Check out this short animation of simplifying a circuit.

https://www.youtube.com/watch?v=5IiFmQQuiO8

Pre-lecture Study Resources

Watch the pre-lecture videos and read through the OpenStax text before doing the pre-lecture homework or attending class.

BoxSand Videos

Learning Objectives

Summary

Summary

Atomistic Goals

Students will be able to...

  1.  

BoxSand Introduction

Subject  |  Lecture


We have previously learned how to analyze circuits using Kirchhoff's laws and Ohm's law. The method is by far the most general and will always work when trying to analyze a circuit. However, there are many different methods of analyzing circuits. In this section we will look at one such alternative method.

Some circuits have their elements connected in such a way that it is easily possible to reduce the circuit into a representative circuit consisting of only a battery and a resistor connected to each other. The ability to reduce a complicated circuit into a simpler one is often an easier way to analyze the circuit. In order to reduce a complex circuit into a simpler one, we need to first identify the two simplest ways of connecting elements and how each way can be reduced. Two of the easiest resistor combinations to reduce are known as series and parallel.

 

Resistors in series

When two or more elements are connected in such a way that no junctions are connected between them, then we refer to this arrangement as a series connection. The elements can be resistors, capacitors, inductors, lightbulbs etc. In this section we will only look at how resistors (and ideal lightbulbs) in series can be simplified.

Below is a circuit diagram with two resistors in series. Using Kirchhoff's law's and Ohm's law we can create an equation, from which emerges a pattern for resistors in series. The pattern is: resistors in series can be replaced by a single resister of equivalent resistance equal to the sum of the individual resistance of each resistor that is in the series arrangement.

This is an image of a circuit loop with a battery and two resistors labeled r one and r two which are all in a sequence. There are three areas of the loop that will have different voltages, the area after the battery and before the first resistor, the area between the two resistors, and the area after the second resistor and to the battery. We can simplify this down to a battery looped to a single resistor that as a magnitude of resistor one and resistor two so that there are only two changes in voltages, one before and after the resistor. The summation of the changes in voltage is equal to zero so that in the first scenario, there were three terms where the voltage of the battery minus the current multiplied by the first resistor minus the current multiplied by the second resistor which is equal to zero. In the second scenario with only one resistor, the equation can be simplified to the voltage of the battery minus the current multiplied by the sum of the resistors which is equal to zero. In general, the total resistance is equal to the summation of all of the resistors present.

Note that as you add more resistors in parallel the total resistance increases, and with Ohm's law, this means that the current decreases.

 

Resistors in parallel

When two or more elements are connected in such a way that the change in voltage across each element is the same, then we refer to this arrangement as a parallel connection. The elements can be resistors, capacitors, inductors, lightbulbs etc. In this section we will only look at how resistors (and ideal lightbulbs) in parallel can be simplified.

Below is a circuit diagram with two resistors in parallel. Using Kirchhoff's law's and Ohm's law we can create an equation, from which emerges a pattern for resistors in parallel. The pattern is: resistors in parallel can be replaced by a single resister of equivalent resistance equal to one over the sum of the inverse of the individual resistance of each resistor that in in the parallel arrangement.

This is an image of a circuit loop with a battery and resistors in a parallel formation. The current flows from the battery to a junction where one leads to resistor one and a second one continues to resistor two and both of them eventually merge back after the resistor to end on the other end of the battery. This can be simplified down to a battery and a single resistor in a single loop where the total resistance is equal to one over the sum of one divided by resistor one plus one divided by resistor two. The current changes when the wire splits to resistor one and two but the total current is equal to current one plus current two. The voltage of the battery is equal to the inverse of the sum of one over resistor one plus one over resistor two multiplied by the current. In general, the total resistance is equal to the inverse of the sum of one over the resistor.

Note that as you add more resistors in parallel the total resistance decreases, and with Ohm's law, this means that the current increases.

The general idea for analyzing circuits with a series and/or parallel resistors is to reduce the circut to the most basic form of just one battery and one equivalent resistance.  Along the way you might have to do multiple steps, for example, you may need to reduce a few reisitors in series first, then reduce more resisitors in parallel until you reach one equivalent resistance.  The trick is to always draw the reduced circuit at each step.  Once you reach the most reduced form of the circuit, use a combination of Ohm's law and Kirchhoff's laws as you work your way backwards through each previous reduced circuit. 

 

Key Equations and Infographics

A representation with the words equivalent resistance: series circuits on the top. There is an equation that shows that the equivalent resistance of a collection of resistors in series is equal to the summation of the individual resistance values. This is written in words below.

 

A representation with the words equivalent resistance: parallel circuits on the top. There is an equation that shows that the inverse of the equivalent resistance of a collection of resistors in parallel is equal to the summation of the inverse of the individual resistors values. This is written in words below.

Now, take a look at the pre-lecture reading and videos below.

OpenStax Reading


OpenStax Section #.# | Title -- From Fundamentals

 

Openstax's section on Resistors in series and parallel.

Openstax College Textbook Icon

 

Equations, definitions, and notation icon Concept Map Icon
Key Terms Icon Student Contributed Content Icon

Additional Study Resources

Use the supplemental resources below to support your post-lecture study.

YouTube Videos

 

Doc Schuster covers Equivalent resistances of resistors and how to analyze parallel and series resistor circuits.

https://www.youtube.com/watch?v=C9P3Jgr779I

https://www.youtube.com/watch?v=CZgqGTxL9cA

Pre-Med Academy's series on the electric current covers Resistors in parallel and in series.

https://www.youtube.com/watch?v=CyyUhPNOjTA

Here are a couple of videos from Step-by-Step Science where they calculate voltage resistance and current for resistors in series, then in parallel.

 

https://www.youtube.com/watch?v=W1UH0jzWjtI

 

https://www.youtube.com/watch?v=WT6wbh39MX4

 

Other Resources

This link will take you to the repository of other content related resources .

Content Icon

Simulations


 

This PhET is their Circuit Construction Kit. See if you can use what you know abour resistors in parallel in series to make predictions on how the circuit will behave.

Phet Interactive Simulations Icon

For additional simulations on this subject, visit the simulations repository.

Simulation Icon

Demos


For additional demos involving this subject, visit the demo repository

Demos Icon

History


Oh no, we haven't been able to write up a history overview for this topic. If you'd like to contribute, contact the director of BoxSand, KC Walsh (walshke@oregonstate.edu).

Physics Fun

 

Other Resources


Resource Repository

Boston University's page on series and aprallel resistance

 

B.U. Physics Icon

 

PPLATO is a complete resource with a lot of information, and several practice questions per subject. This webpage coveres electric current and Circuits, which includes resistors in series and parallel. Use the sidebar of the website for quicker navigation.

 

Other Content Icon

The Hyperphysics reference covers resistance, and resistor combinations.

 

Hyper Physics Icon

 

Other Resources

This link will take you to the repository of other content related resources .

Content Icon

The Physics Classroom has a section on Series, Parallel, and Combination Circuits.

Series Circuits Parallel Circuits Combination Circuits
The Physics Classroom Icon The Physics Classroom Icon The Physics Classroom Icon

Problem Solving Guide

Use the Tips and Tricks below to support your post-lecture study.

Assumptions

 

Checklist

 

Misconceptions & Mistakes

 

Pro Tips

 

Multiple Representations

Multiple Representations is the concept that a physical phenomena can be expressed in different ways.

Physical

Physical Representations describes the physical phenomena of the situation in a visual way.

 

Mathematical

Mathematical Representation uses equation(s) to describe and analyze the situation.

A representation with the words equivalent resistance: series circuits on the top. There is an equation that shows that the equivalent resistance of a collection of resistors in series is equal to the summation of the individual resistance values. This is written in words below.

 

A representation with the words equivalent resistance: parallel circuits on the top. There is an equation that shows that the inverse of the equivalent resistance of a collection of resistors in parallel is equal to the summation of the inverse of the individual resistors values. This is written in words below.

Graphical

Graphical Representation describes the situation through use of plots and graphs.

 

Descriptive

Descriptive Representation describes the physical phenomena with words and annotations.

 

Experimental

Experimental Representation examines a physical phenomena through observations and data measurement.

 

Practice

Use the practice problem sets below to strengthen your knowledge of this topic.

Fundamental examples

 

(1) Consider the simple circuit diagram below. In terms of the variables given on the diagram, what is the current at point A?

This is an image of a battery and two light bulbs in a series circuit. The current moves from the positive end of the battery to the first light bulb, resistor one, and then the second light bulb, resistor two, and then to the a point labeled A and to the negative end of the battery.

(2) Consider the simple circuit diagram below. In terms of the variables given in the diagram, what is the current flowing through the battery?

This is an image of a battery and two light bulbs in a parallel circuit. The current moves from the positive end of the battery and splits at a junction where one moves down to the first light bulb, resistor one, while the other moves on and then moves down to the second light bulb, resistor two. After the resistor, the current merge back together and end at the negative part of the battery.

(3) Consider the same circuit as before but with one additional resistor. If $R_1 =  6 Ω$, $R_2 = 4 Ω$, and $R_3 = 5 Ω$, what is the new current flowing through the battery?

This is an image of a battery that has one point five volts and three light bulbs in a parallel circuit. The current moves from the positive end of the battery and splits at a junction where one moves down to the first light bulb, resistor one, while the other moves on and splits again at another junction where one moves down to the second light bulb, resistor two, while the other moves on and then down to the third light bulb, resistor three. Then all three wires eventually merge to end at the negative part of the battery.

(4) Consider the circuit below. The values of the resistors are $R_1 =  6 Ω$, $R_2 = 4 Ω$, $R_3 = 5 Ω$, and $R_4 = 7 Ω$, and $ε = 1.5 V$.

This is an image of a battery that has three light bulbs in a series and a fourth in a parallel circuit. The current moves from the positive end of the battery to the first light bulb, resistor one, then to a second light bulb, resistor two. Then the current splits at a junction where one moves down to the third light bulb, resistor three, and the other continues then moves down to the fourth light bulb, resistor four. The current eventually merge together to end at the negative part of the battery.

 

(a) What is the equivalent resistance of $R_3$ and $R_4$: $R_{eq34}$?

(b) What is the equivalent resistance of $R_1$ and $R_2$, $R_{eq12}$?

(c) What is the equivalent resistance of the circuit, $R_{eq1234}$?

(d) What is the $\Delta V_{34}$?

Solutions: 1 2 3 4a-c 4d

While we continue to work on making custom content check out these two guided exercises from the University of Wisconsen-Green Bay. 

  • University of Wisconsin-Green Bay

Short foundation building questions, often used as clicker questions, can be found in the clicker questions repository for this subject.

Clicker Questions Icon

Practice Problems

BoxSand practice problems

BoxSand's quantitative practice problems

Recommended example practice problems 

  • OpenStax, problems at bottom of page,**
  • Physics Hypertextbook, Resistors in Circuits, Website Link


For additional practice problems and worked examples, visit the link below. If you've found example problems that you've used please help us out and submit them to the student contributed content section.

Example Problems Icon