6.3: Formatting Microsoft PowerPoint Slides- Layout and Design Principles
By the end of this section, you will be able to:
- Format the layout of each slide
- Understand best practices in design principles
It’s time to transform the five slides from My Life in a Snapshot by manipulating the layout and adding options. Formatting the layout of each slide in Microsoft PowerPoint is the process of adding, subtracting, and/or adjusting the arrangement of elements such as text, images, and shapes on a slide. You may want to format the layout of a slide in PowerPoint to make it more visually appealing and effective for your audience. PowerPoint includes many options for altering the layout of the slide. The slide layout can be changed by using the tool on the Home tab in the Slides command group . From the Home tab, select Layout tab from the Slides command group. Here, you will see a listing and image of the layout options.
We have used three types of layouts in My Life in a Snapshot . Here are some commonly used slide layouts:
- Title Slide : This layout includes a title and subtitle and is typically used for the first slide of a presentation .
- Title and Content: This layout includes a title, subtitle, and one or two content boxes that you can use for text or media. This layout is typically used to give an overview of the presentation and the main topics to be covered.
- Comparison: This layout includes two content boxes, which can be used to present different types of information, such as text and images, or to compare and contrast two pieces of information.
- Section Header: This layout is used to create a slide that can be used as a header for a section of a presentation. It typically includes a title and subtitle, with a distinctive design.
- Content with Caption: This layout includes a content box and a caption box, which can be used to present a single image or other media and provide additional information about it.
These common PowerPoint slide layouts can help you create a clear and effective presentation structure. You can add, remove, or customize placeholders as you need, as well as use combinations of these layouts to create a unique, personalized presentation. PowerPoint also offers a variety of built-in slide layouts that you can use to create different types of slides.
Formatting Layout
When you design your slide layouts, arranging text boxes and other objects becomes key in making sure they are positioned in an effective manner. In this section, we will review the Alignment Guides option within the View tab and discuss the numerous built-in layout designs that PowerPoint can offer.
Alignment Guides
As stated previously, getting things to look exactly how you want them to appear next to each other is crucial to maximizing the design power of PowerPoint. But it can be difficult to align objects with other objects on a single slide, or objects with text, using only your mouse. Under the View tab, you will find a helpful alignment tool that you can access by checking the Guides box. When this box is checked, there will be two dashed lines on the presentation slide, one centered vertically and the other centered horizontally.
When you hover your mouse over one of these lines while holding down the Ctrl key, the cursor turns into a double line with arrows. Drag the line to where you want one guideline to be and let go of the mouse. When you do this, another line is created.
Mac Tip
Hold down the Option key, not the Ctrl key, to turn your cursor into a double line with arrows.
You can continue to add guidelines anywhere on your slide to insert and align objects, text boxes, photos, and so forth. (In the section on Adding Visuals and Features to Microsoft PowerPoint Slides, we will cover inserting objects and images.) See Figure 6.23 for a visual example of what the guides look like after adding them to a slide. (If you see that the Guides box is checked but no lines are apparent, just uncheck it and check it again. That will usually bring the guides back into view.)
Link to Learning
Not every presentation has to be delivered by a person. There are many reasons why creating a self-running presentation is valuable. Many companies will design a presentation that can be left unattended in a booth or kiosk, at a trade show or convention, or saved as a video and sent to a client list. A self-running presentation can also help address time-related constraints. Read Microsoft’s steps for creating a self-running slideshow in PowerPoint to learn more.
Design Principles
In this section, you will learn about some basic design principles that are best practices for designing your own slides or choosing a theme for your presentation. You’ll find out how to use proximity, alignment, repetition, contrast, and white space to make your design elements stand out.
Proximity
In photography, proximity refers to nearness —the distance between the camera and the subject being photographed. In the context of PowerPoint , it refers to the distance between the audience and the subject matter being presented. You can control the relative proximity within a PowerPoint slide. In photography, proximity can affect the composition of the photograph by changing the relative sizes of the elements in the frame. For example, if the camera is positioned close to a small subject, the subject may appear larger in the frame; if the camera is positioned farther away, the subject may appear smaller. The proximity of the camera to the subject can also influence the overall look of the photograph. A photograph taken from a close distance may have a more intimate or detailed appearance, while one taken from farther away may have a more distant or expansive look.
Proximity is an important consideration in designing PowerPoint layouts because it can affect the composition, perspective, and overall look of each slide. In Figure 6.24, you can see two different sizes of the budget sheet. The first one is effective for an overall view of what the document looks like. The second one is more effective if you want the audience to be able to read it. If so, it’s preferable to zoom in as close as possible to that content.
Alignment
Aligning objects or text on a page adds organization and creates a sense of cohesion, making your content in general more usable. When alignment exists on a slide, the human eye knows where to focus, and the slide is more comfortable to view. In PowerPoint, alignment is the way that text, images, and other elements are positioned on a slide. Proper alignment is important because it helps to create a cohesive, professional-looking presentation. When elements on a slide are aligned, they are more visually balanced, which can make the slide look more organized and appealing to the audience. Properly aligned elements can help guide the viewer’s eye and create a natural flow from one element to the next, making the presentation easier to follow and understand. In addition to the Guides checkbox that we reviewed in the section on Formatting Layout , there are also checkboxes for Rulers and Gridlines. Ticking these boxes will show additional lines on the slide that will help you align your slide elements.
Repetition
Repetition is the use of similar or identical elements, such as colors, fonts, or design elements, across multiple slides in a presentation. In a slideshow, repetition—especially when similar elements are repeated across multiple slides—can make the presentation feel more polished and professional and make it easy for the audience to follow and understand. Repetition also promotes a consistent look and feel for the presentation. Repetition of important elements such as headings or key points can establish a visual hierarchy that guides the viewer’s eye and makes your presentation easier to follow.
Repetition of visual elements is a good way of reinforcing the key points you want to establish with the audience because they know where to look. In this way, repetition makes the main message of your presentation more memorable and connected for the audience.
Contrast
In presentations, contrast refers to the use of different elements, such as colors, fonts, and other design elements, to focus attention and create visual interest. You may want to use contrasting colors, such as complementary colors or light and dark shades, or contrasting fonts, such as a bold or decorative font for headings and a simple font for body text. Using contrast helps create a hierarchy and makes your presentation easier to follow.
Using contrasting design elements, such as different shapes or patterns, can help to add visual interest and break up the slide into distinct sections. Overall, contrast is a useful tool in presentations because it can help to draw attention, create visual interest, and make the presentation more effective and engaging for the audience. Notice how in the new title slide of My Life in a Snapshot (Figure 6.21), the title is in large font, the subtitle is in small font, and the colors used are off-white, red, and black. The different font sizes and colors contrast with one another and create an engaging, yet professional, appearance.
White Space
The last design element to consider within this section is white space . White space, also known as negative space, is the unoccupied areas of a slide that are not filled with text or other content. By leaving enough white space around text and other elements, you can make the content easier to read and understand. White space can be used to create visual interest by creating balance and separating different elements on the slide. By surrounding a key point or element with white space, you can draw attention to it and make it stand out. Additionally, using white space consistently throughout a presentation can help to create a cohesive look and feel. It is an important element of slide design and can be used in a variety of ways to enhance the readability, visual appeal, and effectiveness of a presentation. Filling your slides with text or images will make them look too busy and hard for your audience to read. Using the Designer tool to suggest different layouts can help add white space and sustain interest throughout the presentation with aesthetically pleasing slides.
Another principle that underlies all the design principles reviewed in this section is known as the rule of thirds . This is a basic principle of photography and design that suggests that an image can be divided into nine equal parts by two equally spaced horizontal lines and two equally spaced vertical lines. It is essentially a tic-tac-toe game board!
The rule of thirds theory suggests that if you place the important elements of the image along these lines, or at their intersections, your photo or design will be more balanced and will have more visual interest. By placing the main subject of your photo or design along one of the lines or at an intersection, you can create a sense of tension and dynamism that draws the viewer’s eye into the image. Additionally, using the rule of thirds can help you avoid placing the subject of your image dead center every time, which can make for a static and uninteresting composition.
Although the rule of thirds is not a hard-and-fast rule, it is a useful guideline that can help you create more visually appealing and dynamic compositions in your slide creations and layouts. Figure 6.25 provides an example of a grid created according to the rule of thirds.
There are other composition models you can use, as well. The point is that in design, composition is the basis of it all. You want a well-composed layout and placement of text and images, aligned so that the eye moves easily about the slide.


4: Apportionment
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- 4.1: Introduction
- Apportionment is the problem of dividing up a fixed number of things among groups of different sizes. In politics, this takes the form of allocating a limited number of representatives amongst voters. This problem, presumably, is older than the United States, but the best-known ways to solve it have their origins in the problem of assigning each state an appropriate number of representatives in the new Congress when the country was formed.
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- 4.2: Hamilton’s Method
- Alexander Hamilton proposed the method that now bears his name. His method was approved by Congress in 1791, but was vetoed by President Washington. It was later adopted in 1852 and used through 1911. He begins by determining, to several decimal places, how many things each group should get. Since he was interested in the question of Congressional representation, we’ll use the language of states and representatives, so he determines how many representatives each state should get.
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- 4.3: Jefferson’s Method
- After Washington vetoed Hamilton’s method, Jefferson’s method was adopted, and used in Congress from 1791 through 1842. Jefferson, of course, had political reasons for wanting his method to be used rather than Hamilton’s. Primarily, his method favors larger states, and his own home state of Virginia was the largest in the country at the time. He would also argue that it’s the ratio of people to representatives that is the critical thing, and apportionment methods should be based on that.
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- 4.4: Webster’s Method
- Daniel Webster (1782-1852) proposed a method similar to Jefferson’s in 1832. It was adopted by Congress in 1842, but replaced by Hamilton’s method in 1852. It was then adopted again in 1901. The difference is that Webster rounds the quotas to the nearest whole number rather than dropping the decimal parts. If that doesn’t produce the desired results at the beginning, he says, like Jefferson, to adjust the divisor until it does.
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- 4.5: Huntington-Hill Method
- In 1920, no new apportionment was done, because Congress couldn’t agree on the method to be used. They appointed a committee of mathematicians to investigate, and they recommended the Huntington-Hill Method. They continued to use Webster’s method in 1931, but after a second report recommending Huntington-Hill, it was adopted in 1941 and is the current method of apportionment used in Congress.
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- 4.6: Lowndes’ Method
- William Lowndes (1782-1822) was a Congressman from South Carolina (a small state) who proposed a method of apportionment that was more favorable to smaller states. Unlike the methods of Hamilton, Jefferson, and Webster, Lowndes’s method has never been used to apportion Congress.
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- 4.7: Apportionment of Legislative Districts
- In most states, there are a fixed number of representatives to the state legislature. Rather than apportioning each county a number of representatives, legislative districts are drawn so that each legislator represents a district. The apportionment process, then, comes in the drawing of the legislative districts, with the goal of having each district include approximately the same number of constituents.
Thumbnail: Breakdown of US House of Representatives as of January 1, 2022. Blue circles are Democratic Representatives, red are Republican Representatives and white are vacant (Public Domain; AlSmith28 via Wikipedia )


10.5: Estimation by Clustering
- understand the concept of clustering
- be able to estimate the result of adding more than two numbers when clustering occurs using the clustering technique
Cluster
When more than two numbers are to be added, the sum may be estimated using the clustering technique. The rounding technique could also be used, but if several of the numbers are seen to
cluster
(are seen to be close to) one particular number, the clustering technique provides a quicker estimate. Consider a sum such as
\(32 + 68 + 29 + 73\)
Notice two things:
There are more than two numbers to be added.
Clustering occurs.
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Both 68 and 73 cluster around 70, so \(68 + 73\) is close to \(80 + 70 = 2(70) = 140\).
- Both 32 and 29 cluster around 30, so \(32 + 29\) is close to \(30 + 30 = 2(30) = 60\).
The sum may be estimated by
\(\begin{array} {rcl} {(2 \cdot 30) + (2 \cdot 70)} & = & {60 + 140} \\ {} & = & {200} \end{array}\)
In fact, \(32 + 68 + 29 + 73 = 202\).
Estimate each sum. Results may vary.
\(27 + 48 + 31 + 52\).
Solution
27 and 31 cluster near 30. Their sum is about \(2 \cdot 30 = 60\).
48 and 52 cluster near 50. Their sum is about \(2 \cdot 50 = 100\).
Thus, \(27 + 48 + 31 + 52\) is about \(\begin{array} {rcl} {(2 \cdot 30) + (2 \cdot 50)} & = & {60 + 100} \\ {} & = & {160} \end{array}\)
In fact, \(27 + 48 + 31 + 52 = 158.\).
\(88 + 21 + 19 + 91\).
Solution
88 and 91 cluster near 90. Their sum is about \(2 \cdot 90 = 180\).
21 and 19 cluster near 20. Their sum is about \(2 \cdot 20 = 40\).
Thus, \(88 + 21 + 19 + 91\) is about \(\begin{array} {rcl} {(2 \cdot 90) + (2 \cdot 20)} & = & {180 + 40} \\ {} & = & {220} \end{array}\)
In fact, \(88 + 21 + 19 + 91 = 219\).
\(17 + 21 + 48 + 18\).
Solution
17, 21, and 18 cluster near 20. Their sum is about \(3 \cdot 20 = 60\).
48 is about 50.
Thus, \(17 + 21 + 48 + 18\) is about \(\begin{array} {rcl} {(3 \cdot 20) + 50} & = & {60 + 50} \\ {} & = & {110} \end{array}\)
In fact, \(17 + 21 + 48 + 18 = 104.\).
\(61 + 48 + 39 + 57 + 52\).
Solution
61 and 57 cluster near 60. Their sum is about \(2 \cdot 60 = 120\).
48, 49, and 52 cluster near 50. Their sum is about \(3 \cdot 50 = 150\).
Thus, \(61 + 48 + 39 + 57 + 52\) is about \(\begin{array} {rcl} {(2 \cdot 60) + (3 \cdot 50)} & = & {120 + 150} \\ {} & = & {270} \end{array}\)
In fact, \(61 + 48 + 39 + 57 + 52 = 267.\).
\(706 + 321 + 293 + 684\).
Solution
706 and 684 cluster near 700. Their sum is about \(2 \cdot 700 = 1,400\).
321 and 293 cluster near 300. Their sum is about \(2 \cdot 300 = 600\).
Thus, \(706 + 321 + 293 + 684\) is about \(\begin{array} {rcl} {(2 \cdot 700) + (2 \cdot 300)} & = & {1,400 + 600} \\ {} & = & {2,000} \end{array}\)
In fact, \(706 + 321 + 293 + 684 = 2,004.\).
Practice Set A
Use the clustering method to estimate each sum.
\(28 + 51 + 31 + 47\)
- Answer
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\((2 \cdot 30) + (2 \cdot 50) = 60 + 100 = 160\)
Practice Set A
\(42 + 39 + 68 + 41\)
- Answer
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\((3 \cdot 40) + 70 = 120 + 70 = 190\)
Practice Set A
\(37 + 39 + 83 + 42 + 79\)
- Answer
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\((3 \cdot 40) + (2 \cdot 80) = 120 + 160 = 280\)
Practice Set A
\(612 + 585 + 830 + 794\)
- Answer
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\((2 \cdot 600) + (2 \cdot 800) = 1,200 + 1,600 = 2,800\)
Exercises
Use the clustering method to estimate each sum. Results may vary.
Exercise \(\PageIndex{1}\)
\(28 + 51 + 31 + 47\)
- Answer
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\(2(30) + 2(50) = 160 (157)\)
Exercise \(\PageIndex{2}\)
\(42 + 19 + 39 + 23\)
Exercise \(\PageIndex{3}\)
\(88 + 62 + 59 + 90\)
- Answer
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\(2(90) + 2(60) = 300 (299)\)
Exercise \(\PageIndex{4}\)
\(76 + 29 + 33 + 82\)
Exercise \(\PageIndex{5}\)
\(19 + 23 + 87 + 21\)
- Answer
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\(3(20) + 90 = 150 (150)\)
Exercise \(\PageIndex{6}\)
\(41 + 28 + 42 + 37\)
Exercise \(\PageIndex{7}\)
\(89 + 32 + 89 + 93\)
- Answer
-
\(3(90) + 30 = 300 (303)\)
Exercise \(\PageIndex{8}\)
\(73 + 72 + 27 + 71\)
Exercise \(\PageIndex{9}\)
\(43 + 62 + 61 + 55\)
- Answer
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\(40 + 3(60) = 220 (221)\)
Exercise \(\PageIndex{10}\)
\(31 + 77 + 31 + 27\)
Exercise \(\PageIndex{11}\)
\(57 + 34 + 28 + 61 + 62\)
- Answer
-
\(3(60) + 2(30) = 240 (242)\)
Exercise \(\PageIndex{12}\)
\(94 + 18 + 23 + 91 + 19\)
Exercise \(\PageIndex{13}\)
\(103 + 72 + 66 + 97 + 99\)
- Answer
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\(3(100) + 2(70) = 440 (437)\)
Exercise \(\PageIndex{14}\)
\(42 + 121 + 119 + 124 + 41\)
Exercise \(\PageIndex{15}\)
\(19 + 24 + 87 + 23 + 91 + 93\)
- Answer
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\(3(20) + 3(90) = 330 (337)\)
Exercise \(\PageIndex{16}\)
\(108 + 61 + 63 + 96 + 57 + 99\)
Exercise \(\PageIndex{17}\)
\(518 + 721 + 493 + 689\)
- Answer
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\(2(500) + 2(700) = 2,400 (2,421)\)
Exercise \(\PageIndex{18}\)
\(981 + 1208 + 1214 + 1006\)
Exercise \(\PageIndex{19}\)
\(23 + 81 + 77 + 79 + 19 + 81\)
- Answer
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\(2(20) + 4(80) = 360 (360)\)
Exercise \(\PageIndex{20}\)
\(94 + 68 + 66 + 101 + 106 + 71 + 110\)
Exercises for Review
Exercise \(\PageIndex{21}\)
Specify all the digits greater than 6.
- Answer
-
7, 8, 9
Exercise \(\PageIndex{22}\)
Find the product: \(\dfrac{2}{3} \cdot \dfrac{9}{14} \cdot \dfrac{7}{12}\).
Exercise \(\PageIndex{23}\)
Convert 0.06 to a fraction.
- Answer
-
\(\dfrac{3}{50}\)
Exercise \(\PageIndex{24}\)
Write the proportion in fractional form: "5 is to 8 as 25 is to 40."
Exercise \(\PageIndex{25}\)
Estimate the sum using the method of rounding: \(4,882 + 2,704\).
- Answer
-
\(4,900 + 2,700 = 7,600 (7,586)\)


4.1: Mechanical Cooling – General Applications
Click play on the following audio player to listen along as you read this section.
Mechanical cooling has a wide range of applications; it is used in automobiles, rooftop HVAC units, coolers, refrigerators, freezers and even in the chiller units that produce the cold water for hydronic cooling coils. Mechanical cooling is utilized for the large scale cooling of entire buildings, or it can be used to keep the inside of the fridge cold and allow for the long term storage and freezing of food.
Air conditioning units use the basic Direct Expansion (DX) cooling cycle in a process that removes heat from one area where it is not desired, and transfers that heat to an area where it is quickly dissipated. The air conditioner itself does not create heat, nor does it destroy heat—it simply transfers heat from a room into a refrigerant, and then from inside the building to outside.
Hydronic chiller systems may use a DX system to bring large quantities of water below room temperature, and then pump this cooled water throughout the building to lower the average temperature. Often used in association with rooftop water coolers to rapidly release thermal energy collected in the building.
Heat pumps are a type of DX cooling system that allows for the refrigerant to flow in both directions through the system. By reversing the direction of the flow of refrigerant, a heat pump can go from cooling a house in the summer, to warming that house in the winter, all while using the same equipment. This can simplify the installation and allow a single system, rather than separate heating and cooling components, to regulate the temperature of the building.


5.7: Small Business, Large Impact
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- Last updated
- Save as PDF
What are the advantages and disadvantages facing owners of small businesses?
An uncertain economy has not stopped people from starting new companies. The National Federation of Independent Businessesreports that 85 percent of Americans view small businesses as a positive influence on American life. This is not surprising when you consider the many reasons why small businesses continue to thrive in the United States:
- Independence and a better lifestyle: Large corporations no longer represent job security or offer the fast-track career opportunities they once did. Mid-career employees leave the corporate world—either voluntarily or as a result of downsizing—in search of the new opportunities that self-employment provides. Many new college and business school graduates shun the corporate world altogether to start their own companies or look for work in smaller firms.
- Personal satisfaction from work: Many small-business owners cite this as one of the primary reasons for starting their companies. They love what they do.
- Best route to success: Business ownership provides greater advancement opportunities for women and minorities, as we will discuss later in this chapter. It also offers small-business owners the potential for profit.
- Rapidly changing technology: Technology advances and decreased costs provide individuals and small companies with the power to compete in industries that were formerly closed to them.
- Major corporate restructuring and downsizing: These force many employees to look for other jobs or careers. They may also provide the opportunity to buy a business unit that a company no longer wants.
- Outsourcing: As a result of downsizing, corporations may contract with outside firms for services they used to provide in-house. Outsourcing creates opportunities for smaller companies that offer these specialized goods and services.
- Small businesses are resilient: They are able to respond fairly quickly to changing economic conditions by refocusing their operations.
There are several cities and regions that are regarded as the best locations for start-up businesses and entrepreneurs. Among them are Tulsa, Oklahoma; Tampa, Florida; Atlanta, Georgia; Raleigh, North Carolina; Oklahoma City, Oklahoma; Seattle, Washington; Minneapolis, Minnesota; and Austin, Texas. 19
Why Stay Small?
Owners of small businesses recognize that being small offers special advantages. Greater flexibility and an uncomplicated company structure allow small businesses to react more quickly to changing market forces. Innovative product ideas can be developed and brought to market more quickly, using fewer financial resources and personnel than would be needed in a larger company. And operating more efficiently keeps costs down as well. Small companies can also serve specialized markets that may not be cost-effective for large companies. Another feature is the opportunity to provide a higher level of personal service. Such attention brings many customers back to small businesses such as gourmet restaurants, health clubs, spas, fashion boutiques, and travel agencies.
Steve Niewulis played in baseball’s minor leagues before an injury to his rotator cuff cut short his career. Niewulis decided to combine his love of the game with a clever idea that has elevated him to the big leagues. The fact that players had trouble keeping their hands dry while batting inspired his big idea: a sweat-busting rosin bag attached to a wristband so that a player can dry the bat handle between pitches. In less than two years, Niewulis’s Fort Lauderdale, Florida, company, Tap It! Inc., sold thousands of Just Tap It! wristbands. The product, which retails for $12.95, is used by baseball players, basketball players, tennis players, golfers, and even rock climbers. His secret to success? Find a small distribution network that allows small companies, with just one product line, to succeed. 20
On the other hand, being small is not always an asset. The founders may have limited managerial skills or encounter difficulties obtaining adequate financing, potential obstacles to growing a company. Complying with federal regulations is also more expensive for small firms. Those with fewer than 20 employees spend about twice as much per employee on compliance than do larger firms. In addition, starting and managing a small business requires a major commitment by the owner. Long hours, the need for owners to do much of the work themselves, and the stress of being personally responsible for the success of the business can take a toll.
But managing your company’s growing pains doesn’t need to be a one-person job. Four years after he started DrinkWorks (now Whirley DrinkWorks), a company that makes custom drinking cups, Richard Humphrey was logging 100-hour weeks. “I was concerned that if I wasn’t there every minute, the company would fall apart.” Humphrey got sick, lost weight, and had his engagement fall apart. When forced by a family emergency to leave the company in the hands of his five employees, Humphrey was amazed at how well they managed in his absence. “They stepped up to the plate and it worked out,” he says. “After that the whole company balanced out.” 21
CONCEPT CHECK \(\PageIndex{1}\)
- Why are small businesses becoming so popular?
- Discuss the major advantages and disadvantages of small businesses.


11.15: Reading- Quality of Groundwater
As was noted at the very beginning of this chapter, one of the good things about groundwater as a source of water is that it is not as easily contaminated as surface water is. But there are two caveats to that: one is that groundwater can become naturally contaminated because of its very close connection to the materials of its aquifer, and the second is that once contaminated by human activities, groundwater is very difficult to clean up.
Natural Contamination of Groundwater
Groundwater moves slowly through an aquifer, and unlike the surface water of a stream, it has a lot of contact with the surrounding rock or sediment. In most aquifers, the geological materials that make up the aquifer are relatively inert, or are made up of minerals that dissolve very slowly into the groundwater. Over time, however, all groundwater gradually has more and more material dissolved within it as it remains in contact with the aquifer. In some areas, that rock or sediment includes some minerals that could potentially contaminate the water with elements that might make the water less than ideal for human consumption or agricultural use. Examples include copper, arsenic, mercury, fluorine, sodium, and boron. In some cases, contamination may occur because the aquifer material has particularly high levels of the element in question. In other cases, the aquifer material is just normal rock or sediment, but some particular feature of the water or the aquifer allows the contaminant to build up to significant levels.
An example of natural contamination takes place in the bedrock aquifers of the east coast of Vancouver Island and the adjacent Gulf Islands. The aquifer is the Cretaceous (90 Ma to 65 Ma) Nanaimo Group, which is made up of sandstone, mudstone, and conglomerate (Figure 1).
The rocks of the Nanaimo Group are not particularly enriched in any trace elements, but the submarine-fan sandstone that makes up much of the group is a lithic wacke, and therefore has relatively high levels of clay (for a sandstone). This clay is good at adsorbing [1] some elements from the water and desorbing others, and in the process, its pH goes up (it becomes alkaline). At high pH levels (some as high as 9 in the Nanaimo Group), the element fluorine that is present naturally in the rock (as it is in almost any rock) has an increased tendency to dissolve in the water. In some areas, groundwater in the Nanaimo Group has fluorine levels that are well above recommended levels for drinking water. The World Health Organization (WHO) maximum acceptable concentration (MAC) for fluorine is 1.5 mg/L (milligrams per litre). Between 5% and 10% of the domestic wells around Nanaimo and adjacent Gabriola Island have more than that, some as much as 10 mg/L. A small amount of fluorine in the human diet is considered important for maintaining dental health, but high levels can lead to malformation and discolouration of teeth, and long-term exposure can lead to other more serious health effects such as skeletal problems.
Nanaimo Group groundwater can also have elevated levels of boron, again related to pH and adsorption from clay minerals. While boron at the levels found there is not toxic to humans, there is enough boron in some wells to be toxic to plants, and the water cannot be used for irrigation.
Rural residents in the densely populated country of Bangladesh (over 1,000 residents/km2, compared with 3.4/km2 in Canada) used to rely mostly on surface supplies for their drinking water, and many of these were subject to bacterial contamination. Infant mortality rates were among the highest in the world and other illnesses such as diarrhea, dysentery, typhoid, cholera, and hepatitis were common. In the 1970s, international agencies, including UNICEF, started a program of drilling wells to access abundant groundwater supplies at depths of 20 m to 100 m. Eventually over 8 million such wells were drilled. Infant mortality and illness rates dropped dramatically, but it was later discovered that the water from a high proportion of these wells has arsenic above safe levels (Figure 2).
Most of the wells in the affected areas are drilled into relatively recent sediments of the vast delta of the Ganges and Brahmaputra Rivers. While these sediments are not particularly enriched in arsenic, they have enough organic matter in them to use up any oxygen present. This leads to water with a naturally low oxidation potential (anoxic conditions); arsenic is highly soluble under these conditions, and so any arsenic present in the sediments easily gets dissolved into the groundwater. Arsenic poisoning leads to headaches, confusion, and diarrhea, and eventually to vomiting, stomach pain, and convulsions. If not treated, the final outcomes are heart disease, stroke, cancer, diabetes, coma, and death. There are ways to treat arsenic-rich groundwater, but it is a challenge in Bangladesh to implement the simple and effective technology that is available.
Anthropogenic Contamination of Groundwater
Groundwater can become contaminated by pollution at the surface (or at depth), and there are many different anthropogenic (human-caused) sources of contamination.
The vulnerability of aquifers to pollution depends on several factors, including the depth to the water table, the permeability of the material between the surface and the aquifer, the permeability of the aquifer, the slope of the surface, and the amount of precipitation. Confined aquifers tend to be much less vulnerable than unconfined ones, and deeper aquifers are less vulnerable than shallow ones. Steeper slopes mean that surface water tends to run off rather than infiltrate (and this can reduce the possibility of contamination). Contamination risk is also less in dry areas than in areas with heavy rainfall.
Studies of groundwater vulnerability have been completed for various regions of British Columbia. A groundwater vulnerability map for southern Vancouver Island is shown in Figure 3. The yellow to red areas are considered to have high vulnerability to pollution from surface sources, and most of these are where the aquifers are unconfined in quite permeable unconsolidated sediments of either glacial or fluvial origin, where the water table is relatively shallow and the terrain is relatively flat.
The important sources of anthropogenic groundwater contamination include the following:
- Chemicals and animal waste related to agriculture, and chemicals applied to golf courses and domestic gardens
- Landfills
- Industrial operations
- Mines, quarries, and other rock excavations
- Leaking fuel storage tanks (especially those at gas stations)
- Septic systems
- Runoff from roads (e.g., winter salting) or chemical spills of materials being transported
Agriculture
Intensive agricultural operations and golf courses can have a significant impact on the environment, especially where chemicals and other materials are used to enhance growth or control pests. An example of agricultural contamination is in the Abbotsford area of the Fraser Valley, where nitrate levels above the 44 mg/L maximum acceptable level (expressed as nitrate) in the Abbotsford-Sumas aquifer have been observed since the 1950s; however, the problem became much worse as agriculture intensity increased in the 1980s. By 2004, groundwater with nitrate levels in excess of 44 mg/L was reported over an area of about 75 km2 around Abbotsford, and the problem extended across the border into the Sumas area of Washington State.
This region is intensively used for berry crops (especially raspberries and blueberries) and large poultry operations, as well as lesser amounts of grazing and forage crops. Chicken manure is typically stored in fields adjacent to chicken barns, and may release nitrogen to the environment from runoff water, and from releases of ammonia gas. Over decades, both chemical fertilizers and chicken manure and other manures have been applied to the berry crops to provide extra nitrogen to help maximize berry growth. If the fertilizer added is in excess of what the plants need, or is poorly timed compared to when it is needed, then the extra nitrogen may be leached into the groundwater below. Berry crops are irrigated over the summer to help the crops grow. Summer irrigation and winter rainfall may carry excess nitrate from the near surface to the aquifer below.
Since the 1990s, agricultural practices have been tightened up to reduce the rate of groundwater contamination, but it will take decades for nitrate levels to drop in the Abbotsford-Sumas aquifer. Agriculture and Agri-Food Canada and many others are conducting research on better irrigation and nitrate management techniques to reduce the amount of nitrogen that leaches to groundwater.
Landfills
In the past, domestic and commercial refuse was commonly trucked to a “dump” (typically a hole in the ground), and when the hole was filled, it was covered with soil and forgotten. In situations like this, rain and melting snow can easily pass through the soil used to cover the refuse. This water passes into the waste itself, and the resulting landfill leachate that flows from the bottom of the landfill can seriously contaminate the surrounding groundwater and surface water. In the past few decades, regulations around refuse disposal have been significantly strengthened, and important steps have been taken to reduce the amount of landfill waste by diverting recyclable and compostable materials to other locations.
A modern engineered landfill has an impermeable liner (typically heavy plastic, although engineered clay liners or natural clay may be adequate in some cases), a plumbing system for draining leachate (the rainwater that flows through the refuse and becomes contaminated), and a network of monitoring wells both within and around the landfill (Figure 4). Once part or all of a landfill site is full, it is sealed over with a plastic cover, and a system is put in place to extract landfill gas (typically a mixture of carbon dioxide and methane). That gas can be sent to a nearby location where it is burned to create heat or used to generate electricity. The leachate must be treated, and that can be done in a normal sewage treatment plant.
The monitoring wells are used to assess the level of the water table around the landfill and to collect groundwater samples so that any leakage can be detected. Because some leakage is almost inevitable, the ideal placement for landfills is in areas where the depth to the water table is significant (tens of metres if possible) and where the aquifer material is relatively impermeable. Landfills should also be situated far from streams, lakes, or wetlands so that contamination of aquatic habitats can be avoided.
Today there are hundreds of abandoned dumps scattered across the country; most have been left to contaminate groundwater that we might wish to use sometime in the future. In many cases, it’s unlikely that we’ll be able to do so.
Unless you live in a remote rural area, there’s a good chance that the refuse you can’t recycle is picked up at the curb and taken to a landfill. Most landfills are operated by cities or regional districts, and you should be able to find information about yours on the appropriate local government website. See if you can answer some the following questions:
- Which government body operates your landfill?
- Where is the landfill situated?
- Is your waste all placed in a landfill, or are there other processes in use (e.g., incineration or composting)?
- Are landfill gases captured, and, if so, what is done with them?
- What could be changed to improve the waste disposal situation in your community (e.g., more recycling, compost collection, waste-to-energy technology)?
Industrial Operations
Although western Canada doesn’t have the same extent of industrial pollution as other parts of the country, there are still seriously contaminated sites in the west, most with the potential to contaminate groundwater. One example is the lead and zinc smelter at Trail, B.C. The largest in the world, it has been operating for over 100 years and has left a residue of metal contamination around the region (Figure 6). In some parts of Trail, the contamination is serious enough that existing soil has been removed from residential properties and replaced with clean soil brought in from elsewhere. This contaminated soil has contributed to contamination of groundwater in the Trail area. Groundwater beneath the actual smelter site is contaminated, and the operator (Teck Resources) is currently working on plans to prevent that water from reaching the nearby Columbia River.
Mines, Quarries, and Rock Excavations
Mines and other operations that involve the excavation of large amounts of rock (e.g., highway construction) have the potential to create serious environmental damage. The exposure of rock that has previously not been exposed to air and water can lead to the oxidation of sulphide-bearing minerals, such a pyrite (FeS 2 ), within the rock. The combination of pyrite, water, oxygen, and a special type of bacteria ( Acidithiobacillus ferrooxidans ) that thrives in acidic conditions leads to the generation of acidity, in some cases to pH less than 2. Water that acidic is hazardous by itself, but the low pH also has the property of increasing the solubility of certain heavy metals. The water that is generated by this process is known as acid rock drainage (ARD). ARD can occur naturally where sulphide-bearing rocks are near the surface. The issue of ARD is a major environmental concern at both operating mines and abandoned mines. In streams around the Mt. Washington Mine on Vancouver Island (Figure 7), copper levels are high enough to be toxic to fish. Groundwater adjacent to the contaminated streams in the area is very likely contaminated as well.
Leaking Fuel Tanks
Underground storage tanks (USTs) are used to store fuel at gas stations, industrial sites, airports, and anywhere that large volumes of fuel are used. They do not last forever, and eventually they start to leak their contents into the ground. This is a particular problem at older gas stations — although it may also become a future problem at newer gas stations. You may have noticed gas stations that have been closed and then surrounded by chain-link fence (Figure 8). In virtually all such cases the closure has been triggered by the discovery of leaking USTs and the requirement to cease operations and remediate the site.Petroleum fuels are complex mixtures of hydrocarbon compounds and the properties of their components—such as density, viscosity, solubility in water, and volatility — tend to vary widely. As a result, a petroleum spill is like several spills for the price of one. The petroleum liquid slowly settles through the unsaturated zone and then tends to float on the surface of the groundwater (Figure 9). The more readily soluble components of the spill dissolve in the groundwater and are dispersed along with the normal groundwater flow, and the more volatile components of the spill rise toward the surface, potentially contaminating buildings.
There is almost certainly a leaking UST at a former gas station near you. Look for an empty property that is surrounded by a chain-link fence with “No Trespassing” signs. You might see evidence of monitoring wells (like those shown in Figure 9), and there could be some petroleum barrels around that are being used to store contaminated water. Once you’ve identified one of these, you’ll probably start seeing them everywhere!
Septic Systems
In areas that are not served by sewage networks leading to a central sewage treatment plant, most homeowners rely on septic systems for disposal of sewage. There are two primary components to a simple septic system, the septic tank and the drainage field (Figure 10). A typical septic tank is constructed of either concrete or plastic and has a volume of 5,000 L to 10,000 L (5 m3 to 10 m3). This forms the first treatment and is designed to be anaerobic (without oxygen). That promotes the activity of certain bacteria that help break down the waste. As the waste is degraded, some portions tend to sink to form sludge at the base of the tank, and others float to the surface, forming a scum layer. A septic tank may be divided into two parts to keep the sludge at the bottom and the scum on the top from draining out. The water then moves to the drainage field, which provides the right conditions for a different set of bacteria that operate in aerobic conditions. The drainage field includes an array of plastic pipes that are perforated to allow the effluent to drain out over a large area and seep slowly into the ground. In order to install a drainage field, it is first necessary to test the soil below, as it must be sufficiently permeable to allow the effluent to percolate away, but not so permeable that it flows too quickly and the soil is not able to filter out the pathogenic bacteria.
If they are properly installed and used, and if the sludge is periodically removed from the tank, a septic system should be effective in treating the sewage for decades. The anaerobic and aerobic bacteria should be able to break down the incoming waste and there should be little risk to the surface environment or groundwater. But many things can go wrong with a septic system, including the following:
- If inappropriate chemicals are added to the waste stream, they may interfere with the natural breakdown of the sewage.
- If the tank is not periodically pumped out, solids can get into the drainage field and compromise the drainage, resulting in the flow of effluent toward the surface.
- If the soil is either not sufficiently permeable or too permeable, the effluent will not drain away (and will start to pool at the surface) or it will drain too quickly.
- If the drainage field is constructed in an area where the water table is close to surface, some of the effluent is likely to flow into the groundwater without being treated.
Prevention and Mitigation of Groundwater Contamination
As illustrated in the landfill example above, there are two fairly simple ways to significantly reduce the chance and degree of groundwater contamination from surface sources. One is to prevent rainwater from infiltrating down to the water table and picking up contaminants; this can be achieved by simply capping or roofing over the landfill, mine tailings, or spill site. The second is to provide an impermeable barrier beneath the contaminant. Modern landfills and mine tailings impoundments are all built using some combination of clay and engineered plastic barriers. Both of these solutions — caps and liners — are subject to failure due to leaks.
Once contaminants are in the groundwater, the main form of remediation is to pump out the contaminated water and treat it at the surface. This can be a slow process, and preventing the contaminant from travelling significantly during this process can be accomplished by manipulating local groundwater flow through the extraction or injection of water at certain locations. Consider this in the exercise below.
This diagram shows a groundwater contaminant plume in red. The source of the contamination has been removed but if the plume is not dealt with, it will eventually enter the stream and threaten the health of wildlife. Pumping the contaminant from well B for treatment will not be sufficient to prevent some of the contamination from making it to the stream.
What could you do at wells A and C to prevent this? Explain and use the diagram below to illustrate the expected changes to the water table and the movement of the plume.
- “Adsorb” (with a “d”) is not the same as “absorb” (with a “b”). Water can be absorbed by a sponge. Ions dissolved in water can be adsorbed onto—or desorbed from—the surfaces of clay minerals. ↵
Contributors and Attributions
- Groundwater Quality. Authored by : Steven Earle. Provided by : BC Campus. Located at : https://opentextbc.ca/geology/chapter/14-4-groundwater-quality . Project : Physical Geology. License : CC BY: Attribution


16.3: What kind of rubric is best for a particular project?
What is a Rubric?
Different Rubrics
...So which one is BEST for me?
- Answer
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By Ashley Wade
Learning Objectives
So what is a rubric? According to Goodrich, a rubric is “a scoring tool that lists the criteria for a piece of work or what counts.” Through this scoring tool, teachers are able to determine how to grade their students work fairly and reasonably. With the implementation of a rubric, students and their parents will also be able to “figure out how their [or their child's] project will be evaluated.” Rubrics allow everyone to see what is being asked and what an individual is supposed to be doing. One student even stated that they disliked rubrics because “if you get something wrong, your teacher can prove you knew what you were supposed to do.” Usually “rubrics specify level of performance expected for several levels of quality.” These are either titled as ratings, from excellent to poor, or as numerical scores, from zero to four. From these two types of ratings, a teacher or instructor can total the scores to determine a final grade, ranging from an A to a F. (What is a Rubric?, 2008)
There are many different types of rubrics that have been developed. Listed below are some of the main types of rubrics that can be used in the classroom.
Holistic
• Usually judges the assignment as a whole entity
• Usually very easy to use and can be created very quickly
• Associated with less feedback
Analytical
• Usually judges the assignment by its individual parts
• Usually harder and more time consuming to create
• Has more specific feedback
General or Generic
• Usually scores assignments that have similar tasks
• Better for regular events or assignments, i.e. papers or science projects
• Easier to compare an individual’s progress over time
Task-specific
• Usually scores assignments that are unique and specific
• Better for just assigning grades
• Harder to create, yet get nothing in return either, such as feedback
Note
"...rubrics help teachers as well as evaluate student work. Further, creating rubrics with your students can be powerfully instructive." (Andrade, H., 2000)
By examining the various rubrics listed above, one can see that each is very specific to what kind of needs it can accommodate. By using a general rubric, a teacher can grade a student's paper quickly, make the grading process easier, as well as making the student's expectations known at the same time. If a teacher asked their student to put on a skit in front of the class, a more task-specific rubric could be developed in order to accommodate the specific ideas the teacher would want the student to present. With rubrics, the teacher can make sure he/she grades all the students with the same intentions in mind, as well as lets the student know ahead of time what they will be graded on. Simply put, rubrics are something that take time to get used to and create. By looking at various rubrics that have already been made, a teacher can then determine which ones might benefit them most. Even by including one's students in the process of creating a rubric can help to benefit all sides of the spectrum.
Exercise \(\PageIndex{1}\)
1.) Who benefits from rubrics?
a) The Parent
b) The Student
c) The Teacher
d) All of the Above
2.) Which of the following is NOT a type of rubric?
a) Analytical
b) General
c) Holistic
d) Major
3.) Mrs. Smith might use a task-specific rubric because...
a) She is having her students do a very general assignment.
b) She will easily be able to compare her students progress.
c) She is having her students do a very unique assignment.
d) It is harder to create, yet gives great feedback.
4.) Mrs. White might use a holistic rubric in order to...
a) Judge her student's assignment by its individual parts
b) Judge her student's assignment as a whole.
c) Obtain more feedback for her students
d) Take more time in making a rubric for her students
1) D
2) D
3) C
4) B
What is a Rubric? (2008) In Rubistar/4 Teachers. Retrieved October 25, 2008, from http://rubistar.4teachers.org/index.php?screen=WhatIs&module=Rubistar
Andrade, H. (2000) Using Rubrics to Promote Thinking and Learning. Educational Leadership . 57(5),1-7.
Types of Rubrics (2000) In Virtual Assessment Center. Retrieved October 25, 2008, from www.carla.umn.edu/assessment/VAC/Evaluation/p_6.html
Kidd, J. (2008). L7 performance assessment. Retrieved October 26, 2008, from www.blackboard.odu.edu/bbcswebdav/users/jkemp002/Wiki%20Lectures/Lecture%207.pdf


11: Conventional and Sustainable Agriculture
Learning Outcomes
- Describe the components of soils
- Discuss how land use affects global ecosystem conditions
- Identify environmental effects of pesticides
- Recognize the relationship between exposure to POPs and human health
- Explain alternative practices in farming and soil management
-
- Page 11.1: Soil Profiles and Processes
- The word “soil” has been defined differently by different scientific disciplines. In agriculture and horticulture, soil generally refers to the medium for plant growth, typically material within the upper meter or two. Soil consists predominantly of mineral matter, but also contains organic matter (humus) and living organisms. The pore spaces between mineral grains are filled with varying proportions of water and air.
-
- Page 11.2: Soil-Plant Interactions
- Soil plays a key role in plant growth. Beneficial aspects to plants include providing physical support, water, heat, nutrients, and oxygen (Figure 1). Mineral nutrients from the soil can dissolve in water and then become available to plants. Although many aspects of soil are beneficial to plants, excessively high levels of trace metals (either naturally occurring or anthropogenically added) or applied herbicides can be toxic to some plants.
-
- Page 11.3: Conventional Agriculture
- The prevailing agricultural system, variously called “conventional farming,” “modern agriculture,” or “industrial farming,” has delivered tremendous gains in productivity and efficiency. Food production worldwide has risen in the past 50 years; the World Bank estimates that between 70 percent and 90 percent of the recent increases in food production are the result of conventional agriculture rather than greater acreage under cultivation.
-
- Page 11.4: Pests and Pesticides
- Pests are organisms that occur where they are not wanted or that cause damage to crops or humans or other animals. Thus, the term “pest” is a highly subjective term. A pesticide is a term for any substance intended for preventing, destroying, repelling, or mitigating any pest. Though often misunderstood to refer only to insecticides, the term pesticide also applies to herbicides, fungicides, and various other substances used to control pests.
Thumbnail image - Women farmers planting a rice field in West Sumatra.


6.3: Emotional and Social Intelligence in Leadership
Overview
The position of either leader or follower does not hold power. Rather, it is how we respond when we are in these roles, based on our emotional intelligence, that gives power to each role. Emotional intelligence has been described as the “ability to monitor and discriminate among emotions and to the use the data to guide thought and action” (Pangman & Pangman, 2010, p. 146). Goleman (1998), a researcher who has completed excellent work in the area of work performance, studied the importance of emotional intelligence in achieving personal excellence. He defines emotional intelligence in greater depth, stating that it is composed of “abilities such as being able to motivate oneself and persist in the face of frustrations; to control impulse and delay gratification; to regulate one’s moods and keep distress from swamping the ability to think; to empathise and to hope” (Goleman, 1995, p. 21). Goleman’s model of emotional intelligence contains five skills that comprise personal and social competencies (see Table \(\PageIndex{1}\) below). The three skills of self-awareness, self-regulation, and motivation relate to the individual’s personal competence. The remaining skills of empathy and social skills are classified as social competencies (Sadri, 2012, p. 537). Goleman stressed that all of the skills can be learned.
|
Competency |
Skill Area |
Description |
|---|---|---|
|
Personal |
Self-awareness |
Knowing one’s self |
|
Self-regulation |
Managing one’s self |
|
|
Motivation |
Sentiments and passions that facilitate the attainment of goals | |
|
Social |
Empathy |
Understanding of others and compassion toward them |
|
Social skills |
Expertise in inspiring others to be in agreement |
Developing Emotional and Social Intelligence
Students are at an ideal stage of their lives and careers to check their emotional intelligence. Completion of the emotional intelligence quiz at the link below may help you identify areas for growth.
Essential Learning Activity \(\PageIndex{1}\)
Visit Queendom.com to access an emotional intelligence assessment.
Now that you have identified an area for growth, you may ask, “How can I increase my emotional intelligence?” Your brain has been developing neural pathways in response to your environment since early childhood. Over time these pathways become hard-wired in your brain, allowing you to respond rapidly to circumstances in your environment. In fact, it is believed that emotional responses occur faster than cognitive responses, thus you seem to act before you think. Siegel’s (2012) research in the area of interpersonal neurobiology shows that there is a way to change your brain’s response to stressors. Increasing your “mindfulness” can provide you with an opportunity to “break the link between environmental stimuli and habitual responses” (Gerardi, 2015, p. 60) and to choose a different course of action. Daniel Siegel (2010) coined the term mindsight to refer to the phenomenon of becoming aware of emotional reactions and changing them in real time. Gerardi (2015) stressed that working on developing mindsight is hard but valuable work for those who wish to become successful leaders.
From the Field
It is important to step back, take a few deep breaths, and look at all aspects of the situation before reacting.
As a nurse, gaining emotional and social intelligence and using mindsight are all critical to becoming a successful leader in the field. You will encounter and be required to cope with many different types of people, both colleagues and patients. It is extremely important to be self-aware, reflect on your feelings, and think about how emotions can influence both actions and relationships (or social interactions). That is, you must learn to reflect on your clinical experiences and think of how you could have changed a situation by using self-awareness or mindsight. In the words of Pattakos, “Between stimulus and response, there is a space. In that space lies our freedom and our power to choose our response. In our response lies our growth and our happiness” (as cited in Gerardi, 2015, p. 60).


1.3: Foundations of Modern Cell Theory
- Explain the key points of cell theory and the individual contributions of Hooke, Schleiden, Schwann, Remak, and Virchow
- Explain the contributions of Semmelweis, Snow, Pasteur, Lister, and Koch to the development of germ theory
While some scientists were arguing over the theory of spontaneous generation, other scientists were making discoveries leading to a better understanding of what we now call the cell theory. Modern cell theory has three basic tenets:
- All organisms are made of cells
- All cells only come from other cells (the principle of biogenesis).
- Cells are the fundamental units of structure and function in organisms.
Today, these tenets are fundamental to our understanding of life on earth. However, modern cell theory grew out of the collective work of many scientists.
The Origins of Cell Theory
The English scientist Robert Hooke first used the term “cells” in 1665 to describe the small chambers within cork that he observed under a microscope of his own design. To Hooke, thin sections of cork resembled “Honey-comb,” or “small Boxes or Bladders of Air.” He noted that each “Cavern, Bubble, or Cell” was distinct from the others (Figure \(\PageIndex{1}\)). At the time, Hooke was not aware that the cork cells were long dead and, therefore, lacked the internal structures found within living cells.
Despite Hooke’s early description of cells, their significance as the fundamental unit of life was not yet recognized. Around the same time, a Dutch cloth merchant named Antonie van Leeuwenhoek (1632–1723) was the first to develop a lens powerful enough to view microbes. In 1675, using a simple but powerful microscope of his own making, Leeuwenhoek was able to observe single-celled organisms, which he described as “animalcules” or “wee little beasties,” swimming in a drop of rain water. From his drawings of these little organisms, we now know he was looking at bacteria and protists. For this reason he is known as the "Father of Microbiology."
Nearly 200 years later, in 1838, Matthias Schleiden (1804–1881), a German botanist who made extensive microscopic observations of plant tissues, described them as being composed of cells. Visualizing plant cells was relatively easy because plant cells are clearly separated by their thick cell walls. Schleiden believed that cells formed through crystallization, rather than cell division.
Theodor Schwann (1810–1882), a noted German physiologist, made similar microscopic observations of animal tissue. In 1839, after a conversation with Schleiden, Schwann realized that similarities existed between plant and animal tissues. This laid the foundation for the idea that cells are the fundamental components of plants and animals.
In the 1850's, two Polish scientists living in Germany pushed this idea further, culminating in what we recognize today as the modern cell theory. In 1852, Robert Remak (1815–1865), a prominent neurologist and embryologist, published convincing evidence that cells are derived from other cells as a result of cell division. However, this idea was questioned by many in the scientific community. Three years later, Rudolf Virchow (1821–1902), a well-respected pathologist, published an editorial essay entitled “Cellular Pathology,” which popularized the concept of cell theory using the Latin phrase omnis cellula a cellula (“all cells arise from cells”), which is essentially the second tenet of modern cell theory. 1 Given the similarity of Virchow’s work to Remak’s, there is some controversy as to which scientist should receive credit for articulating cell theory. See the following Eye on Ethics feature for more about this controversy.
Rudolf Virchow, a prominent, Polish-born, German scientist, is often remembered as the “Father of Pathology.” Well known for innovative approaches, he was one of the first to determine the causes of various diseases by examining their effects on tissues and organs. He was also among the first to use animals in his research and, as a result of his work, he was the first to name numerous diseases and created many other medical terms. Over the course of his career, he published more than 2,000 papers and headed various important medical facilities, including the Charité – Universitätsmedizin Berlin, a prominent Berlin hospital and medical school. But he is, perhaps, best remembered for his 1855 editorial essay titled “Cellular Pathology,” published in Archiv für Pathologische Anatomie und Physiologie , a journal that Virchow himself cofounded and still exists today.
Despite his significant scientific legacy, there is some controversy regarding this essay, in which Virchow proposed the central tenet of modern cell theory—that all cells arise from other cells. Robert Remak, a former colleague who worked in the same laboratory as Virchow at the University of Berlin, had published the same idea three years before. Though it appears Virchow was familiar with Remak’s work, he neglected to credit Remak’s ideas in his essay. When Remak wrote a letter to Virchow pointing out similarities between Virchow’s ideas and his own, Virchow was dismissive. In 1858, in the preface to one of his books, Virchow wrote that his 1855 publication was just an editorial piece, not a scientific paper, and thus there was no need to cite Remak’s work.
By today’s standards, Virchow’s editorial piece would certainly be considered an act of plagiarism, since he presented Remak’s ideas as his own. However, in the 19 th century, standards for academic integrity were much less clear. Virchow’s strong reputation, coupled with the fact that Remak was a Jew in a somewhat anti-Semitic political climate, shielded him from any significant repercussions. Today, the process of peer review and the ease of access to the scientific literature help discourage plagiarism. Although scientists are still motivated to publish original ideas that advance scientific knowledge, those who would consider plagiarizing are well aware of the serious consequences.
In academia, plagiarism represents the theft of both individual thought and research—an offense that can destroy reputations and end careers. 2 3 4 5
- What are the key points of the cell theory?
- What contributions did Rudolf Virchow and Robert Remak make to the development of the cell theory?
The Germ Theory of Disease
Prior to the discovery of microbes during the 17 th century, other theories circulated about the origins of disease. For example, the ancient Greeks proposed the miasma theory, which held that disease originated from particles emanating from decomposing matter, such as that in sewage or cesspits. Such particles infected humans in close proximity to the rotting material. Diseases including the Black Death, which ravaged Europe’s population during the Middle Ages, were thought to have originated in this way.
In 1546, Italian physician Girolamo Fracastoro proposed, in his essay De Contagione et Contagiosis Morbis , that seed-like spores may be transferred between individuals through direct contact, exposure to contaminated clothing, or through the air. We now recognize Fracastoro as an early proponent of the germ theory of disease, which states that diseases may result from microbial infection. However, in the 16 th century, Fracastoro’s ideas were not widely accepted and would be largely forgotten until the 19 th century.
In 1847, Hungarian obstetrician Ignaz Semmelweis (Figure \(\PageIndex{3}\)) observed that mothers who gave birth in hospital wards staffed by physicians and medical students were more likely to suffer and die from puerperal fever after childbirth (10%–20% mortality rate) than were mothers in wards staffed by midwives (1% mortality rate). Semmelweis observed medical students performing autopsies and then subsequently carrying out vaginal examinations on living patients without washing their hands in between. He suspected that the students carried disease from the autopsies to the patients they examined. His suspicions were supported by the untimely death of a friend, a physician who contracted a fatal wound infection after a postmortem examination of a woman who had died of a puerperal infection. The dead physician’s wound had been caused by a scalpel used during the examination, and his subsequent illness and death closely paralleled that of the dead patient.
Although Semmelweis did not know the true cause of puerperal fever, he proposed that physicians were somehow transferring the causative agent to their patients. He suggested that the number of puerperal fever cases could be reduced if physicians and medical students simply washed their hands with chlorinated lime water before and after examining every patient. When this practice was implemented, the maternal mortality rate in mothers cared for by physicians dropped to the same 1% mortality rate observed among mothers cared for by midwives. This demonstrated that handwashing was a very effective method for preventing disease transmission. Despite this great success, many discounted Semmelweis’s work at the time, and physicians were slow to adopt the simple procedure of handwashing to prevent infections in their patients because it contradicted established norms for that time period.
Around the same time Semmelweis was promoting handwashing, in 1848, British physician John Snow conducted studies to track the source of cholera outbreaks in London. By tracing the outbreaks to two specific water sources, both of which were contaminated by sewage, Snow ultimately demonstrated that cholera bacteria were transmitted via drinking water. Snow’s work is influential in that it represents the first known epidemiological study, and it resulted in the first known public health response to an epidemic. The work of both Semmelweis and Snow clearly refuted the prevailing miasma theory of the day, showing that disease is not only transmitted through the air but also through contaminated items. Although the work of Semmelweis and Snow successfully showed the role of sanitation in preventing infectious disease, the cause of disease was not fully understood. The subsequent work of Louis Pasteur, Robert Koch, and Joseph Lister would further substantiate the germ theory of disease.
While studying the causes of beer and wine spoilage in 1856, Pasteur discovered properties of fermentation by microorganisms. He had demonstrated with his swan-neck flask experiments that airborne microbes, not spontaneous generation, were the cause of food spoilage. He also suggested that if microbes were responsible for food spoilage and fermentation, they could also be responsible for causing infection. This laid the foundation for the germ theory of disease.
Meanwhile, British surgeon Joseph Lister (Figure \(\PageIndex{4}\)) was trying to determine the causes of post-surgical infections. Many physicians did not give credence to the idea that microbes on their hands, on their clothes, or in the air could infect patients’ surgical wounds, despite the fact that 50% of surgical patients, on average, were dying of post-surgical infections. 10 Lister, however, was familiar with the work of Semmelweis and Pasteur; therefore, he insisted on handwashing and extreme cleanliness during surgery. In 1867, to further decrease the incidence of post-surgical wound infections, Lister began using carbolic acid (phenol) spray disinfectant/antiseptic during surgery. His extremely successful efforts to reduce post-surgical infection caused his techniques to become a standard medical practice.
A few years later, Robert Koch (Figure \(\PageIndex{4}\)) proposed a series of postulates (Koch’s postulates) based on the idea that the cause of a specific disease could be attributed to a specific microbe. Using these postulates, Koch and his colleagues were able to definitively identify the causative pathogens of specific diseases, including anthrax, tuberculosis, and cholera. 11 Koch’s “one microbe, one disease” concept was the culmination of the 19 th century’s paradigm shift away from miasma theory and toward the germ theory of disease.
- Compare and contrast the miasma theory of disease with the germ theory of disease.
- How did Joseph Lister’s work contribute to the debate between the miasma theory and germ theory and how did this increase the success of medical procedures?
- How did the discovery of microbes change human understanding of disease?
Key Concepts and Summary
- Although cells were first observed in the 1660's by Robert Hooke, cell theory was not well accepted for another 200 years. The work of scientists such as Schleiden, Schwann, Remak, and Virchow contributed to its acceptance.
- The miasma theory of disease was widely accepted until the 19 th century, when it was replaced by the germ theory of disease thanks to the work of Semmelweis, Snow, Pasteur, Lister, and Koch, and others.
Footnotes
- M. Schultz. “Rudolph Virchow.” Emerging Infectious Diseases 14 no. 9 (2008):1480–1481.
- B. Kisch. “Forgotten Leaders in Modern Medicine, Valentin, Gouby, Remak, Auerbach.” Transactions of the American Philosophical Society 44 (1954):139–317.
- H. Harris. The Birth of the Cell . New Haven, CT: Yale University Press, 2000:133.
- C. Webster (ed.). Biology, Medicine and Society 1840-1940 . Cambridge, UK; Cambridge University Press, 1981:118–119.
- C. Zuchora-Walske. Key Discoveries in Life Science . Minneapolis, MN: Lerner Publishing, 2015:12–13.
- T. Embley, W. Martin. “Eukaryotic Evolution, Changes, and Challenges.” Nature Vol. 440 (2006):623–630.
- O.G. Berg, C.G. Kurland. “Why Mitochondrial Genes Are Most Often Found in Nuclei.” Molecular Biology and Evolution 17 no. 6 (2000):951–961.
- L. Sagan. “On the Origin of Mitosing Cells.” Journal of Theoretical Biology 14 no. 3 (1967):225–274.
- A.E. Douglas. “The Microbial Dimension in Insect Nutritional Ecology.” Functional Ecology 23 (2009):38–47.
- Alexander, J. Wesley. “The Contributions of Infection Control to a Century of Progress” Annals of Surgery 201:423-428, 1985.
- S.M. Blevins and M.S. Bronze. “Robert Koch and the ‘Golden Age’ of Bacteriology.” International Journal of Infectious Diseases . 14 no. 9 (2010): e744-e751. doi:10.1016/j.ijid.2009.12.003.
Contributor
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Nina Parker, (Shenandoah University), Mark Schneegurt (Wichita State University), Anh-Hue Thi Tu (Georgia Southwestern State University), Philip Lister (Central New Mexico Community College), and Brian M. Forster (Saint Joseph’s University) with many contributing authors. Original content via Openstax (CC BY 4.0; Access for free at https://openstax.org/books/microbiology/pages/1-introduction )


23.3: RLC Series AC Circuits
Learning Objectives
By the end of this section, you will be able to:
- Calculate the impedance, phase angle, resonant frequency, power, power factor, voltage, and/or current in a RLC series circuit.
- Draw the circuit diagram for an RLC series circuit.
- Explain the significance of the resonant frequency.
Impedance
When alone in an AC circuit, inductors, capacitors, and resistors all impede current. How do they behave when all three occur together? Interestingly, their individual resistances in ohms do not simply add. Because inductors and capacitors behave in opposite ways, they partially to totally cancel each other’s effect. Figure shows an RLC series circuit with an AC voltage source, the behavior of which is the subject of this section. The crux of the analysis of an RLC circuit is the frequency dependence of \(X_L\) and \(X_C\), and the effect they have on the phase of voltage versus current (established in the preceding section). These give rise to the frequency dependence of the circuit, with important “resonance” features that are the basis of many applications, such as radio tuners.
The combined effect of resistance \(R\), inductive reactance \(X_L\), and capacitive reactance \(X_C\) is defined to be impedance , an AC analogue to resistance in a DC circuit. Current, voltage, and impedance in an RLC circuit are related by an AC version of Ohm’s law:
\[I_0 = \dfrac{V_0}{Z} \, or \, I_{rms} = \dfrac{V_{rms}}{Z}.\]
Here \(I_0\) is the peak current, \(V_0\) the peak source voltage, and \(Z\) is the impedance of the circuit. The units of impedance are ohms, and its effect on the circuit is as you might expect: the greater the impedance, the smaller the current. To get an expression for \(Z\) in terms of \(R\), \(X_L\), and \(X_C\), we will now examine how the voltages across the various components are related to the source voltage. Those voltages are labeled \(V_R\), \(V_L\) and \(V_C\) in Figure .
Conservation of charge requires current to be the same in each part of the circuit at all times, so that we can say the currents in \(R\), \(L\), and \(C\) are equal and in phase. But we know from the preceding section that the voltage across the inductor \(V_L\) leads the current by one-fourth of a cycle, the voltage across the capacitor \(V_C\) follows the current by one-fourth of a cycle, and the voltage across the resistor \(V_R\) is exactly in phase with the current. Figure shows these relationships in one graph, as well as showing the total voltage around the circuit \(V = V_R + V_L + V_C\), where all four voltages are the instantaneous values. According to Kirchhoff’s loop rule, the total voltage around the circuit \(V\) is also the voltage of the source.
You can see from Figure that while \(V_R\) is in phase with the current, \(V_L\) leads by \(90^o\), and \(V_C\) follows by \(90^o\). Thus \(V_L\) and \(V_C\) are \(180^o\) out of phase (crest to trough) and tend to cancel, although not completely unless they have the same magnitude. Since the peak voltages are not aligned (not in phase), the peak voltage \(V_0\) of the source does not equal the sum of the peak voltages across \(R\), \(L\), and \(C\). The actual relationship is
\[V_0 = \sqrt{V_{0R}^2 + (V_{0L} - V_{0C})^2},\] where \(V_{0R}\), \(V_{0L}\), and \(V_{0C}\) are the peak voltages across \(R\), \(L\), and \(C\), respectively. Now, using Ohm’s law and definitions from Reactance, Inductive and Capacitive , we substitute \(V_0 = I_0Z\) into the above, as well as \(V_{0R} = I_0R\), \(V_{0L} = I_0X_L\), and \(V_{0C} = I_0X_C\), yielding
\[I_0Z = \sqrt{I_0^2R^2 + (I_0X_L - I_0X_C)^2} = I_0\sqrt{R^2 + (X_L - X_C)^2}.\]
\(I_0\) cancels to yield an expression for \(Z\):
\[Z = \sqrt{R^2 + (X_L - X_C)^2},\] which is the impedance of an RLC series AC circuit. For circuits without a resistor, take \(R = 0\); for those without an inductor, take \(X_L = 0\); and for those without a capacitor, take \(X_C = 0\).
Example \(\PageIndex{1}\) : Calculating Impedance and Current
An RLC series circuit has a \(40.0 \, \Omega\) resistor, a 3.00 mH inductor, and a \(5.00 \, \mu F\) capacitor. (a) Find the circuit’s impedance at 60.0 Hz and 10.0 kHz, noting that these frequencies and the values for \(L\) and \(C\) are the same as in [link] and [link] . (b) If the voltage source has \(V_{rms} = 120 \, V\), what is \(I_{rms}\) at each frequency?
Strategy
For each frequency, we use \(Z = \sqrt{R^2 + (X_L - X_C)^2}\) to find the impedance and then Ohm’s law to find current. We can take advantage of the results of the previous two examples rather than calculate the reactances again.
Solution for (a)
At 60.0 Hz, the values of the reactances were found in [link] to be \(X_L = 1.13 \, \Omega\) and in [link] to be \(X_C = 531 \, \Omega\). Entering these and the given \(40.0 \, \Omega\) for resistance into \(Z = \sqrt{R^2 +(X_L - X_C)^2}\) yields
\[Z = \sqrt{R^2 +(X_L - X_C)^2}\]
\[= \sqrt{(40.0 \, )^2 + (1.13 \, \Omega - 531 \, \Omega)^2}\]
\[= 531 \, \Omega \, at \, 60.0 \, Hz.\]
Similarly, at 10.0 kHz, \(X_L = 188 \, \Omega\) and \(X_C = 3.18 \, \Omega\), so that
\[Z = \sqrt{(40.0 \, \Omega)^2 + (188 \, \Omega - 3.18 \, \Omega)^2}\]
\[= 190 \, \Omega \, at \, 10.0 \, kHz.\]
Discussion for (a)
In both cases, the result is nearly the same as the largest value, and the impedance is definitely not the sum of the individual values. It is clear that \(X_L\) dominates at high frequency and \(X_C\) dominates at low frequency.
Solution for (b)
The current \(I_{rms}\) can be found using the AC version of Ohm’s law in Equation \(I_{rms} = V_{rms}/Z\).
\[I_{rms} = \dfrac{V_{rms}}{Z} = \dfrac{120 \, V}{531 \, \Omega} = 0.226 \, A \, at \, 60.0 \, Hz.\]
Finally, at 10.0 kHz we find
\[I_{rms} = \dfrac{V_{rms}}{Z} = \dfrac{120 \, V}{180 \, \Omega} = 0.633 \, A \, at \, 10.0 \, Hz.\]
Discussion for (a)
The current at 60.0 Hz is the same (to three digits) as found for the capacitor alone in [link] . The capacitor dominates at low frequency. The current at 10.0 kHz is only slightly different from that found for the inductor alone in [link] . The inductor dominates at high frequency.
Resonance in RLC Series AC Circuits
How does an RLC circuit behave as a function of the frequency of the driving voltage source? Combining Ohm’s law, \(I_{rms} = V_{rms}/Z\), and the expression for impedance \(Z\) from \(Z = \sqrt{R^2 + (X_L - X_C)^2}\) gives
\[I_{rms} = \dfrac{V_{rms}}{\sqrt{R^2 + (X_L = X_C)^2}}.\]
The reactances vary with frequency, with \(X_L\) large at high frequencies and \(X_C\) large at low frequencies, as we have seen in three previous examples. At some intermediate frequency \(f_0\), the reactances will be equal and cancel, giving \(Z = R\) —this is a minimum value for impedance, and a maximum value for \(I_{rms}\) results. We can get an expression for \(f_0\) by taking
\[X_L = X_C.\]
Substituting the definitions of \(X_L\) and \(X_C\),
\[2\pi f_0 L = \dfrac{1}{2\pi f_0 C}.\]
Solving this expression for \(f_0\) yields
\[f_0 = \dfrac{1}{2\pi \sqrt{LC}},\] where \(F_0\) is the resonant frequency of an RLC series circuit. This is also the natural frequency at which the circuit would oscillate if not driven by the voltage source. At \(f_0\), the effects of the inductor and capacitor cancel, so that \(Z = R\), and \(I_{rms}\) is a maximum.
Resonance in AC circuits is analogous to mechanical resonance, where resonance is defined to be a forced oscillation—in this case, forced by the voltage source—at the natural frequency of the system. The receiver in a radio is an RLC circuit that oscillates best at its \(f_0\). A variable capacitor is often used to adjust \(f_0\) to receive a desired frequency and to reject others. Figure is a graph of current as a function of frequency, illustrating a resonant peak in \(I_{rms}\) at \(f_0\). The two curves are for two different circuits, which differ only in the amount of resistance in them. The peak is lower and broader for the higher-resistance circuit. Thus the higher-resistance circuit does not resonate as strongly and would not be as selective in a radio receiver, for example.
Example \(\PageIndex{2}\): Calculating Resonant Frequency and Current
For the same RLC series circuit having a \(40.0 \, \Omega\) resistor, a 3.00 mH inductor, and a \(5.00 \, \mu F\) capacitor: (a) Find the resonant frequency. (b) Calculate \(I_{rms}\) at resonance if \(V_{rms}\) is 120 V.
Strategy
The resonant frequency is found by using the expression in \(f_0 = \frac{1}{2\pi \sqrt{LC}}\).
The current at that frequency is the same as if the resistor alone were in the circuit.
Solution for (a)
Entering the given values for \(L\) and \(C\) into the expression given for \(f_0\) in \(f_0 = \frac{1}{2\pi \sqrt{LC}}\) yields
\[f_0 = \dfrac{1}{2\pi \sqrt{LC}}\]
\[= \dfrac{1}{2 \pi \sqrt{(3.00 \times 10^{-3} \, H)(5.00 \times 10^{-6} \, F)}} = 1.30 \, kHz.\]
Discussion for (a)
We see that the resonant frequency is between 60.0 Hz and 10.0 kHz, the two frequencies chosen in earlier examples. This was to be expected, since the capacitor dominated at the low frequency and the inductor dominated at the high frequency. Their effects are the same at this intermediate frequency.
Solution for (b)
The current is given by Ohm’s law. At resonance, the two reactances are equal and cancel, so that the impedance equals the resistance alone. Thus,
\[I_{rms} = \dfrac{V_{rms}}{Z} = \dfrac{120 \, V}{40.0 \, \Omega} = 3.00 \, A.\]
Discussion for (b)
At resonance, the current is greater than at the higher and lower frequencies considered for the same circuit in the preceding example.
Power in RLC Series AC Circuits
If current varies with frequency in an RLC circuit, then the power delivered to it also varies with frequency. But the average power is not simply current times voltage, as it is in purely resistive circuits. As was seen in Figure , voltage and current are out of phase in an RLC circuit. There is a phase angle \(\phi\) between the source voltage \(V\) and the current \(I\), which can be found from
\[cos \, \phi = \dfrac{R}{Z}.\]
For example, at the resonant frequency or in a purely resistive circuit \(Z = R\), so that \(cos \, \phi = 1\). This implies that \(\phi = 0^o\) and that voltage and current are in phase, as expected for resistors. At other frequencies, average power is less than at resonance. This is both because voltage and current are out of phase and because \(I_{rms}\) is lower. The fact that source voltage and current are out of phase affects the power delivered to the circuit. It can be shown that the average power is
\[P_{ave} = I_{rms}V_{rms}cos \, \phi,\] thus \(cos \, \phi\) is called the power factor , which can range from 0 to 1. Power factors near 1 are desirable when designing an efficient motor, for example. At the resonant frequency, \(cos \, \phi = 1\).
Example \(\PageIndex{3}\) : Calculating the Power Factor and Power
For the same RLC series circuit having a \(40.0 \, \Omega\) resistor, a 3.00 mH inductor, a \(5.00 \, \mu F\) capacitor, and a voltage source with a \(V_{rms}\) of 120 V: (a) Calculate the power factor and phase angle for \(f = 60.0 \, Hz.\)
(b) What is the average power at 50.0 Hz? (c) Find the average power at the circuit’s resonant frequency.
Strategy and Solution for (a)
The power factor at 60.0 Hz is found from
\[cos \, \phi = \dfrac{R}{Z}.\]
We know \(Z = 531 \, \Omega \) from Example , so that
\[cos \, \phi = \dfrac{40.0 \, \Omega}{531 \, \Omega} = 0.0753 \, at \, 60.0 \, Hz.\]
This small value indicates the voltage and current are significantly out of phase. In fact, the phase angle is
\[\phi = cos^{-1} \, 0.0753 = 85.7^o \, at \, 60.0 \, Hz.\]
Discussion for (a)
The phase angle is close to \(90^o\), consistent with the fact that the capacitor dominates the circuit at this low frequency (a pure RC circuit has its voltage and current \(90^o\) out of phase).
Strategy and Solution for (b)
The average power at 60.0 Hz is
\[P_{ave} = I_{rms}V_{rms}cos \, \phi.\]
\(I_{rms}\) was found to be 0.226 A in Example . Entering the known values gives
\[P_{ave} = (0.226 \, A)(120 \, V)(0.0753) = 2.04 \, W \, at \, 60.0 \, Hz.\]
Strategy and Solution for (c)
At the resonant frequency, we know \(cos \, \phi = 1\), and \(I_{rms}\) was found to be 6.00 A in Example . Thus,
\[P_{ave} = (3.00 \, A)(120 \, V)(1) = 360 \, W\)
at resonance (1.30 kHz)
Discussion
Both the current and the power factor are greater at resonance, producing significantly greater power than at higher and lower frequencies.
Power delivered to an RLC series AC circuit is dissipated by the resistance alone. The inductor and capacitor have energy input and output but do not dissipate it out of the circuit. Rather they transfer energy back and forth to one another, with the resistor dissipating exactly what the voltage source puts into the circuit. This assumes no significant electromagnetic radiation from the inductor and capacitor, such as radio waves. Such radiation can happen and may even be desired, as we will see in the next chapter on electromagnetic radiation, but it can also be suppressed as is the case in this chapter. The circuit is analogous to the wheel of a car driven over a corrugated road as shown in Figure . The regularly spaced bumps in the road are analogous to the voltage source, driving the wheel up and down. The shock absorber is analogous to the resistance damping and limiting the amplitude of the oscillation. Energy within the system goes back and forth between kinetic (analogous to maximum current, and energy stored in an inductor) and potential energy stored in the car spring (analogous to no current, and energy stored in the electric field of a capacitor). The amplitude of the wheels’ motion is a maximum if the bumps in the road are hit at the resonant frequency.
A pure LC circuit with negligible resistance oscillates at \(f_0\), the same resonant frequency as an RLC circuit. It can serve as a frequency standard or clock circuit—for example, in a digital wristwatch. With a very small resistance, only a very small energy input is necessary to maintain the oscillations. The circuit is analogous to a car with no shock absorbers. Once it starts oscillating, it continues at its natural frequency for some time. Figure shows the analogy between an LC circuit and a mass on a spring.
PHET EXPLORATIONS: CIRCUIT CONSTRUCTION KIT (AC + DC), VIRTUAL LAB
Build circuits with capacitors, inductors, resistors and AC or DC voltage sources, and inspect them using lab instruments such as voltmeters and ammeters.
Summary
- The AC analogy to resistance is impedance \(Z\), the combined effect of resistors, inductors, and capacitors, defined by the AC version of Ohm’s law: \[I_0 = \dfrac{V_0}{Z} \, or \, I_{rms} = \dfrac{V_{rms}}{Z},\] where \(I_0\) is the peak current and \(V_0\) is the peak source voltage.
- Impedance has units of ohms and is given by \(Z = \sqrt{R^2 + (X_L - X_C)^2}\).
- The resonant frequency \(f_0\), at which \(X_L = X_C\), is \[f_0 = \dfrac{1}{2\pi \sqrt{LC}}.\]
- In an AC circuit, there is a phase angle \(\phi\) between source voltage \(V\) and the current \(I\) which can be found from \[cos \, \phi = \dfrac{R}{Z},\]
- \(\phi = 0^o\) for a purely resistive circuit or an RLC circuit at resonance.
- The average power delivered to an RLC circuit is affected by the phase angle and is given by \[P_{ave} = I_{rms}V_{rms} \, cos \, \phi,\] \(cos \, \phi\) is called the power factor, which ranges from 0 to 1.
Glossary
- impedance
- the AC analogue to resistance in a DC circuit; it is the combined effect of resistance, inductive reactance, and capacitive reactance in the form \(Z=\sqrt{R^2+(X_L−X_C)^2}\)
- resonant frequency
- the frequency at which the impedance in a circuit is at a minimum, and also the frequency at which the circuit would oscillate if not driven by a voltage source; calculated by \(f_0=\frac{1}{2π\sqrt{LC}}\)
- phase angle
- denoted by \(ϕ\), the amount by which the voltage and current are out of phase with each other in a circuit
- power factor
- the amount by which the power delivered in the circuit is less than the theoretical maximum of the circuit due to voltage and current being out of phase; calculated by (cosϕ\)


9.2: My R script does not work!
What if your script does not work?
Most likely, there is some message (which you probably do not understand) but outputs nothing or something inappropriate. You will need to debug your script!
- First is to find where exactly your script fails. If you run source() command with echo=TRUE option, this is possible just by looking into output. If this is still not clear, run the script piece by piece : open it in any simple text editor and copy-paste pieces of the script from the beginning to the end into the R window.
- Above mentioned is related with one of the most important principles of debugging: minimize your code as much as possible, and find the minimal example which still does not work . It is likely that you will see the mistake after minimization. If not, that minimal example will be appropriate to post somewhere with a question.
- Related with the above is that if you want to ask somebody else about your R problem, make not only minimal, but minimal self-contained example. If your script loads some data, attach it to your question, or use some embedded R data (like trees or iris ), or generate data with sample() , runif() , seq() , rep() , rnorm() or other command. Even R experts are unable to answer questions without data.
- Back to the script. In R , many expressions are “Russian dolls” so to understand how they work (or why they do not work), you will need to take them to pieces, “undress”, removing parentheses from the outside and going deeper to the core of expression like:
Code \(\PageIndex{1}\) (R):
This research could be interleaved with occasional calls to the help like ?log or ?nrow .
- To make smaller script, do not remove pieces forever. Use commenting instead, both one-line and multi-line. The last is not defined in R directly but one can use:
Code \(\PageIndex{2}\) (R):
- If your problem is likely within the large function, especially within the cycle, use some way to “look inside”. For example, with print() :
Code \(\PageIndex{3}\) (R):
Of course, R has much more advanced functions for debugging but frequently minimization and analysis of print() ’s (this is called tracing ) are enough to solve the problem.
- The most common problems are mismatched parentheses or square brackets, and missing commas. Using a text editor with syntax highlighting can eliminate many of these problems. One of useful precautions is always count open and close brackets. These counts should be equal.
- Scripts or command sets downloaded from Internet could suffer from automatic tools which, for example, convert quotes into quotes-like (but not readable in R ) symbols. The only solution is to carefully replace them with the correct R quotes. By the way, this is another reason why not to use office document editors for R .
-
Sometimes, your script does not work because
your data changed
and now conflicts with your script.
This should not happen if your script was made using “paranoid mode”, commands which are generally safe for all kinds of data, like mat.or.vec() which makes vector if only one column is specified, and matrix otherwise.
Another useful “paranoid” custom is to make checks like if(is.matrix) { ... } everywhere. These precautions allow to avoid situations when you updated data start to be of another type, for example, you had in the past one column, and now you have two.
Of course, something always should be left to chance, but this means that you should be ready to conflicts of this sort.
-
Sometimes, script does not work because there were
changes
in
R
.
For example, in the beginning of its history, R used underscore ( _ ) for the left assignment, together with <- . The story is when S language was in development, on some keyboards underscore was located where on other keyboards there was left arrow (as one symbol). These two assignment operators were inherited in R . Later, R team decided to get rid of underscore as an assignment. Therefore, older scripts might not work in newer R .
Another, more recent example was to change clustering method="ward" to method="ward.D" . This was because initial implementation of Ward’s method worked well but did not reflect the original description. Consequently, in older versions of R newer scripts might stop to work.
Fortunately, in R cases like first (broken backward compatibility ) or second (broken forward compatibility ) are rare. They are more frequent in R packages though.
- If you downloaded the script and do not understand what it is doing, use minimization and other principles explained above. But even more important is to play with a script, change options, change order of commands, feed it with different data, and so on. Remember that (almost) everything what is made by one human could be deciphered by another one.