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12.1 Resistors in series and parallel  (Page 7/17)

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Section summary

  • 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 + . . . . 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 + . . . . 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.

Conceptual questions

A switch has a variable resistance that is nearly zero when closed and extremely large when open, and it is placed in series with the device it controls. Explain the effect the switch in [link] has on current when open and when closed.

The diagram shows a circuit with a voltage source and internal resistance small r connected in series with a resistance R and a switch.
A switch is ordinarily in series with a resistance and voltage source. Ideally, the switch has nearly zero resistance when closed but has an extremely large resistance when open. (Note that in this diagram, the script E represents the voltage (or electromotive force) of the battery.)

What is the voltage across the open switch in [link] ?

There is a voltage across an open switch, such as in [link] . Why, then, is the power dissipated by the open switch small?

Why is the power dissipated by a closed switch, such as in [link] , small?

A student in a physics lab mistakenly wired a light bulb, battery, and switch as shown in [link] . Explain why the bulb is on when the switch is open, and off when the switch is closed. (Do not try this—it is hard on the battery!)

This diagram shows a circuit with a voltage source and internal resistance small r. A resistance R and an open switch are connected in parallel to it.
A wiring mistake put this switch in parallel with the device represented by R size 12{R} {} . (Note that in this diagram, the script E represents the voltage (or electromotive force) of the battery.)

Knowing that the severity of a shock depends on the magnitude of the current through your body, would you prefer to be in series or parallel with a resistance, such as the heating element of a toaster, if shocked by it? Explain.

Would your headlights dim when you start your car’s engine if the wires in your automobile were superconductors? (Do not neglect the battery’s internal resistance.) Explain.

Some strings of holiday lights are wired in series to save wiring costs. An old version utilized bulbs that break the electrical connection, like an open switch, when they burn out. If one such bulb burns out, what happens to the others? If such a string operates on 120 V and has 40 identical bulbs, what is the normal operating voltage of each? Newer versions use bulbs that short circuit, like a closed switch, when they burn out. If one such bulb burns out, what happens to the others? If such a string operates on 120 V and has 39 remaining identical bulbs, what is then the operating voltage of each?
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Source:  OpenStax, Introductory physics - for kpu phys 1100 (2015 edition). OpenStax CNX. May 30, 2015 Download for free at http://legacy.cnx.org/content/col11588/1.13
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