Series Circuits

Voltage Drops Add

the battery budget is spent

A source gives each coulomb an energy budget. Series resistors spend that budget in drops that add back to the source voltage.

Example

A source gives each coulomb an energy budget. Series resistors spend that budget in drops that add back to the source voltage. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The battery gives an energy budget

The battery gives each coulomb 12 joules of energy. In a series loop, the resistors spend that budget one after the other.

Vsource=12 VV_{\text{source}} = 12\ \text{V}
Series resistors split the battery budgetA single loop with a battery, an ammeter, and two resistors in the only path; voltage markers show the energy drops across the resistors.12 VA2 A2 ohm4 ohm2 A2 A2 A

Each resistor spends part of the budget

The current is the same through both resistors, so each voltage drop is current times resistance.

Vdrop=IRV_{\text{drop}} = I R

At the same current, bigger resistance drops more voltage

Hold the current at 2 amperes and compare resistor values. The drop grows with resistance because each coulomb spends more energy in the larger resistor.

IRVdrop2 A1 ohm2 V2 A2 ohm4 V2 A4 ohm8 V\begin{array}{c|c|c}I & R & V_{\text{drop}} \\2\ \text{A} & 1\ \text{ohm} & 2\ \text{V} \\2\ \text{A} & 2\ \text{ohm} & 4\ \text{V} \\2\ \text{A} & 4\ \text{ohm} & 8\ \text{V}\end{array}
Series resistors split the battery budgetA single loop with a battery, an ammeter, and two resistors in the only path; voltage markers show the energy drops across the resistors.12 VA2 A2 ohm4 ohm+-4 V+-8 V

Compute the two drops

Ra drops 4 volts, and Rb drops 8 volts. The table values are the same values drawn on the two voltage markers.

partIRVRa2 A2 ohm4 VRb2 A4 ohm8 V\begin{array}{c|c|c|c}\text{part} & I & R & V \\\text{Ra} & 2\ \text{A} & 2\ \text{ohm} & 4\ \text{V} \\\text{Rb} & 2\ \text{A} & 4\ \text{ohm} & 8\ \text{V}\end{array}
Series resistors split the battery budgetA single loop with a battery, an ammeter, and two resistors in the only path; voltage markers show the energy drops across the resistors.12 VA2 A2 ohm4 ohm+-4 V+-8 V

The drops add back to the source

The two drops are 4 volts plus 8 volts, which uses the full 12 volt battery budget. The diagram and the table are making the same claim.

VRa+VRb=4 V+8 V=12 VV_{\text{Ra}} + V_{\text{Rb}} = 4\ \text{V} + 8\ \text{V} = \hl{12}\ \text{V}
Series resistors split the battery budgetA single loop with a battery, an ammeter, and two resistors in the only path; voltage markers show the energy drops across the resistors.12 VA2 A2 ohm4 ohm+-4 V+-8 V
electricity With 12 V across 2 ohm and 4 ohm in series, the drops are 4 V and 8 V, and those drops add back to the source.