Sections

Section Summary
# Section Summary

### 4.1 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:
$${R}_{\text{s}}={R}_{1}+{R}_{2}+{R}_{3}+\text{.}\text{.}\text{.}\text{.}$$
- 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
$$\frac{1}{{R}_{\text{p}}}=\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}+\frac{1}{{R}_{3}}+\text{.}\text{.}\text{.}\text{.}$$
- 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.

### 4.2 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 $r\text{.}$
- 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 $V=\text{emf}-\text{Ir}\text{,}$ 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.

### 4.3 Kirchhoff's Rules

- 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.

### 4.4 DC Voltmeters and Ammeters

- 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.

### 4.5 Null Measurements

- 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.

### 4.6 DC Circuits Containing Resistors and Capacitors

- An $\text{RC}$ circuit is one that has both a resistor and a capacitor.
- The time constant $\tau $ for an $\text{RC}$ circuit is $\tau =\text{RC}\text{.}$
- 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,
$$V=\text{emf}(1-{e}^{-t/\text{RC}})\text{(charging).}$$
- Within the span of each time constant $\tau \text{,}$ 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 $t=0\text{,}$ then its voltage decreases exponentially as given by
$$V={V}_{0}{e}^{-t/\text{RC}}\text{(discharging).}$$
- In each time constant $\tau \text{,}$ the voltage falls by 0.368 of its remaining initial value, approaching zero asymptotically.