Why the number printed on a battery is never the number a voltmeter shows once the circuit is switched on — and how NEET builds a dozen question types out of that one gap.
NCERT §3.10 · fully in syllabus · very high frequency
Part 1 · The idea, told simply
A cell is a pump. It picks charge up at the low end and lifts it to the high end, so that it can fall back through the circuit and do something useful on the way.
Story track
Picture a water pump at the bottom of a garden, pushing water up to a tank on a hill. The pump is rated: "I can lift every litre by 12 metres." That rating is the emf, written ε. It is a promise about how much energy the pump gives to each unit of charge.
But here is the catch. The pump has pipes inside it, and those pipes are narrow and rusty. Some of the lift is spent just getting the water out of the pump itself. So although the pump lifts every litre by 12 metres internally, what actually arrives at the outlet is a bit less. That internal rustiness is the internal resistance, written r, and what arrives at the outlet is the terminal voltage, written V.
Now the whole topic in one line: V = ε − Ir. The faster the water flows, the more gets wasted inside, and the less arrives outside.
The three words that must never be mixed up
emf (ε) — the full energy the cell gives per unit charge. Fixed. A property of the chemistry inside.
Terminal voltage (V) — what is actually available across the cell's two terminals. Changes with the current drawn.
Lost volts (Ir) — the difference, eaten by the cell's own internal resistance.
NCERT is emphatic on one point: emf is not a force. The name is a historical accident from a time when the phenomenon was not understood. It is a potential difference, measured in volts, and defined as the work done per unit charge in carrying charge from one terminal to the other inside the source.
Animation 1 · Inside the cell — where the lost volts go
Drag the external resistance. Watch the energy bar split into "delivered outside" and "wasted inside".
Open circuit: the only time a voltmeter tells you the emf
Story track
Switch the circuit off. No current flows, so I = 0, so Ir = 0, so V = ε. Nothing is being wasted inside because nothing is moving.
This is why a nearly dead AA cell can still read a healthy 1.5V on a multimeter and yet fail to run a toy. The multimeter draws almost no current, so it reads the emf. The toy draws real current, the internal resistance has grown large with age, and the terminal voltage collapses.
Short circuit: the other extreme
Now make R = 0 by joining the terminals with a thick wire. The only thing left in the circuit is the cell's own r, so the current is as large as it can possibly be:
I_max = ε/r
NCERT adds a practical warning: in most cells the maximum allowed current is far below this, because that much current permanently damages the cell.
Animation 2 · The V–I line of a cell
Every cell has one straight line. The intercept is ε, the slope is −r. Slide R and watch the operating point travel down it.
Charging: the sign flips
Story track
Everything so far assumed the cell was pushing current out. When you charge a battery you force current backwards into it — a bigger supply pushes current in through the positive terminal.
Now the internal resistance is being fought against you as well, so the outside supply must provide the emf plus the internal drop. The terminal voltage during charging is therefore larger than the emf:
Discharging (cell supplies): V = ε − Ir V < ε
Charging (cell absorbs): V = ε + Ir V > ε
This is the standard NEET trap: a question says "a battery is being charged" and every student who reflexively writes V = ε − Ir loses the mark.
Animation 3 · Discharging vs charging
Toggle the mode. Watch the current reverse and the terminal voltage cross over the emf line.
Part 2 · The physics underneath
Maths track · deriving V = ε − Ir
NCERT §3.10 sets it up carefully. The positive electrode P sits at a potential V₊ above the electrolyte next to it (point A); the negative electrode N sits at −V₋ relative to the electrolyte next to it (point B). With no current, the electrolyte is at a single potential throughout, so
ε = V₊ + V₋ (the open-circuit terminal difference)
Once current I flows, the electrolyte itself has resistance r, and the current inside the cell runs from B to A. Crossing that stretch costs a drop Ir. Therefore
V = V₊ + V₋ − Ir = ε − Ir
Combining with Ohm's law across the external resistor, V = IR:
IR = ε − Ir ⇒ I = ε/(R + r)
This single equation generates almost every numerical in this topic. Note that ε and r sit inside the cell and never change with the circuit; only I and V respond to R.
Extracting r from measurements
Two standard experimental routes, both examined:
One reading (ε and R known): r = R(ε − V)/V = ε/I − R
Two readings (V₁ at I₁, V₂ at I₂): r = (V₁ − V₂)/(I₂ − I₁)
Two loads (I₁ through R₁, I₂ through R₂): r = (I₂R₂ − I₁R₁)/(I₁ − I₂)
The middle form is simply the magnitude of the slope of the V–I line, which is why the graph question in this topic is always about a slope.
Power: three different powers, and NEET asks all three
Total power generated by the cell: P_total = εI
Power delivered to the external R: P_ext = I²R = VI
Power wasted inside the cell: P_int = I²r
Efficiency: η = P_ext/P_total = R/(R + r) = V/ε
Maths track · maximum power transfer
How should R be chosen to squeeze the most power out of a given cell? Write the external power as a function of R:
P = I²R = ε²R/(R + r)²
Differentiate and set to zero, or rewrite the denominator as (R − r)² + 4Rr and note the numerator is fixed. Either way P peaks when
R = r and then P_max = ε²/4r
Worth knowing but worth doubting too: at that point the efficiency is only R/(R+r) = 50%. Half the energy is being cooked inside the cell. Maximum power and maximum efficiency are different goals, and NEET has an assertion–reason question sitting exactly on that distinction.
What actually decides r
Beyond NCERT but standard and occasionally asked:
r increases with the distance between the electrodes.
r decreases as the area of the electrodes dipped in the electrolyte increases.
r decreases as the concentration of the electrolyte increases, and decreases with rising temperature (ions move more freely).
r rises steeply as a cell ages — the practical reason old batteries fail under load.
NCERT notes that dry cells have a much higher internal resistance than ordinary electrolytic cells.
Trap · the five that cost marks
Using V = ε − Ir for a battery that is being charged. It is + Ir.
Reading "the voltmeter reads 1.5V" as the emf when the circuit is closed. It is V, not ε.
Computing I with ε/R instead of ε/(R + r) when r is given and not negligible.
Quoting the efficiency at maximum power as 100%. It is 50%.
Assuming ε changes when R changes. Never. Only I and V move.
Animation 4 · The maximum power hump
Power delivered outside versus external resistance. The peak sits exactly at R = r — and efficiency there is only half.
Animation 5 · The energy ledger
Every joule the chemistry releases goes to exactly two places. Watch the split move as the load changes.
Part 3 · Formula sheet
The master equations
Discharging: V = ε − Ir (V < ε)
Charging: V = ε + Ir (V > ε)
Circuit current: I = ε/(R + r)
Ohm's law outside: V = IR
Open circuit (I = 0): V = ε
Short circuit (R = 0): I_max = ε/r, V = 0
Finding the internal resistance
r = R(ε − V)/V [one load, ε known]
r = ε/I − R
r = (V₁ − V₂)/(I₂ − I₁) [two terminal-voltage readings]
r = (I₂R₂ − I₁R₁)/(I₁ − I₂) [two loads]
r = magnitude of the slope of the V–I line
Power and efficiency
P_total (by cell) = εI
P_external = I²R = VI = V²/R
P_internal = I²r (heats the cell)
η = P_ext/P_total = R/(R + r) = V/ε
Maximum external power at R = r: P_max = ε²/4r
Efficiency at maximum power = 50%
Useful ratios
V/ε = R/(R + r)
Lost-volts fraction = Ir/ε = r/(R + r)
If R = r: V = ε/2
If R = 4r: V = 0.8ε
If R ≫ r: V → ε (ideal cell behaviour)
Sanity checks before you tick an option
Discharging: V must be less than ε (always)
Charging: V must be greater than ε (always)
I can never exceed ε/r
Efficiency can never exceed 100%, and is 50% at peak power
ε and r never change when R changes
Part 4 · 50 NEET-pattern questions with full solutions
Includes 4 graph-based questions, 3 assertion–reason questions, and questions built on repeatedly-examined NEET patterns and NCERT exercises (tagged accordingly). Year labels are deliberately not attached: the patterns are authentic, the exact wording is mine.