21.1 Resistors in Series and Parallel

The total resistance of an electrical circuit with resistors wired in a series is the sum of the individual resistances: $R_{s} = R_{1} + R_{2} + R_{3} + . . . .$
Each resistor in a series circuit has the same amount of current flowing through it.
The voltage drop, or power dissipation, across each individual resistor in a series is different, and their combined total adds up to the power source input.
The total resistance of an electrical circuit with resistors wired in parallel is less than the lowest resistance of any of the components and can be determined using the formula:
$\frac{1}{R_{p}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + . . . .$
Each resistor in a parallel circuit has the same full voltage of the source applied to it.
The current flowing through each resistor in a parallel circuit is different, depending on the resistance.
If a more complex connection of resistors is a combination of series and parallel, it can be reduced to a single equivalent resistance by identifying its various parts as series or parallel, reducing each to its equivalent, and continuing until a single resistance is eventually reached.

21.2 Electromotive Force: Terminal Voltage

All voltage sources have two fundamental parts—a source of electrical energy that has a characteristic electromotive force (emf), and an internal resistance $r$ .
The emf is the potential difference of a source when no current is flowing.
The numerical value of the emf depends on the source of potential difference.
The internal resistance $r$ of a voltage source affects the output voltage when a current flows.
The voltage output of a device is called its terminal voltage $V$ and is given by $V = emf - Ir$ , where $I$ is the electric current and is positive when flowing away from the positive terminal of the voltage source.
When multiple voltage sources are in series, their internal resistances add and their emfs add algebraically.
Solar cells can be wired in series or parallel to provide increased voltage or current, respectively.

Kirchhoff’s rules can be used to analyze any circuit, simple or complex.
Kirchhoff’s first rule—the junction rule: The sum of all currents entering a junction must equal the sum of all currents leaving the junction.
Kirchhoff’s second rule—the loop rule: The algebraic sum of changes in potential around any closed circuit path (loop) must be zero.
The two rules are based, respectively, on the laws of conservation of charge and energy.
When calculating potential and current using Kirchhoff’s rules, a set of conventions must be followed for determining the correct signs of various terms.
The simpler series and parallel rules are special cases of Kirchhoff’s rules.

Voltmeters measure voltage, and ammeters measure current.
A voltmeter is placed in parallel with the voltage source to receive full voltage and must have a large resistance to limit its effect on the circuit.
An ammeter is placed in series to get the full current flowing through a branch and must have a small resistance to limit its effect on the circuit.
Both can be based on the combination of a resistor and a galvanometer, a device that gives an analog reading of current.
Standard voltmeters and ammeters alter the circuit being measured and are thus limited in accuracy.

Null measurement techniques achieve greater accuracy by balancing a circuit so that no current flows through the measuring device.
One such device, for determining voltage, is a potentiometer.
Another null measurement device, for determining resistance, is the Wheatstone bridge.
Other physical quantities can also be measured with null measurement techniques.

An $RC$ circuit is one that has both a resistor and a capacitor.
The time constant $τ$ for an $RC$ circuit is $τ = RC$ .
When an initially uncharged ( $V_{0} = 0$ at $t = 0$ ) capacitor in series with a resistor is charged by a DC voltage source, the voltage rises, asymptotically approaching the emf of the voltage source; as a function of time,
$V = emf (1 - e^{- t / RC}) (charging).$
Within the span of each time constant $τ$ , the voltage rises by 0.632 of the remaining value, approaching the final voltage asymptotically.
If a capacitor with an initial voltage $V_{0}$ is discharged through a resistor starting at $t = 0$ , then its voltage decreases exponentially as given by
$V = V_{0} e^{- t / RC} (discharging).$
In each time constant $τ$ , the voltage falls by 0.368 of its remaining initial value, approaching zero asymptotically.