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Discussion for (e)

Note, coincidentally, that the total power dissipated by the resistors is also 7.20 W, the same as the power put out by the source. That is,

P 1 + P 2 + P 3 = ( 0 . 360 + 2 . 16 + 4 . 68 ) W = 7 . 20 W . size 12{P rSub { size 8{1} } +P rSub { size 8{2} } +P rSub { size 8{3} } = \( 0 "." "360"+2 "." "16"+4 "." "68" \) " W"=7 "." "20"" W"} {}

Power is energy per unit time (watts), and so conservation of energy requires the power output of the source to be equal to the total power dissipated by the resistors.

Major features of resistors in series

  1. Series resistances add: 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} } + "." "." "." "." } {}
  2. The same current flows through each resistor in series.
  3. Individual resistors in series do not get the total source voltage, but divide it.

Resistors in parallel

[link] shows resistors in parallel    , wired to a voltage source. Resistors are in parallel when each resistor is connected directly to the voltage source by connecting wires having negligible resistance. Each resistor thus has the full voltage of the source applied to it.

Each resistor draws the same current it would if it alone were connected to the voltage source (provided the voltage source is not overloaded). For example, an automobile’s headlights, radio, and so on, are wired in parallel, so that they utilize the full voltage of the source and can operate completely independently. The same is true in your house, or any building. (See [link] (b).)

Part a shows two electrical circuits which are compared. The first electrical circuit is arranged with resistors in parallel. The circuit has three paths, with a voltage source V at one end. Just after the voltage source, the circuit has current I. The first path has resistor R sub one and current I sub one after the resistor. The second path has resistor R sub two and current I sub two after the resistor. The third path has resistor R sub three with current I sub three after the resistor. The first circuit is equivalent to the second circuit. The second circuit has a voltage source V and an equivalent parallel resistance R sub p. Part b shows a complicated electrical wiring diagram of a distribution board that supplies electricity to a house.
(a) Three resistors connected in parallel to a battery and the equivalent single or parallel resistance. (b) Electrical power setup in a house. (credit: Dmitry G, Wikimedia Commons)

To find an expression for the equivalent parallel resistance R p size 12{R rSub { size 8{p} } } {} , let us consider the currents that flow and how they are related to resistance. Since each resistor in the circuit has the full voltage, the currents flowing through the individual resistors are I 1 = V R 1 size 12{I rSub { size 8{1} } = { {V} over {R rSub { size 8{1} } } } } {} , I 2 = V R 2 size 12{I rSub { size 8{2} } = { {V} over {R rSub { size 8{2} } } } } {} , and I 3 = V R 3 size 12{I rSub { size 8{3} } = { {V} over {R rSub { size 8{3} } } } } {} . Conservation of charge implies that the total current I size 12{I} {} produced by the source is the sum of these currents:

I = I 1 + I 2 + I 3 . size 12{I=I rSub { size 8{1} } +I rSub { size 8{2} } +I rSub { size 8{3} } } {}

Substituting the expressions for the individual currents gives

I = V R 1 + V R 2 + V R 3 = V 1 R 1 + 1 R 2 + 1 R 3 . size 12{I= { {V} over {R rSub { size 8{1} } } } + { {V} over {R rSub { size 8{2} } } } + { {V} over {R rSub { size 8{3} } } } =V left ( { {1} over {R rSub { size 8{1} } } } + { {1} over {R rSub { size 8{2} } } } + { {1} over {R rSub { size 8{3} } } } right )} {}

Note that Ohm’s law for the equivalent single resistance gives

I = V R p = V 1 R p . size 12{I= { {V} over {R rSub { size 8{p} } } } =V left ( { {1} over {R rSub { size 8{p} } } } right )} {}

The terms inside the parentheses in the last two equations must be equal. Generalizing to any number of resistors, the total resistance R p size 12{R rSub { size 8{p} } } {} of a parallel connection is related to the individual resistances by

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} } } } + "." "." "." "." } {}

This relationship results in a total resistance R p size 12{R rSub { size 8{p} } } {} that is less than the smallest of the individual resistances. (This is seen in the next example.) When resistors are connected in parallel, more current flows from the source than would flow for any of them individually, and so the total resistance is lower.

Calculating resistance, current, power dissipation, and power output: analysis of a parallel circuit

Let the voltage output of the battery and resistances in the parallel connection in [link] be the same as the previously considered series connection: V = 12 . 0 V size 12{V="12" "." 0" V"} {} , R 1 = 1 . 00 Ω size 12{R rSub { size 8{1} } =1 "." "00" %OMEGA } {} , R 2 = 6 . 00 Ω size 12{R rSub { size 8{2} } =6 "." "00" %OMEGA } {} , and R 3 = 13 . 0 Ω size 12{R rSub { size 8{3} } ="13" "." 0 %OMEGA } {} . (a) What is the total resistance? (b) Find the total current. (c) Calculate the currents in each resistor, and show these add to equal the total current 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.

Questions & Answers

advantages of CRO over ordinary voltmeter
Dismas Reply
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Daniel Reply
what is equilibrium
Sade Reply
If a system is said to be under equilibrium whenever there is no force act upon it... And it remain in its initial stage..
soniya
What is conductivity
Saud Reply
It is the ease with which electrical charges or heat can be transmitted through a material or a solution.
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how to find magnitude and direction
Arjune Reply
how to caclculate for speed
Arjune
derivation of ohms law
Kazeem Reply
derivation of resistance
Kazeem
R=v/I where R=resistor, v=voltage, I=current
Kazeem
magnitude
Arjune
A puck is moving on an air hockey table. Relative to an x, y coordinate system at time t 0 s, the x components of the puck’s ini￾tial velocity and acceleration are v0x 1.0 m/s and ax 2.0 m/s2 . The y components of the puck’s initial velocity and acceleration are v0y 2.0 m/s and ay 2.0
Arjune
Electric current is the flow of electrons
Kelly Reply
is there really flow of electrons exist?
babar
Yes It exists
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babar
if electron flows from where first come and end the first one
babar
an electron will flow accross a conductor because or when it posseses kinectic energy
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ghulam
free electrons of conductor
ankita
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Serah Reply
electric means the flow of heat current in a circuit.
Serah
What is electric
Manasseh Reply
electric means?
ghulam
electric means the flow of heat current in a circuit.
Serah
electric means the flow of electric current through conductor
Sade
the continuos flow of electrons in a circuit is called electric
ANUBHA
electric means charge
ghulam
electric means current
Sade
a boy cycles continuously through a distance of 1.0km in 5minutes. calculate his average speed in ms-1(meter per second). how do I solve this
Jenny Reply
speed = distance/time be sure to convert the km to m and minutes to seconds check my utube video "mathwithmrv speed"
PhysicswithMrV
d=1.0km÷1000=0.001 t=5×60=300s s=d\t s=0.001/300=0.0000033m\s
Serah
A puck is moving on an air hockey table. Relative to an x, y coordinate system at time t 0 s, the x components of the puck’s ini￾tial velocity and acceleration are v0x 1.0 m/s and ax 2.0 m/s2 . The y components of the puck’s initial velocity and acceleration are v0y 2.0 m/s and ay 2.0
Arjune
D=1km=1000m t=5mins×60secs=300sec s=d/t=3.333m/s
Daniel
I think Daniel Glorious is ryt
Amalia
why we cannot use DC instead of AC in a transformer
kusshaf Reply
becuse the d .c cannot travel for long distance trnsmission
ghulam
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Chiwetalu Reply
branch of science which deals with matter energy and their relationship between them
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Life science
the
what is heat and temperature
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how does sound affect temperature
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sound is directly proportional to the temperature.
juny
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Wisdom Reply
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Emma
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Talhatu
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Chuks
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mahmoud
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babatunde
scaler quantity are quanties that have only direction and no magnitude
Natsu
ice Point
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scalar quantity are quantities that have magnitude but no direction
Ehigiamusoe
Practice Key Terms 9

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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