Current Electricity | Class 12 CBSE Physics | Concept Notes
Why combine cells at all?
A single cell has a fixed EMF and an internal resistance determined by its construction and chemical properties. These are properties of the cell itself and cannot normally be changed simply by connecting an external circuit.
However, practical devices often require a voltage or current that a single cell cannot provide efficiently.
For example:
- A device may require a higher voltage than one cell can supply.
- A circuit may need a larger current for a given load.
- A source may need a particular combination of EMF and internal resistance to operate a load effectively.
By connecting cells in different ways, we can change the effective EMF and effective internal resistance of the battery combination.
The two most important arrangements are:
- Series connection — cells are connected end-to-end.
- Parallel connection — cells are connected across the same two terminals.
The important idea is not simply to memorize that “EMFs add in series.” Instead, we can derive the result directly from Kirchhoff’s loop rule and see why it happens.
Cells in series
Setup
Consider n identical cells connected in series.
Each cell has:
- EMF = ε
- Internal resistance = r
The positive terminal of one cell is connected to the negative terminal of the next cell. Therefore, the same current I flows through every cell and through the external resistance R.
Because the cells are connected in series, the current passes successively through the internal resistance of every cell.
Series arrangement
For n identical cells:
- Total EMF = nε
- Total internal resistance = nr
The external resistance is R.
Deriving the equivalent EMF
Apply Kirchhoff’s loop rule to the complete circuit.
As we move through the cells in their aiding direction, every cell provides a potential rise of ε.
Therefore, the total EMF supplied by n cells is:
Total EMF = ε + ε + ε + … n times
So,
Effective EMF = nε
The EMFs add because the cells are connected so that the potential rises produced by the individual cells reinforce one another.
Deriving the equivalent internal resistance
Each cell has internal resistance r.
Since the cells are in series, the same current passes through every internal resistance. Series resistances simply add:
Total internal resistance = r + r + r + … n times
Therefore,
Effective internal resistance = nr
Thus, n identical cells connected in series behave, as far as the external circuit is concerned, like a single source having:
Effective EMF = nε
Effective internal resistance = nr
Current supplied to the external resistance
The external resistance is R.
The total resistance encountered by the current is therefore:
Total resistance = external resistance + internal resistance
So,
Total resistance = R + nr
Using the loop equation:
nε = I(R + nr)
Hence,
I = nε / (R + nr)
This is the current delivered to the external resistance.
A useful physical interpretation
Connecting cells in series increases both the useful EMF and the internal resistance by the same factor.
This has an important consequence.
If the external resistance R is much larger than the total internal resistance nr, then the increase in EMF can substantially increase the current through the load.
But if the load resistance is small, the increased internal resistance also becomes significant.
So, adding more cells in series does not automatically mean that the current will increase in direct proportion to the number of cells. The external resistance must also be considered.
What happens to the terminal voltage?
The EMF of the series combination is nε, but the terminal voltage is not necessarily equal to nε when current is being drawn.
When the battery supplies current, there is a voltage drop across its internal resistance.
For the complete series combination:
Terminal voltage = nε − I(nr)
Since the external resistance R receives this terminal voltage,
Terminal voltage = IR
Therefore,
IR = nε − I(nr)
This distinction between EMF and terminal voltage is important.
- EMF represents the energy supplied per unit charge by the source.
- Terminal voltage is the potential difference available at the external terminals while the source is delivering current.
The terminal voltage is smaller than the EMF when the battery is supplying current because of the internal resistance.
What if one cell is connected in the opposite direction?
The simple result “effective EMF = nε” assumes that all the cells are connected in the same, or aiding, orientation.
Suppose one cell is reversed.
Its EMF now opposes the EMF of the other cells.
This is where the sign convention in Kirchhoff’s loop rule becomes important.
A cell traversed from its negative terminal to its positive terminal contributes +ε.
A cell traversed in the opposite direction contributes −ε.
Therefore, the effective EMF of a series combination is more generally:
Effective EMF = algebraic sum of the individual EMFs
The word algebraic is important because each EMF must be given a sign according to its orientation.
One reversed cell among identical cells
Suppose there are n identical cells, each having EMF ε, and one cell is reversed.
Then:
- n − 1 cells aid the circuit.
- 1 cell opposes the circuit.
Therefore,
Effective EMF = (n − 1)ε − ε
So,
Effective EMF = (n − 2)ε
Notice that the result is not simply (n − 1)ε.
Why?
Because the reversed cell does two things simultaneously:
- It fails to contribute the positive EMF that it would have contributed in the correct orientation.
- It contributes an equal negative EMF in the opposite direction.
Thus, relative to the correctly connected arrangement, reversing one cell reduces the effective EMF by 2ε.
Example: five cells with one reversed
Suppose five identical cells are connected in series, each having an EMF of 1.5 V.
If all five cells are correctly oriented:
Effective EMF = 5 × 1.5
Effective EMF = 7.5 V
Now suppose one of the five cells is reversed.
Four cells aid the circuit and one opposes them:
Effective EMF = 4 × 1.5 − 1.5
Effective EMF = 4.5 V
Equivalently:
Effective EMF = (5 − 2) × 1.5
Effective EMF = 4.5 V
The reversed cell therefore reduces the effective EMF from 7.5 V to 4.5 V.
Does reversing a cell also reverse its internal resistance?
No.
Resistance has no direction or polarity.
The internal resistance of every cell is still present in the circuit regardless of how the cell is oriented.
Therefore, even when one or more cells are reversed, their internal resistances continue to add because all the internal resistances remain connected in series.
For n identical cells:
Effective internal resistance = nr
This remains true whether the cells are all aiding, some are opposing, or their EMFs have different magnitudes.
The orientation affects the EMF, not the way the resistances add.
General result for cells in series
For cells connected in series, the most general way to think about the combination is:
Effective EMF
Effective EMF = algebraic sum of all individual EMFs
Cells aiding the chosen direction contribute positively, while cells opposing it contribute negatively.
Effective internal resistance
Effective internal resistance = sum of all individual internal resistances
Unlike EMF, resistance does not acquire a negative sign merely because a cell is reversed.
For identical cells all connected in the same direction:
Effective EMF = nε
Effective internal resistance = nr
and
Current = nε / (R + nr)
What if the cells are not identical?
The same reasoning works even when the cells have different EMFs and different internal resistances.
Suppose the cells have:
- EMFs ε₁, ε₂, ε₃, …
- Internal resistances r₁, r₂, r₃, …
For a series combination:
Effective EMF = algebraic sum of the individual EMFs
and
Effective internal resistance = r₁ + r₂ + r₃ + …
Therefore, the current through an external resistance R is:
Current = effective EMF / (R + effective internal resistance)
This is the more general result from which the identical-cell formula follows as a special case.
Key idea to remember
The safest way to remember series combinations is not simply:
“EMFs add and resistances add.”
Instead, remember the underlying principle:
In series, the same current passes through every cell, so the internal resistances add. The EMFs combine algebraically according to their orientations.
Therefore:
- Aiding cells → EMFs add
- Opposing cells → EMFs subtract
- All internal resistances → add
For n identical, correctly oriented cells:
Effective EMF = nε
Effective internal resistance = nr
Current = nε / (R + nr)
This gives a complete picture of why cells connected in series behave like a single source with a larger EMF and a larger internal resistance.
Cells in parallel
Setup
Consider n identical cells connected in parallel.
Each cell has:
- EMF = ε
- Internal resistance = r
All the positive terminals are connected to one common node, and all the negative terminals are connected to another common node. The combination is connected to an external resistance R.
The same two terminals are therefore shared by every cell.
The key question is:
What happens to the effective EMF and internal resistance when identical cells are connected in parallel?
The answer is different from the series combination:
Effective EMF = ε
Effective internal resistance = r/n
The EMF does not increase, but the internal resistance decreases.
Why doesn’t the EMF increase?
This is one of the most common points of confusion.
Students often see n cells and assume that the EMFs should add to give nε. That is true for cells connected in series with the same orientation, but not for identical cells connected in parallel.
In a parallel combination, every cell is connected across the same two nodes.
Therefore, every cell has the same terminal potential difference.
An individual cell tends to establish a potential difference of ε across its terminals when no current is being drawn. Since all the positive terminals are joined together and all the negative terminals are joined together, the cells are all connected across the same potential difference.
The cells therefore do not provide separate voltage rises one after another, as they do in series.
Instead, they provide additional current-carrying paths at the same voltage.
Therefore, for n identical cells connected in parallel:
Effective EMF = ε
Not nε.
Not ε/n.
Simply:
ε_eff = ε
An intuitive way to understand it
Think of voltage as the “push” provided by the source and current as the amount of charge flow that the source can supply.
Connecting identical cells in series increases the available voltage push:
Series → higher EMF
Connecting identical cells in parallel keeps the same voltage but provides more paths for current:
Parallel → same EMF, lower internal resistance
This distinction is fundamental:
Series connection increases the source voltage; parallel connection increases the current-supplying capability of the source.
Deriving the Effective Internal Resistance
The important change in a parallel combination occurs in the internal resistance.
Each cell has internal resistance r.
Because all the positive terminals are connected together and all the negative terminals are connected together, the internal resistance of every cell lies between the same two nodes.
Therefore, the internal resistances are connected in parallel.
For n identical resistances, each of value r:
1 / r_eff = 1 / r + 1 / r + … n times
Therefore:
1 / r_eff = n / r
Taking the reciprocal:
r_eff = r / n
Thus, for n identical cells connected in parallel:
Effective internal resistance = r/n
The internal resistance becomes smaller as more identical cells are connected in parallel.
Why does the internal resistance decrease?
This result has a useful physical interpretation.
Suppose one cell has to supply a large current to an external circuit. The current must pass through that cell’s internal resistance, producing an internal voltage drop.
With several identical cells connected in parallel, the required total current can be shared among the cells.
For example, if n identical cells share the current equally, each cell supplies approximately:
Current from each cell = I/n
where I is the total current supplied to the external circuit.
The current through each cell’s internal resistance is therefore smaller.
Consequently, the voltage drop inside each cell is reduced.
This is why a parallel combination can supply a larger current while maintaining its terminal voltage more effectively.
Current Delivered to the External Resistance
The equivalent source has:
- Effective EMF = ε
- Effective internal resistance = r/n
The external resistance is R.
Therefore, the total resistance in the complete circuit is:
Total resistance = R + r/n
Applying Kirchhoff’s loop rule:
ε = I(R + r/n)
Hence:
I = ε / (R + r/n)
This is the total current supplied to the external resistance.
Terminal Voltage of the Parallel Combination
The terminal voltage of a source supplying current is less than its EMF because of the voltage drop across its internal resistance.
For the parallel combination:
Terminal voltage = ε − I(r/n)
The same terminal voltage appears across the external resistance.
Therefore:
Terminal voltage = IR
This gives another useful way of understanding the effect of adding cells in parallel.
As n increases, the effective internal resistance r/n decreases. Therefore, for a given load, the internal voltage drop becomes smaller and the terminal voltage remains closer to the EMF.
Example
Suppose four identical cells are connected in parallel.
Each cell has:
- EMF = 1.5 V
- Internal resistance = 2 Ω
The effective EMF is:
Effective EMF = 1.5 V
The effective internal resistance is:
Effective internal resistance = 2/4 = 0.5 Ω
Thus, the four-cell combination behaves like a source having:
EMF = 1.5 V
Internal resistance = 0.5 Ω
Notice what happened:
The EMF remained 1.5 V, rather than becoming 6 V.
But the internal resistance decreased from 2 Ω to 0.5 Ω.
If the external resistance is R, the total current is:
I = 1.5 / (R + 0.5)
What happens to the current supplied by each cell?
For identical cells operating under ideal symmetric conditions, the total current is shared equally among the cells.
If the total current supplied to the external circuit is I, then the current supplied by each cell is:
Current per cell = I/n
For example, if four identical cells together supply 8 A, each cell supplies approximately:
8/4 = 2 A
This current sharing is one of the main advantages of parallel connection.
Each cell does not have to provide the entire load current by itself.
A crucial condition: the cells must be identical
The simple result
Effective EMF = ε
and
Effective internal resistance = r/n
applies to identical cells connected correctly in parallel.
For cells with different EMFs, directly connecting them in parallel can cause unwanted circulating currents between the cells.
For example, if one cell has a slightly higher EMF than another, the higher-EMF cell can drive current through the lower-EMF cell even when there is little or no external load.
Therefore, the simple parallel-cell formula should not be applied blindly to cells with different EMFs.
Series vs Parallel: The Essential Difference
The difference can now be understood from the circuit arrangement rather than memorized as two unrelated formulas.
| Connection | Effective EMF | Effective internal resistance | Main effect |
| n identical cells in series | nε | nr | Increases voltage |
| n identical cells in parallel | ε | r/n | Reduces internal resistance |
Series
The cells are arranged one after another, so their EMFs reinforce one another and their internal resistances add.
Effective EMF = nε
Effective internal resistance = nr
Parallel
The cells are connected across the same two terminals, so the EMF remains unchanged while their internal resistances act in parallel.
Effective EMF = ε
Effective internal resistance = r/n
The Deeper Principle
The most useful way to remember the parallel combination is:
Parallel cells do not add their voltage; they add their ability to supply current.
The common terminal voltage remains the same for all the cells, while the current is shared among them.
Thus, for n identical cells in parallel:
Effective EMF = ε
Effective internal resistance = r/n
and the current through an external resistance R is:
I = ε / (R + r/n)
As the number of parallel cells increases, r/n becomes smaller.
In the ideal mathematical limit of very large n:
r/n approaches zero
so the combination approaches an ideal source with:
EMF = ε
Internal resistance ≈ 0
The voltage does not become infinitely large. Instead, the source becomes increasingly capable of supplying current without a large internal voltage drop.
Key takeaway
Series connection: increase the EMF.
Parallel connection: reduce the effective internal resistance.
For identical cells:
Series → nε and nr
Parallel → ε and r/n
The formulas follow directly from the way the cells are connected, so understanding the circuit arrangement is more reliable than memorizing the results.
Worked Example 1 — series
Three identical cells, each ε=2V, r=0.5Ω, are connected in series and drive an external resistance of 4.5Ω. Find the current.
Solution:
Worked Example 2 — parallel
Four identical cells, each ε=1.5V, r=1Ω, are connected in parallel and drive an external resistance of 2Ω. Find the current.
Solution:
Worked Example 3 — the trap question
Two cells with different EMFs, 1.5 V and 2 V (both internal resistance 1Ω), are connected in parallel. A student computes the combined EMF as the simple average, (1.5+2)/2 = 1.75 V. Is this correct?
Answer: No — and this is a genuinely different situation from identical cells in parallel.
The lesson isn’t “the average is wrong” — it’s that simple averaging works only by coincidence when resistances are matched, and the general formula is what actually generalizes.
Common errors to watch for
Combining cells is a topic where a few apparently reasonable shortcuts can lead to incorrect results. The safest approach is to understand what the circuit arrangement does to the EMF and internal resistance.
| Common error | What is actually true |
| Adding EMFs for identical cells in parallel | Identical cells connected in parallel have effective EMF = ε, not nε. Their internal resistance decreases to r/n. |
| Dividing the EMF by n for cells in parallel | The EMF does not become ε/n. All cells are connected across the same two terminals, so the effective EMF remains ε. |
| Averaging the EMFs of unequal cells connected in parallel | A simple average is generally incorrect. The effective EMF depends on both the individual EMFs and their internal resistances. If the cells have EMFs ε₁, ε₂, … and internal resistances r₁, r₂, …, the effective EMF is a resistance-weighted average: ε_eff = (ε₁/r₁ + ε₂/r₂ + …)/(1/r₁ + 1/r₂ + …). A simple average is obtained only when all the internal resistances are equal. |
| Forgetting the internal resistance of a reversed cell in series | Reversing a cell changes the sign of its EMF contribution, but it does not remove its internal resistance. The resistance still contributes positively to the total internal resistance. |
| Assuming a reversed cell simply means “one less cell” | A reversed identical cell actually reduces the total EMF by 2ε compared with the correctly oriented arrangement. For n cells with one reversed cell, the effective EMF is (n − 2)ε. |
| Assuming parallel cells always supply more current than series cells | There is no universal answer. The result depends on the external resistance R compared with the cell resistance r. For the same n identical cells, the series and parallel combinations deliver the same current when R = r. |
| Ignoring terminal-voltage drop | The EMF is not necessarily the voltage available at the terminals while current is flowing. Internal resistance causes a voltage drop, so the terminal voltage is lower than the EMF when the source is delivering current. |
| Connecting unequal cells directly in parallel without considering circulating current | Cells with different EMFs can drive current into one another even without a significant external load. The simple formula r/n applies only to identical cells. |
Quick Practice
Q1. Five identical cells, each ε=1.2V, r=0.4Ω, are connected in series. Find the effective EMF and internal resistance of the combination.
Answer: ε_eff = 5×1.2 = 6V, r_eff = 5×0.4 = 2Ω
Q2. The same five cells are instead connected in parallel. Find the effective EMF and internal resistance.
Answer: ε_eff = 1.2V (unchanged), r_eff = 0.4/5 = 0.08Ω
Q3. A series combination of n identical cells has one cell inserted backwards. In terms of a single cell’s EMF ε, what is the combination’s effective EMF?
Answer: ε_eff = (n−2)ε — the reversed cell’s own contribution flips sign, so instead of contributing +ε like the rest, it contributes −ε, a net loss of 2ε compared to all cells facing the same way.
Summary
Series vs Parallel: Summary
For n identical cells, each having EMF ε and internal resistance r:
| Property | Series connection | Parallel connection |
| Effective EMF | nε | ε |
| Effective internal resistance | nr | r/n |
| Current through external resistance R | nε / (R + nr) | ε / (R + r/n) |
| Main effect | Increases the available voltage | Reduces the effective internal resistance |
| Current sharing | The same current flows through every cell | The total current is shared among the cells |
| Typical application | When a higher voltage is required | When greater current-supplying capability and smaller voltage sag are required |
When would you actually choose series vs. parallel?
It is tempting to say that series connection is always better for voltage and parallel connection is always better for current. The more precise statement is that the load resistance determines which arrangement delivers the larger current for the same number of identical cells.
For n identical cells:
Series combination
I_series = nε / (R + nr)
Parallel combination
I_parallel = ε / (R + r/n)
Setting the two currents equal gives:
R = r
This provides a useful comparison.
If R > r
The series combination delivers the larger current.
If R < r
The parallel combination delivers the larger current.
If R = r
Both combinations deliver the same current.
This result shows why there is no universal rule that one arrangement is always superior. The appropriate connection depends on the characteristics of the external load.
A Useful Physical Picture
The difference between the two arrangements can be understood without memorizing formulas.
Cells in series
The voltage contributions of the cells are placed one after another.
Therefore:
EMFs add → higher effective voltage
But the internal resistances are also placed one after another:
Internal resistances add → higher effective internal resistance
Cells in parallel
All cells are connected across the same two terminals.
Therefore:
The EMF remains the same
But their internal resistances provide multiple parallel paths:
Effective internal resistance decreases
This allows the cells to share the load current.
The Three Rules Worth Remembering
For identical cells, the following three rules are enough to reconstruct most results:
1. Series EMFs
Cells connected in series have EMFs that combine algebraically.
Aiding cells add; opposing cells subtract.
2. Series internal resistances
Internal resistances connected in series always add, regardless of the orientation of the cells.
3. Parallel identical cells
Identical cells connected in parallel have the same effective EMF as one cell, while their internal resistances combine in parallel.
Therefore:
Series: effective EMF = nε, effective internal resistance = nr
Parallel: effective EMF = ε, effective internal resistance = r/n
Final Takeaway
The key is to understand what changes and what does not.
Series connection:
- EMF increases.
- Internal resistance increases.
- The same current flows through every cell.
Parallel connection:
- EMF remains unchanged for identical cells.
- Effective internal resistance decreases.
- The total current is shared among the cells.
So, rather than memorizing two isolated formulas, think of the physical arrangement:
Series cells combine their voltage contributions. Parallel cells share the current-supplying task.
That single idea provides the basis for deriving the effective EMF, internal resistance, terminal voltage, and current for both arrangements.
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