Series and Parallel Circuits: Combined E.M.F.

This section explains how to calculate the combined electromotive force (e.m.f.) of multiple voltage sources connected in series. Understanding series and parallel circuits is fundamental to electricity.

Series Circuits

In a series circuit, components are connected one after the other, so the same current flows through each component. The total resistance in a series circuit is the sum of the individual resistances.

Calculating Total Resistance in a Series Circuit

The total resistance ($R_{total}$) of a series circuit is calculated as:

$$R_{total} = R_1 + R_2 + R_3 + ... + R_n$$

Calculating Combined E.M.F. in a Series Circuit

When multiple voltage sources are connected in series, their e.m.f.s add up. The combined e.m.f. ($V_{total}$) is the sum of the individual e.m.f.s.

$$V_{total} = V_1 + V_2 + V_3 + ... + V_n$$

Important Note: The total current in a series circuit is the same through all components, but the voltage drop across each component will be different depending on its resistance.

Parallel Circuits

In a parallel circuit, components are connected across each other, so the voltage across each component is the same. The total current in a parallel circuit is the sum of the currents through each branch.

Calculating Total Resistance in a Parallel Circuit

The reciprocal of the total resistance ($R_{total}$) of a parallel circuit is the sum of the reciprocals of the individual resistances.

$$\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n}$$

Calculating Combined E.M.F. in a Parallel Circuit

When multiple voltage sources are connected in parallel, the voltage across each source is the same. Therefore, the combined e.m.f. is equal to the voltage of any individual source.

$$V_{total} = V_1 = V_2 = V_3 = ... = V_n$$

Important Note: The total current in a parallel circuit is the sum of the currents in each branch, and the voltage across each branch is constant.

Combined E.M.F. of Sources in Series

This section focuses specifically on calculating the combined e.m.f. when multiple voltage sources are connected in series.

Formula for Combined E.M.F. in Series

If you have $n$ voltage sources connected in series, the combined e.m.f. ($V_{total}$) is the sum of the individual e.m.f.s:

Source	E.M.F. (V)
1	$V_1$
2	$V_2$
3	$V_3$
...	$V_n$

$$V_{total} = V_1 + V_2 + V_3 + ... + V_n$$

Example:

Consider three voltage sources connected in series with e.m.f.s of 12V, 24V, and 36V respectively. The combined e.m.f. is:

$V_{total} = 12V + 24V + 36V = 72V$

Practice Questions:

1. Three batteries with e.m.f.s of 9V, 12V, and 18V are connected in series. Calculate the total e.m.f.
2. Four voltage sources with e.m.f.s of 5V, 10V, 15V, and 20V are connected in series. Calculate the combined e.m.f.