Series resistor circuits

Resistors are in series when they lie on the same unbranched path. Charge passes through each resistor in turn, so the same current passes through every resistor.

Each resistor has a potential difference across it. The total potential difference across the combination is the sum of the potential differences across the individual resistors:

For each resistor, $$V = IR$$. Since the current $$I$$ is the same through all the resistors, the potential difference across each is proportional to its resistance. A resistor with twice the resistance of another has twice the potential difference across it.

Total resistance in series

The total resistance of resistors in series is the sum of their individual resistances:

This follows from adding the potential differences: because $$V = IR$$ for each resistor and the same current passes through them all, the total potential difference is $$I$$ multiplied by the sum of the resistances. Adding another resistor in series therefore increases the total resistance.

For example, resistors of $$2,\Omega$$ and $$4,\Omega$$ in series have a total resistance of $$6,\Omega$$. If the total potential difference across them is $$6,\text{V}$$, the current through both is $$1,\text{A}$$. The potential differences are then $$2,\text{V}$$ and $$4,\text{V}$$ respectively: they add to $$6,\text{V}$$ and are in the same ratio as the resistances.

Parallel resistor circuits

Resistors are in parallel when each resistor is connected across the same two circuit points. The current has a separate path, or branch, through each resistor.

Because every branch connects the same two points, the potential difference across each branch is the same. The total current entering the parallel combination divides between the branches and is equal to the sum of the branch currents.

For a given potential difference, a branch with lower resistance carries a greater current. This follows from $$I = V/R$$: with the same $$V$$ across each branch, a smaller $$R$$ gives a larger $$I$$.

Total resistance of parallel resistors

The total resistance $$R_T$$ is the resistance of a single resistor that would draw the same total current at the same potential difference. Since each branch provides a path for current, the total current is greater than the current through any one branch alone. The total resistance is therefore less than the resistance of any individual branch.

For resistors with resistances $$R_1$$, $$R_2$$, $$R_3$$ and so on, the total resistance is given by:

This relationship follows from adding the branch currents. Each branch current is its common potential difference divided by its resistance; the total current is that same potential difference divided by $$R_T$$.

For example, if resistors of $$6,\Omega$$ and $$3,\Omega$$ are connected in parallel:

Taking the reciprocal gives $$R_T = 2,\Omega$$, which is less than either branch resistance. At a potential difference of $$12,\mathrm{V}$$, the $$6,\Omega$$ branch carries $$2,\mathrm{A}$$ and the $$3,\Omega$$ branch carries $$4,\mathrm{A}$$. The total current is therefore $$6,\mathrm{A}$$.

Cell arrangements in circuits

A cell provides a potential difference (voltage) that drives charge around a circuit. When more than one cell is used, their combined potential difference depends on how the cells are connected and which way they face.

Cells in series

Cells are connected in series when they are placed one after another in the same path. Their orientations determine whether their potential differences add or subtract.

• When the cells aid one another, the positive terminal of one is connected to the negative terminal of the next. Their potential differences add. For example, two $$1.5\ \text{V}$$ cells provide $$3.0\ \text{V}$$ together.
• When cells oppose one another, their potential differences subtract. For example, two $$1.5\ \text{V}$$ cells facing in opposite directions provide a combined potential difference of $$0\ \text{V}$$. If their voltages differ, the larger voltage determines the direction of the combined potential difference.

Cells in parallel

Identical cells are connected in parallel with the same polarity when their positive terminals are connected together and their negative terminals are connected together. They provide the same potential difference as one cell, not the sum of their voltages. For example, two identical $$1.5\ \text{V}$$ cells in parallel provide $$1.5\ \text{V}$$.

The parallel cells share the current supplied to the circuit. Because each cell supplies only part of the current, the arrangement can supply the circuit for longer than a single cell.

Electrical energy and power

A component transfers electrical energy when current passes through it. Electrical power is the rate at which that energy is transferred: a power of one watt means one joule is transferred each second.

For a constant power, the relationship between power, energy transferred and time is:

Power from current and potential difference

Current is the rate at which charge passes through a component. Potential difference is the energy transferred per unit of charge. Multiplying them therefore gives the energy transferred per second, or power:

Here, $$I$$ is the current through the component and $$V$$ is the potential difference across it. If both remain constant for a time $$t$$, the power is constant, so the energy transferred is:

Power in a resistor

For a resistor obeying $$V = IR$$, the power can also be expressed using its resistance. Substituting $$IR$$ for $$V$$ in $$P = IV$$ gives $$P = I^2R$$. Alternatively, substituting $$V/R$$ for $$I$$ gives $$P = V^2/R$$:

Use the expression that matches the quantities known. In these equations:

• $$E$$ is energy transferred, in joules (J).
• $$P$$ is power, in watts (W).
• $$I$$ is current, in amperes (A).
• $$V$$ is potential difference, in volts (V).
• $$R$$ is resistance, in ohms ($$\Omega$$).
• $$t$$ is time, in seconds (s).

Conservation of charge in circuits

In a dc circuit, electric charge flows through the components, but the components do not use it up. This is conservation of charge: charge cannot disappear as it passes through a component.

Current is the rate at which charge flows. For a steady current, the charge passing a point is related to the time taken by:

Here, $$I$$ is current in amperes, $$Q$$ is charge in coulombs and $$t$$ is time in seconds.

Current in unbranched sections and at junctions

In an unbranched section, charge has only one path to follow. Since charge is not used up or accumulating in the section, the same amount passes every point each second. The current is therefore the same throughout that section, including on either side of a component.

At a junction, charge can take different paths, so the current can divide between branches. Where charge does not accumulate at the junction, the total current entering it equals the total current leaving it. The currents recombine when the branches meet again.

For example, if $$0.6,\text{A}$$ enters a junction and $$0.2,\text{A}$$ leaves through one of two branches, the other branch carries $$0.4,\text{A}$$. Together, the two leaving currents equal the entering current.

Conservation of energy in DC circuits

In a DC circuit, a cell transfers energy from its chemical store to charges in the circuit. Other components transfer energy from the circuit to energy stores or the surroundings. For example, a lamp transfers energy to the surroundings by light and heating.

Energy is conserved: the energy supplied by the cell is not created or used up. It is transferred to other stores or the surroundings.

Potential difference and energy transfer

Potential difference is the energy transferred per unit charge. For a component, it is given by:

Here, $$V$$ is potential difference in volts, $$E$$ is energy transferred in joules, and $$Q$$ is charge in coulombs. A potential difference of $$1\ \mathrm{V}$$ means that $$1\ \mathrm{J}$$ of energy is transferred per coulomb of charge.

Energy around a complete loop

As charge passes through a cell, it gains energy. As it passes through other components, energy is transferred from the circuit. Around a complete loop, the energy supplied per unit charge equals the energy transferred per unit charge by the other components.

This relationship can be stated in terms of voltage. A voltage rise represents energy supplied per unit charge, while a voltage drop represents energy transferred away per unit charge:

For example, suppose a cell provides a voltage rise of $$6\ \mathrm{V}$$ in a loop containing two components. If the voltage drop across one component is $$2\ \mathrm{V}$$, the drop across the other is $$4\ \mathrm{V}$$. Together, the components transfer $$6\ \mathrm{J}$$ for each coulomb of charge, equal to the energy supplied per coulomb by the cell.