirement or choose a lower- r source or add parallel rows. Key terms recap EMF Ideal source voltage of a cell; open-circuit terminal voltage. Internal resistance Effective resistance inside a cell causing drop Ir when current flows. Terminal voltage Actual voltage at the cell terminals under load or charge. External resistance connected to the cell. Load resistance Efficiency Fraction of chemical power delivered to the load, = V/ = R/(R + r) .