21.1 Resistors in series and parallel  (Page 2/17)

 Page 2 / 17

These energies must be equal, because there is no other source and no other destination for energy in the circuit. Thus, $\mathit{qV}={\mathit{qV}}_{1}+{\mathit{qV}}_{2}+{\mathit{qV}}_{3}$ . The charge $q$ cancels, yielding $V={V}_{1}+{V}_{2}+{V}_{3}$ , as stated. (Note that the same amount of charge passes through the battery and each resistor in a given amount of time, since there is no capacitance to store charge, there is no place for charge to leak, and charge is conserved.)

Now substituting the values for the individual voltages gives

$V={\text{IR}}_{1}+{\text{IR}}_{2}+{\text{IR}}_{3}=I\left({R}_{1}+{R}_{2}+{R}_{3}\right).$

Note that for the equivalent single series resistance ${R}_{\text{s}}$ , we have

$V={\text{IR}}_{\text{s}}.$

This implies that the total or equivalent series resistance ${R}_{\text{s}}$ of three resistors is ${R}_{\text{s}}={R}_{1}+{R}_{2}+{R}_{3}$ .

This logic is valid in general for any number of resistors in series; thus, the total resistance ${R}_{\text{s}}$ of a series connection is

${R}_{\text{s}}={R}_{1}+{R}_{2}+{R}_{3}+\text{.}\text{.}\text{.},$

as proposed. Since all of the current must pass through each resistor, it experiences the resistance of each, and resistances in series simply add up.

Calculating resistance, current, voltage drop, and power dissipation: analysis of a series circuit

Suppose the voltage output of the battery in [link] is $\text{12}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{V}$ , and the resistances are ${R}_{1}=1\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\Omega$ , ${R}_{2}=6\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\Omega$ , and ${R}_{3}=\text{13}\text{.}0\phantom{\rule{0.25em}{0ex}}\Omega$ . (a) What is the total resistance? (b) Find the current. (c) Calculate the voltage drop in each resistor, and show these add to equal the voltage output of the source. (d) Calculate the power dissipated by each resistor. (e) Find the power output of the source, and show that it equals the total power dissipated by the resistors.

Strategy and Solution for (a)

The total resistance is simply the sum of the individual resistances, as given by this equation:

$\begin{array}{lll}{R}_{\text{s}}& =& {R}_{1}+{R}_{2}+{R}_{3}\\ & =& 1.00 \Omega +6.00 \Omega +\text{13.0 Ω}\\ & =& \text{20.0 Ω.}\end{array}$

Strategy and Solution for (b)

The current is found using Ohm’s law, $V=\text{IR}$ . Entering the value of the applied voltage and the total resistance yields the current for the circuit:

$I=\frac{V}{{R}_{\text{s}}}=\frac{\text{12}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{V}}{\text{20}\text{.}\text{0}\phantom{\rule{0.25em}{0ex}}\Omega }=0\text{.}\text{600}\phantom{\rule{0.25em}{0ex}}\text{A}.$

Strategy and Solution for (c)

The voltage—or $\text{IR}$ drop—in a resistor is given by Ohm’s law. Entering the current and the value of the first resistance yields

${V}_{1}={\mathit{IR}}_{1}=\left(0\text{.}\text{600}\phantom{\rule{0.25em}{0ex}}\text{A}\right)\left(1\text{.}0\phantom{\rule{0.25em}{0ex}}\Omega \right)=0\text{.}\text{600}\phantom{\rule{0.25em}{0ex}}\text{V}.$

Similarly,

${V}_{2}={\mathit{IR}}_{2}=\left(0\text{.}\text{600}\phantom{\rule{0.25em}{0ex}}\text{A}\right)\left(6\text{.}0\phantom{\rule{0.25em}{0ex}}\Omega \right)=3\text{.}\text{60}\phantom{\rule{0.25em}{0ex}}\text{V}$

and

${V}_{3}={\mathit{IR}}_{3}=\left(0\text{.}\text{600}\phantom{\rule{0.25em}{0ex}}\text{A}\right)\left(\text{13}\text{.}0\phantom{\rule{0.25em}{0ex}}\Omega \right)=7\text{.}\text{80}\phantom{\rule{0.25em}{0ex}}\text{V}.$

Discussion for (c)

The three $\text{IR}$ drops add to $\text{12}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{V}$ , as predicted:

${V}_{1}+{V}_{2}+{V}_{3}=\left(0\text{.}\text{600}+3\text{.}\text{60}+7\text{.}\text{80}\right)\phantom{\rule{0.25em}{0ex}}\text{V}=\text{12}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{V}.$

Strategy and Solution for (d)

The easiest way to calculate power in watts (W) dissipated by a resistor in a DC circuit is to use Joule’s law    , $P=\text{IV}$ , where $P$ is electric power. In this case, each resistor has the same full current flowing through it. By substituting Ohm’s law $V=\text{IR}$ into Joule’s law, we get the power dissipated by the first resistor as

${P}_{1}={I}^{2}{R}_{1}=\left(0\text{.}\text{600}\phantom{\rule{0.25em}{0ex}}\text{A}{\right)}^{2}\left(1\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\Omega \right)=0\text{.}\text{360}\phantom{\rule{0.25em}{0ex}}\text{W}.$

Similarly,

${P}_{2}={I}^{2}{R}_{2}=\left(0\text{.}\text{600}\phantom{\rule{0.25em}{0ex}}\text{A}{\right)}^{2}\left(6\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\Omega \right)=2\text{.}\text{16}\phantom{\rule{0.25em}{0ex}}\text{W}$

and

${P}_{3}={I}^{2}{R}_{3}=\left(0\text{.}\text{600}\phantom{\rule{0.25em}{0ex}}\text{A}{\right)}^{2}\left(\text{13}\text{.}0\phantom{\rule{0.25em}{0ex}}\Omega \right)=4\text{.}\text{68}\phantom{\rule{0.25em}{0ex}}\text{W}.$

Discussion for (d)

Power can also be calculated using either $P=\text{IV}$ or $P=\frac{{V}^{2}}{R}$ , where $V$ is the voltage drop across the resistor (not the full voltage of the source). The same values will be obtained.

Strategy and Solution for (e)

The easiest way to calculate power output of the source is to use $P=\text{IV}$ , where $V$ is the source voltage. This gives

$P=\left(0\text{.}\text{600}\phantom{\rule{0.25em}{0ex}}\text{A}\right)\left(\text{12}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{V}\right)=7\text{.}\text{20}\phantom{\rule{0.25em}{0ex}}\text{W}.$

Calculate the Newton's the weight of a 2.5 Kilogram of melon. What is its weight in pound?
calculate the tension of the cable when a buoy with 0.5m and mass of 20kg
what is displacement
it's the time rate of change of distance
Mollamin
distance in a given direction is diplacement
Musa
Distance in a spacified direction
you shouldn't say distance,displacement and distance are two different things .distance can be lopped curved but displacement is always in a straight line so you can't use distance to define it. displacement is the change of position in a specified direction.
Joshua
Well stayed josh👍
Joshua
well explained
Mary
what is the meaning of physics
to study objects in motion and how they interact or take part in the natural phenomenon of the universe.
Phill
an object that has a small mass and an object has a large mase have the same momentum which has high kinetic energy
The with smaller mass
how
Faith
Since you said they have the same momentum.. So meaning that there is more like an inverse proportionality in the quantities used to find the momentum. We are told that the the is a larger mass and a smaller mass., so we can conclude that the smaller mass had higher velocity as compared to other one
Mathamaticaly correct
Mathmaticaly correct :)
I have proven it by using my own values
Larger mass=4g Smaller mass=2g Momentum of both=8 Meaning V for L =2 and V for S=4 Now find there kinetic energies using the data presented
grateful soul...thanks alot
Faith
Welcome
2 stones are thrown vertically upward from the ground, one with 3 times the initial speed of the other. If the faster stone takes 10 s to return to the ground, how long will it take the slower stone to return? If the slower stone reaches a maximum height of H, how high will the faster stone go
30s
how can i calculate it's height
Julliene
is speed the same as velocity
no
Nebil
in a question i ought to find the momentum but was given just mass and speed
Faith
just multiply mass and speed then you have the magnitude of momentem
Nebil
Yes
Consider speed to be velocity
it worked our . . thanks
Faith
Distinguish between semi conductor and extrinsic conductors
Suppose that a grandfather clock is running slowly; that is, the time it takes to complete each cycle is longer than it should be. Should you (@) shorten or (b) lengthen the pendulam to make the clock keep attain the preferred time?
I think you shorten am not sure
Uche
shorten it, since that is practice able using the simple pendulum as experiment
Silvia
it'll always give the results needed no need to adjust the length, it is always measured by the starting time and ending time by the clock
Paul
it's not in relation to other clocks
Paul
wat is d formular for newton's third principle
Silvia
okay
Silvia
shorten the pendulum string because the difference in length affects the time of oscillation.if short , the time taken will be adjusted.but if long ,the time taken will be twice the previous cycle.
discuss under damped
resistance of thermometer in relation to temperature
how
Bernard
that resistance is not measured yet, it may be probably in the next generation of scientists
Paul
Is fundamental quantities under physical quantities?
please I didn't not understand the concept of the physical therapy
physiotherapy - it's a practice of exercising for healthy living.
Paul
what chapter is this?
Anderson
this is not in this book, it's from other experiences.
Paul
am new in the group
Daniel
Sure
What is Boyce law
Boyles law states that the volume of a fixed amount of gas is inversely proportional to pressure acting on that given gas if the temperature remains constant which is: V<k/p or V=k(1/p)