Sections
Section Summary

# Section Summary

### 4.1Resistors in Series and Parallel

• The total resistance of an electrical circuit with resistors wired in a series is the sum of the individual resistances:
$Rs=R1+R2+R3+....Rs=R1+R2+R3+.... size 12{R rSub { size 8{s} } =R rSub { size 8{1} } +R rSub { size 8{2} } +R rSub { size 8{3} } + "." "." "." "." } {}$
• 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
$1 R p = 1 R 1 + 1 R 2 + 1 R 3 + . . . . 1 R p = 1 R 1 + 1 R 2 + 1 R 3 + . . . . size 12{ { {1} over {R rSub { size 8{p} } } } = { {1} over {R rSub { size 8{1} } } } + { {1} over {R rSub { size 8{2} } } } + { {1} over {R rSub { size 8{3} } } } + "." "." "." "." } {}$
• 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.

### 4.2Electromotive 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 $r.r. size 12{r} {}$
• 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 $rr size 12{r} {}$ of a voltage source affects the output voltage when a current flows.
• The voltage output of a device is called its terminal voltage $VV size 12{V} {}$ and is given by $V=emf−Ir,V=emf−Ir, size 12{V="emf" - ital "Ir"} {}$ where $II size 12{I} {}$ 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.

### 4.3Kirchhoff'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.

### 4.4DC 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.

### 4.5Null 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.

### 4.6DC Circuits Containing Resistors and Capacitors

• An $RCRC size 12{ ital "RC"} {}$ circuit is one that has both a resistor and a capacitor.
• The time constant $ττ size 12{τ} {}$ for an $RCRC size 12{ ital "RC"} {}$ circuit is $τ=RC.τ=RC. size 12{τ= ital "RC"} {}$
• When an initially uncharged ($V0=0V0=0 size 12{V rSub { size 8{0} } =0} {}$ at $t=0t=0 size 12{t=0} {}$) 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).V=emf(1−e−t/RC) (charging). size 12{V="emf" $$1 - e rSup { size 8{ - t/ ital "RC"} }$$ } {}$
• Within the span of each time constant $τ,τ, size 12{τ} {}$ the voltage rises by 0.632 of the remaining value, approaching the final voltage asymptotically.
• If a capacitor with an initial voltage $V0V0 size 12{V rSub { size 8{0} } } {}$ is discharged through a resistor starting at $t=0,t=0, size 12{t=0} {}$ then its voltage decreases exponentially as given by
$V=V0e−t/RC (discharging).V=V0e−t/RC (discharging). size 12{V=V rSub { size 8{0} } e rSup { size 8{ - t/ ital "RC"} } \) } {}$
• In each time constant $τ,τ, size 12{τ} {}$ the voltage falls by 0.368 of its remaining initial value, approaching zero asymptotically.