MHT-CET Physics · Current Electricity

E.m.f., Internal Resistance and Cells Together

A cell's e.m.f. is the voltage it would show with no current; drawing a current I through its internal resistance r lowers the terminal voltage to E − Ir, cells in series add their e.m.f.s and internal resistances, and identical cells in parallel keep their e.m.f. but share the current.

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Why this matters

Four PYQs on cells, two of them HARD: the internal resistance from a voltmeter reading, the external resistance that makes one of two series cells show zero terminal voltage, the reading of a voltmeter across two unequal cells in parallel, and the steady current and capacitor charge in a network. One card.

Concept 1 of 1: Terminal Voltage and Cells in Series or Parallel

A real cell is an ideal e.m.f. E with a small resistance r inside it. With current I flowing, the terminals show V = E − Ir, less than E by the drop inside. A voltmeter of resistance R_V across a cell reads E·R_V/(R_V + r), which is how r is found from a reading. Two cells in series driving R carry I = (E₁ + E₂)/(R + r₁ + r₂); one of them shows zero terminal voltage when its own drop Ir equals its E. Two unequal cells in parallel settle at a common voltage that is the resistance-weighted mean of their e.m.f.s. In the steady state a capacitor carries no current, so its branch drops out of the current calculation but it still charges to the voltage across it.

Definition

$V = E - Ir$ ; voltmeter of resistance R across the cell: $r = \dfrac{E - V}{V/R}$ (3 V, reads 2.5 V on 150 Ω ⇒ 30 Ω).
Series: $E = E_1 + E_2$ , $r = r_1 + r_2$ . Equal cells E with $r_1 > r_2$ : the first shows zero terminal voltage when $R = r_1 - r_2$ .
Parallel (same polarity): $V = \dfrac{E_1/r_1 + E_2/r_2}{1/r_1 + 1/r_2}$ (12 V, 2 Ω with 6 V, 1 Ω ⇒ 8 V).
Steady state with a capacitor: its branch carries no current; it charges to the voltage across it, $Q = CV$ .

Terminal voltage

V = E - Ir, \qquad I = \frac{E}{R + r}

Worked example

A 9 V cell of internal resistance 1 Ω drives a 2 Ω resistor. Current and terminal voltage?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2025 · 19 April Shift II · Q42Moderate

Example 1 · Current Electricity · Ohm's Law, Cells, EMF, and Internal Resistance

Two cells

E_{1}

and

E_{2}

having equal e.m.

f^{'}E

' and internal resistances

r_{1}

and

r_{2}\left( r_{1}>r_{2} \right)

respectively are connected in series. This combination is connected to an external resistance ' R '. It is observed that the potential difference across the cell

E_{1}

becomes zero. The value of

R

will be

Reading e.m.f. and terminal voltage as the same

A voltmeter across a cell that is delivering current reads E − Ir, less than the e.m.f. The e.m.f. appears only when no current flows.

Drill 3 more on terminal voltage and cells in series or parallel

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Terminal Voltage and Cells in Series or Parallel
Terminal voltage
$V = E - Ir, \qquad I = \frac{E}{R + r}$

Watch out for (1)

Reading e.m.f. and terminal voltage as the same
→ Terminal Voltage and Cells in Series or Parallel

Test yourself on Current Electricity

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