Series and parallel circuits

Cambridge·Physics·4.3.2·Extended

Series and parallel circuits

A circuit needs a complete conducting path through a source for current to flow. Components can be connected in one path or in separate branches between junctions.

Series circuits

In a series circuit, components are connected one after another in a single conducting path through the source. There are no branches where the current can divide, so the current is the same at every point in the path.

Parallel circuits

In a parallel circuit, components are connected in separate branches between common junctions. Each branch provides a path for current. At a junction, the current from the source divides between the branches; at the next junction, the branch currents recombine.

The source current equals the sum of the branch currents. For two branches:

Isource=I1+I2I_{\text{source}} = I_1 + I_2

Here, $$I_1$$ and $$I_2$$ are the currents in the two branches. When more than one branch carries current, the source current is greater than the current in any one branch.

Lamps in parallel

To connect lamps in parallel, place each lamp in its own branch between the same two junctions, with a complete path through the source. Each lamp has the supply potential difference across it.

If one lamp fails and breaks its branch, the other lamps can remain lit: their paths through the source are still complete. A switch placed in each separate branch can control its lamp independently. Opening one switch breaks only that branch’s path, leaving the other branches complete.

Equivalent resistance in series and parallel

A group of resistors can be treated as a single resistor when considering the current through the group. Its equivalent resistance, or combined resistance, is the resistance of a single resistor that would draw the same current when the same potential difference is applied across it.

Resistors in series

Resistors in series are connected along one path, so the same current passes through each one. The potential differences across the individual resistors add up to the potential difference across the whole group. Their combined resistance is the sum of their individual resistances:

Rtotal=R1+R2+⋯R_{\text{total}} = R_1 + R_2 + \cdots

For example, resistors of $$2\ \Omega$$ and $$3\ \Omega$$ in series have a combined resistance of $$5\ \Omega$$. Adding another resistor in series increases the combined resistance.

Two resistors in parallel

Two resistors in parallel provide separate paths for current. Each path has the same potential difference across it, and the total current is the sum of the currents in the two paths.

For the same potential difference, the two paths together carry more current than either resistor would carry alone. The combined resistance is therefore less than the resistance of either resistor on its own.

Combined e.m.f. of sources in series

A source transfers energy to charge. Its electromotive force, or e.m.f., is the energy transferred per unit charge. It is measured in volts; one volt is one joule per coulomb.

When sources are connected in series, a charge passes through each source in turn. The combined e.m.f. depends on whether the sources drive charge in the same direction or in opposite directions.

Adding e.m.f.s in series

Choose one direction around the series connection as positive. Give an e.m.f. a positive sign if its source drives charge in that direction, and a negative sign if it drives charge in the opposite direction. The combined e.m.f. is the algebraic sum of these signed values:

Ecombined=E1+E2+⋯\mathcal{E}_{\text{combined}} = \mathcal{E}_1 + \mathcal{E}_2 + \cdots

For example, two sources of $$1.5,\text{V}$$ and $$1.0,\text{V}$$ oriented to drive charge in the same direction have a combined e.m.f. of $$2.5,\text{V}$$ in that direction. If the $$1.0,\text{V}$$ source is reversed, it opposes the other source, giving $$1.5,\text{V} - 1.0,\text{V} = 0.5,\text{V}$$ in the direction of the $$1.5,\text{V}$$ source.

If the signed sum is negative, the combined e.m.f. acts opposite to the direction chosen as positive.

Conservation of current at a junction

A junction is a point in a circuit where branches meet. Current carries electric charge into and out of the junction. In a steady circuit, charge does not build up at the junction, so the amount of charge entering each second must equal the amount leaving each second.

Current is the rate of flow of charge. Therefore, the total current entering a junction equals the total current leaving it:

∑Ientering=∑Ileaving\sum I_{\text{entering}} = \sum I_{\text{leaving}}

This rule follows from conservation of charge. It applies whether there is one incoming branch or several, and whether there is one outgoing branch or several.

Finding an unknown current

Suppose currents of $$2,\mathrm{A}$$ and $$3,\mathrm{A}$$ enter a junction. Two currents leave: one is $$1,\mathrm{A}$$ and the other is unknown.

The total entering is $$2 + 3 = 5,\mathrm{A}$$. The total leaving must also be $$5,\mathrm{A}$$, so the unknown current is $$5 - 1 = 4,\mathrm{A}$$.

Potential difference in series and parallel circuits

Potential difference (voltage) describes the energy transferred per unit charge between two points in a circuit. A potential difference of 1 volt means that 1 joule of energy is transferred for each coulomb of charge passing between those points.

Potential difference across components in series

Components in series are connected one after another. As charge passes through successive components, energy is transferred in each one. The total energy transferred per unit charge across the arrangement is therefore the sum of the transfers across the individual components.

Vtotal=V1+V2+⋯V_{\text{total}} = V_1 + V_2 + \cdots

For example, if the total potential difference across two components in series is $$12\ \text{V}$$ and the potential difference across the first is $$5\ \text{V}$$, that across the second is $$12 - 5 = 7\ \text{V}$$.

Potential difference across parallel branches

Parallel branches are connected across the same two junctions. Each branch therefore has the same potential difference as the whole parallel arrangement.

Vparallel arrangement=Vbranch 1=Vbranch 2=⋯V_{\text{parallel arrangement}} = V_{\text{branch 1}} = V_{\text{branch 2}} = \cdots

For example, if the potential difference across a parallel arrangement is $$12\ \text{V}$$, the potential difference across each branch is also $$12\ \text{V}$$. The potential differences across the branches are not added together.

Calculating resistance of two parallel resistors

Two resistors in parallel provide separate paths for current. Their equivalent resistance, $$R$$, is the resistance of a single resistor that would draw the same total current at the same potential difference.

For two parallel resistors with resistances $$R_1$$ and $$R_2$$, the equivalent resistance is given by:

1R=1R1+1R2\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}

Rearranging gives a direct way to calculate the resistance:

R=R1R2R1+R2R=\frac{R_1R_2}{R_1+R_2}

For example, if the resistors are $$6,\Omega$$ and $$3,\Omega$$, their equivalent resistance is:

R=6×36+3=2 ΩR=\frac{6\times 3}{6+3}=2\,\Omega

Why parallel resistances combine this way

Both branches have the same potential difference, $$V$$. Using $$I=V/R$$, the branch currents are $$V/R_1$$ and $$V/R_2$$. These currents add to give the total current. The equivalent resistor has the same potential difference and carries that total current, so:

VR=VR1+VR2\frac{V}{R}=\frac{V}{R_1}+\frac{V}{R_2}

Dividing by $$V$$ gives the reciprocal relationship above. The extra path allows more total current to flow at the same potential difference, so the equivalent resistance is less than either individual resistance.