Current Electricity | Cells in Series | Internal Resistance

Current Electricity - Combination of Cells in Series

current electricity combination of cells in series is explained with total EMF, equivalent internal resistance, terminal potential difference, Thevenin concept, V-I graphs, power transfer, SVG circuits and exam-level questions for CBSE, NEET, JEE Main, JEE Advanced, IB, AP, IGCSE, ICSE, A-Level and Olympiad Physics.

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1. Formula Sheet First

Eeq = nETotal EMF of n identical cells in series.
req = nrTotal internal resistance.
I = nE/(R+nr)Current through external resistance R.
V = IRTerminal potential difference across load.
V = nER/(R+nr)Load voltage after substitution.
VAB = nE - I(nr)Terminal voltage of battery combination.
VAB = nE - I(nr)Graph equation.
Y-intercept = nEVAB vs I graph.
slope = -nrMagnitude tanθ = nr.
P = I²RPower delivered to external resistance.
Ploss = I²nrPower loss inside cells.
η = R/(R+nr)Efficiency of series combination.

2. What Is Combination of Cells in Series?

Cells are connected in series when the positive terminal of one cell is connected to the negative terminal of the next cell. The EMFs add, but internal resistances also add.

Why EMF increases: each cell gives additional energy per unit charge, so total EMF becomes nE.
Why internal resistance increases: current passes through the internal resistance of every cell.
Practical use: series cells are useful when high voltage is required and load resistance is not too small.

3. SVG Diagram of n Cells in Series

n Identical Cells in Series With External Resistance RE,rE,r...E,rRABIVAB across load R

4. Method 1 - Direct Circuit Method

1
For n identical cells in series, total EMF is Eeq = nE.
2
Total internal resistance is req = nr.
3
Total resistance in circuit is Rtotal = R + nr.
4
Therefore current is I = nE/(R+nr).

5. Terminal Potential Difference

1
Internal voltage loss in all cells is I(nr).
2
Terminal voltage is VAB = nE - I(nr).
3
Across external resistance, VAB = IR.
4
Substitute I = nE/(R+nr): VAB = nER/(R+nr).

6. Method 2 - Thevenin Style Concept

This is a useful advanced method for JEE Main, JEE Advanced, IB and Olympiad-level thinking.

Open load: I = 0, so Vopen = nE.
Equivalent resistance: short ideal voltage sources internally; resistance seen from A-B is Rth = nr.
Thevenin EquivalentnEnrRI = nE/(R+nr)

7. Special Cases

R >> nr: I ≈ nE/R, so series cells are very useful.
R << nr: internal resistance dominates and current does not increase effectively.
Open circuit: I = 0 and VAB = nE.
Short circuit: R = 0, so I = nE/nr = E/r.
Important: short-circuit current does not increase with n for identical cells in series.
Reversed cell: its EMF subtracts but its internal resistance still adds.

8 and 19. Graphs and Required Diagrams

VAB vs IIVABnEslope = -nr Power vs Load RR = nrP Open vs Closed CircuitOpenVAB = nEClosedVAB = nE - Inr Reversed CellEreversed EEMFs subtract, resistances add

For the VAB-I graph, VAB = nE - I(nr). The y-intercept is nE and the magnitude of slope is nr, so the graph gives total internal resistance.

9-10. Power and Maximum Power Transfer

1
Load power is P = I²R.
2
Substitute I = nE/(R+nr): P = n²E²R/(R+nr)².
3
Differentiate P with respect to R for maximum power.
4
The maximum condition is R = nr.
5
At R = nr, Pmax = nE²/(4r).

11-12. Numerical and Conceptual Practice

Current: use I = nE/(R+nr).
Terminal voltage: use VAB = IR or nE - Inr.
Power delivered: P = I²R.
Graph r: slope magnitude gives nr, so divide by n to get r.
Series useful when: external resistance is much larger than internal resistance.
One cell reversed: net EMF reduces, total internal resistance still increases.

13. Common Student Errors

14-18. Exam Question Bank With Accordion Solutions

Click any question to open the answer and explanation.

20. Final Revision Sheet

Total EMF: Eeq = nE.
Total internal resistance: req = nr.
Current: I = nE/(R+nr).
Terminal voltage: VAB = nE - Inr = IR.
V-I graph: intercept nE and slope -nr.
Maximum power: R = nr and Pmax = nE²/(4r).

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