Calculate the combined e.m.f. of several sources in series
4.3.2 Series and Parallel Circuits
Learning Objectives
- Distinguish between series and parallel connections.
- State the fundamental current and voltage rules for series and parallel circuits (junction rule, same‑current, same‑voltage).
- Calculate the combined e.m.f. of several ideal sources in series, taking polarity into account.
- Determine the equivalent resistance of resistors in series and in parallel.
- Explain why domestic lighting is normally wired in parallel.
Key Concepts
| Concept | Definition / Formula |
|---|---|
| Electromotive force (e.m.f.) | The open‑circuit voltage a source would deliver if no current were drawn. |
| Polarity | Direction of the e.m.f.; shown by the “+’’ (positive) and “–’’ (negative) terminals of a cell. |
| Series connection | Positive terminal of one source is joined to the negative terminal of the next; the same current flows through every element. |
| Parallel connection | All positive terminals are joined together and all negative terminals are joined together; the voltage across each branch is the same. |
| Junction (Kirchhoff) rule | At any junction, the total current entering equals the total current leaving: \(\displaystyle\sum I{\text{in}} = \sum I{\text{out}}\). |
| Equivalent resistance (series) | \(R{\text{eq}} = R1 + R2 + \dots + Rn\) |
| Equivalent resistance (parallel) | \(\displaystyle\frac{1}{R{\text{eq}}}= \frac{1}{R1}+ \frac{1}{R2}+ \dots + \frac{1}{Rn}\) |
| Parallel‑resistance rule (syllabus wording) | The total resistance of a parallel network is always less than the smallest individual resistance. |
Series Circuits
Fundamental Rules
- Same‑current rule: In a series loop the current is identical at every point (conservation of charge).
- Voltage (p.d.) rule: The algebraic sum of the potential differences across the elements equals the e.m.f. of the source(s).
- Junction rule: Because a series loop has only one path, the current entering a junction equals the current leaving it.
Construction of a Simple Series Loop
Combined e.m.f. of Ideal Sources in Series
When internal resistances are negligible, the total e.m.f. is the algebraic sum of the individual e.m.f.s, the sign being determined by polarity relative to a chosen current direction.
\[
\mathcal{E}{\text{total}} = \sum{i=1}^{n} \pm \mathcal{E}_i
\]
- “+’’ if the source’s polarity aids the chosen current direction.
- “–’’ if it opposes the direction.
Step‑by‑Step Procedure
- Draw the series circuit and label the + and – terminals of every source.
- Choose a direction for the current (normally from the most positive terminal toward the most negative).
- Assign a sign to each e.m.f.:
- “+’’ when the arrow of the source points in the same direction as the chosen current.
- “–’’ when it points opposite.
- Add the signed values.
- State the polarity of the resulting e.m.f.; the end with the net positive sign is the positive terminal.
Example 1 – All Sources Aiding
Three identical cells, each 1.5 V, are connected in series with the same polarity.
| Cell | e.m.f. (V) | Sign |
|---|---|---|
| 1 | 1.5 | + |
| 2 | 1.5 | + |
| 3 | 1.5 | + |
\[
\mathcal{E}_{\text{total}} = 1.5 + 1.5 + 1.5 = 4.5\ \text{V}
\]
Example 2 – One Source Opposed
Two 2.0 V cells and one 1.0 V cell are in series; the 1.0 V cell is reversed.
| Cell | e.m.f. (V) | Sign |
|---|---|---|
| 1 | 2.0 | + |
| 2 | 2.0 | + |
| 3 | 1.0 | – |
\[
\mathcal{E}_{\text{total}} = 2.0 + 2.0 - 1.0 = 3.0\ \text{V}
\]
Real Cells – Internal Resistance
If a cell has an internal resistance \(r\) and a current \(I\) flows, the terminal voltage is reduced by \(Ir\). For a series string of real cells:
\[
\mathcal{E}{\text{total}} = \sum \mathcal{E}i \;-\; I\!\left(\sum r_i\right)
\]
This formula is needed when the question gives a load current or mentions “real” cells.
Equivalent Resistance (Series)
\[
R{\text{eq}} = R1 + R2 + \dots + Rn
\]
Parallel Circuits
Fundamental Rules
- Same‑voltage rule: Every branch of a parallel network is connected directly across the source, so each branch experiences the full source voltage.
- Current rule: The total current supplied by the source equals the sum of the branch currents: \(\displaystyle I{\text{total}} = I1 + I2 + \dots + In\).
- Junction rule: At the node where the branches split, \(\displaystyle\sum I{\text{in}} = \sum I{\text{out}}\).
- Resistance rule (syllabus wording): The total resistance of a parallel network is always less than the smallest individual resistance.
Construction of a Simple Parallel Network
Equivalent Resistance (Parallel)
\[
\frac{1}{R{\text{eq}}}= \frac{1}{R1}+ \frac{1}{R2}+ \dots + \frac{1}{Rn}
\]
Because the voltage across each branch is identical, the branch with the smallest resistance draws the largest current, but the total resistance is always smaller than the smallest individual resistance.
Why Lamps (or Other Loads) Are Wired in Parallel in Domestic Circuits
- Each lamp receives the full supply voltage, so its brightness is independent of the other lamps.
- If one lamp fails (opens), the remaining lamps continue to operate because the paths are separate.
- The total current is the sum of the individual lamp currents; therefore the main fuse must be rated for the combined load.
Practice Questions
- Three batteries have e.m.f.s of 1.2 V, 1.5 V and 2.0 V. All are connected in series with the same polarity. What is the combined e.m.f.?
- Two 3.0 V cells are connected in series, but one is reversed. Calculate the net e.m.f.
- Four cells are connected in series: 0.9 V, 1.5 V, 1.5 V, and 0.9 V. The second 1.5 V cell is opposite to the others. Find the total e.m.f. and indicate its polarity.
- Three resistors of 10 Ω, 20 Ω and 30 Ω are connected in parallel across a 12 V supply.
- Calculate the equivalent resistance.
- Find the total current drawn from the supply.
- A string of four identical 1.5 V cells (ideal) powers a lamp. If one cell develops an internal resistance of 0.2 Ω while the others remain ideal, and the lamp draws 0.5 A, determine the terminal voltage of the battery pack.
Common Mistakes to Avoid
- Forgetting to assign a sign to each source based on its polarity.
- Adding magnitudes without recognising that a reversed source reduces the total e.m.f.
- Confusing series and parallel rules – in parallel the voltage is the same, not the e.m.f. sum.
- Neglecting internal resistance when a problem explicitly mentions “real” cells.
- Assuming the total resistance in parallel is simply the smallest resistance; use the reciprocal formula instead.
- Omitting the same‑current rule for series circuits or the same‑voltage rule for parallel circuits.
Suggested Diagrams
Summary Checklist
- Identify whether the circuit is series or parallel.
- Series:
- Current is the same everywhere.
- Write each e.m.f. with a sign according to polarity; add them.
- Equivalent resistance: \(R{\text{eq}} = R1+R_2+\dots\).
- Parallel:
- Voltage across each branch equals the source voltage.
- Total current is the sum of the branch currents.
- Equivalent resistance: \(\displaystyle\frac{1}{R{\text{eq}}}= \sum\frac{1}{Ri}\) (always less than the smallest \(R_i\)).
- If the question involves real cells, subtract the internal‑resistance loss:
\(\displaystyle\mathcal{E}{\text{total}} = \sum\mathcal{E}i - I\!\left(\sum r_i\right).\)
- Apply the junction rule (\(\sum I{\text{in}} = \sum I{\text{out}}\)) at every node.