21.1 Resistors in series and parallel  (Page 8/17)

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If two household lightbulbs rated 60 W and 100 W are connected in series to household power, which will be brighter? Explain.

Suppose you are doing a physics lab that asks you to put a resistor into a circuit, but all the resistors supplied have a larger resistance than the requested value. How would you connect the available resistances to attempt to get the smaller value asked for?

Before World War II, some radios got power through a “resistance cord” that had a significant resistance. Such a resistance cord reduces the voltage to a desired level for the radio’s tubes and the like, and it saves the expense of a transformer. Explain why resistance cords become warm and waste energy when the radio is on.

Some light bulbs have three power settings (not including zero), obtained from multiple filaments that are individually switched and wired in parallel. What is the minimum number of filaments needed for three power settings?

Problem exercises

Note: Data taken from figures can be assumed to be accurate to three significant digits.

(a) What is the resistance of ten $\text{275-Ω}$ resistors connected in series? (b) In parallel?

(a) $2\text{.}\text{75}\phantom{\rule{0.25em}{0ex}}\text{k}\Omega$

(b) $\text{27}\text{.}5\phantom{\rule{0.25em}{0ex}}\Omega$

(a) What is the resistance of a $\text{1.00}×{10}^{2}-\Omega$ , a $2\text{.}\text{50-kΩ}$ , and a $4\text{.}\text{00-k}\Omega$ resistor connected in series? (b) In parallel?

What are the largest and smallest resistances you can obtain by connecting a $\text{36}\text{.}0-\Omega$ , a $\text{50}\text{.}0-\Omega$ , and a $\text{700-Ω}$ resistor together?

(a) $\text{786}\phantom{\rule{0.25em}{0ex}}\Omega$

(b) $\text{20}\text{.}3\phantom{\rule{0.25em}{0ex}}\Omega$

An 1800-W toaster, a 1400-W electric frying pan, and a 75-W lamp are plugged into the same outlet in a 15-A, 120-V circuit. (The three devices are in parallel when plugged into the same socket.). (a) What current is drawn by each device? (b) Will this combination blow the 15-A fuse?

Your car’s 30.0-W headlight and 2.40-kW starter are ordinarily connected in parallel in a 12.0-V system. What power would one headlight and the starter consume if connected in series to a 12.0-V battery? (Neglect any other resistance in the circuit and any change in resistance in the two devices.)

$\text{29}\text{.}6\phantom{\rule{0.25em}{0ex}}\text{W}$

(a) Given a 48.0-V battery and $\text{24}\text{.}0-\Omega$ and $\text{96}\text{.}0-\Omega$ resistors, find the current and power for each when connected in series. (b) Repeat when the resistances are in parallel.

Referring to the example combining series and parallel circuits and [link] , calculate ${I}_{3}$ in the following two different ways: (a) from the known values of $I$ and ${I}_{2}$ ; (b) using Ohm’s law for ${R}_{3}$ . In both parts explicitly show how you follow the steps in the Problem-Solving Strategies for Series and Parallel Resistors .

(a) 0.74 A

(b) 0.742 A

Referring to [link] : (a) Calculate ${P}_{3}$ and note how it compares with ${P}_{3}$ found in the first two example problems in this module. (b) Find the total power supplied by the source and compare it with the sum of the powers dissipated by the resistors.

Refer to [link] and the discussion of lights dimming when a heavy appliance comes on. (a) Given the voltage source is 120 V, the wire resistance is $0\text{.}\text{400}\phantom{\rule{0.25em}{0ex}}\Omega$ , and the bulb is nominally 75.0 W, what power will the bulb dissipate if a total of 15.0 A passes through the wires when the motor comes on? Assume negligible change in bulb resistance. (b) What power is consumed by the motor?

(a) 60.8 W

(b) 3.18 kW

A 240-kV power transmission line carrying $5.00×{10}^{2}\phantom{\rule{0.25em}{0ex}}\text{A}$ is hung from grounded metal towers by ceramic insulators, each having a $1\text{.}\text{00}×{\text{10}}^{9}-\Omega$ resistance. [link] . (a) What is the resistance to ground of 100 of these insulators? (b) Calculate the power dissipated by 100 of them. (c) What fraction of the power carried by the line is this? Explicitly show how you follow the steps in the Problem-Solving Strategies for Series and Parallel Resistors .

Show that if two resistors ${R}_{1}$ and ${R}_{2}$ are combined and one is much greater than the other ( ${R}_{1}\text{>>}{R}_{2}$ ): (a) Their series resistance is very nearly equal to the greater resistance ${R}_{1}$ . (b) Their parallel resistance is very nearly equal to smaller resistance ${R}_{2}$ .

(a) $\begin{array}{}{R}_{\text{s}}={R}_{1}+{R}_{2}\\ ⇒{R}_{\text{s}}\phantom{\rule{0.25em}{0ex}}\approx \phantom{\rule{0.25em}{0ex}}{R}_{1}\left({R}_{1}\text{>>}{R}_{2}\right)\end{array}$

(b) $\frac{1}{{R}_{p}}=\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}=\frac{{R}_{1}+{R}_{2}}{{R}_{1}{R}_{2}}$ ,

so that

$\begin{array}{}{R}_{p}=\frac{{R}_{1}{R}_{2}}{{R}_{1}+{R}_{2}}\approx \frac{{R}_{1}{R}_{2}}{{R}_{1}}={R}_{2}\left({R}_{1}\text{>>}{R}_{2}\right)\text{.}\end{array}$

Unreasonable Results

Two resistors, one having a resistance of $1\text{45}\phantom{\rule{0.25em}{0ex}}\Omega$ , are connected in parallel to produce a total resistance of $150\phantom{\rule{0.25em}{0ex}}\Omega$ . (a) What is the value of the second resistance? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Unreasonable Results

Two resistors, one having a resistance of $9\text{00 kΩ}$ , are connected in series to produce a total resistance of $0\text{.}\text{500 MΩ}$ . (a) What is the value of the second resistance? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

(a) $-\text{400 k}\Omega$

(b) Resistance cannot be negative.

(c) Series resistance is said to be less than one of the resistors, but it must be greater than any of the resistors.

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