Calculate the combined e.m.f. of several sources in series

4.3.2 Series and Parallel Circuits

Learning Objective

Calculate the combined electromotive force (e.m.f.) of several sources when they are connected in series.

Key Concepts

• Electromotive force (e.m.f.) – the voltage supplied by a source when no current flows.
• Series connection – the positive terminal of one source is connected to the negative terminal of the next.
• Polarity – the direction of the e.m.f.; must be taken into account when adding sources.

Combined e.m.f. in Series

When ideal sources (negligible internal resistance) are connected in series, the total e.m.f. is the algebraic sum of the individual e.m.f.s, taking polarity into account.

\$\$

\mathcal{E}{\text{total}} = \sum{i=1}^{n} \mathcal{E}_i

\$\$

If a source is opposed to the direction of the others, its e.m.f. is subtracted.

\$\$

\mathcal{E}{\text{total}} = \mathcal{E}1 + \mathcal{E}2 - \mathcal{E}3 + \dots

\$\$

Step‑by‑Step Procedure

1. Identify the polarity of each source (positive + and negative – ends).
2. Write the e.m.f. of each source with a sign: + for sources aiding the current, – for those opposing.
3. Add the signed e.m.f. values using the formula above.
4. State the direction of the resulting e.m.f. (the polarity of the net source).

Example 1 – All Sources Aiding

Three identical cells, each with an e.m.f. of \$1.5\ \text{V}\$, are connected in series with the same polarity.

Combined e.m.f.

\$\$

\mathcal{E}_{\text{total}} = 1.5 + 1.5 + 1.5 = 4.5\ \text{V}

\$\$

Example 2 – One Source Opposed

Two cells of \$2.0\ \text{V}\$ and one cell of \$1.0\ \text{V}\$ are in series. The \$1.0\ \text{V}\$ cell is reversed.

\$\$

\mathcal{E}_{\text{total}} = 2.0 + 2.0 - 1.0 = 3.0\ \text{V}

\$\$

Practice Questions

1. Three batteries have e.m.f.s of \$1.2\ \text{V}\$, \$1.5\ \text{V}\$ and \$2.0\ \text{V}\$. All are connected in series with the same polarity. What is the combined e.m.f.?

2. Two \$3.0\ \text{V}\$ cells are connected in series, but one is reversed. Calculate the net e.m.f.

3. Four cells are connected in series: \$0.9\ \text{V}\$, \$1.5\ \text{V}\$, \$1.5\ \text{V}\$, and \$0.9\ \text{V}\$. The second \$1.5\ \text{V}\$ cell is opposite to the others. Find the total e.m.f. and indicate its polarity.

Common Mistakes to Avoid

• Forgetting to assign a sign to each source based on its polarity.
• Adding magnitudes without considering that a reversed source reduces the total e.m.f.
• Confusing series with parallel connections; in parallel the e.m.f.s are equal (if ideal), not added.

Suggested Diagram

Summary

To find the combined e.m.f. of sources in series:

1. Determine the polarity of each source.
2. Assign a positive sign to sources that aid the current and a negative sign to those that oppose it.
3. Sum the signed e.m.f. values using \$\mathcal{E}{\text{total}} = \sum \mathcal{E}i\$.
4. The resulting value gives both the magnitude and direction (polarity) of the net e.m.f.