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College Physics

Section Summary

College PhysicsSection Summary

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: 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.

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 rr 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=emfIrV=emfIr 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.

21.3 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.

21.4 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.

21.5 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.

21.6 DC 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=RCτ=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(1et/RC) (charging).V=emf(1et/RC) (charging). size 12{V="emf" \( 1 - e rSup { size 8{ - t/ ital "RC"} } \) } {}
  • Within the span of each time constant ττ 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=0t=0 size 12{t=0} {}, then its voltage decreases exponentially as given by
    V=V0et/RC (discharging).V=V0et/RC (discharging). size 12{V=V rSub { size 8{0} } e rSup { size 8{ - t/ ital "RC"} } \) } {}
  • In each time constant ττ size 12{τ} {}, the voltage falls by 0.368 of its remaining initial value, approaching zero asymptotically.
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