3.1: Introduction
Light Pollution and Observing
Module Introduction
Some of your older family members and friends probably tell “tales” of seeing billions of stars, every night, in the nighttime sky. While highly exaggerated, these tales are based in reality. Overall light pollution has dramatically increased over the years, as the pervasiveness of artificial light has brightened our streets, homes, and businesses yet, unfortunately, dimmed our chances of having the spectacular views of yesteryear.
In this lab, you will examine light pollution and issues surrounding light pollution, taking photos of light pollution around your home, local shopping centers, and on highways. You will also learn how the Moon creates a lot of light – natural light pollution – when trying to observe stars and fainter objects. (1)
Objectives
At the end of this module, students will be able to:
- Record data pertaining to light pollution and viewing of stars
- Research the effects of light pollution
- Catalogue sources of man-made light pollution
- Compare and contrast conditions of natural light pollution (1)
Outcomes
The material in this module includes content designed to meet the following course outcomes:
- Demonstrate knowledge of scientific method.
- Communicate scientific ideas through oral or written assignments.
- Interpret scientific models such as formulas, graphs, tables and schematics, draw inferences from them and recognize their limitations.
- Demonstrate the ability to think critically.
- Demonstrate the ability to use scientific and quantitative reasoning. (1)
Assigned Readings
Learning Unit 3
Assignments
- The Effects of Light Pollution Exercise
- Local Light Pollution and Observing Activity
- Lab 3 Quiz (1)
- Authored by : Florida State College at Jacksonville. License : CC BY: Attribution


Here’s to those that love us!
And for those that do not love us,
may God turn their hearts.
And if he cannot turn their hearts,
may he turn their ankles;
so that we may know them,
by their limping.
Irish Blessing/Curse
Cheers
,
slainte
,
skal
,
prost
, and
salud
! A toast speech is a type of tribute speech meant to honor someone. Your goal as a speaker should be to make that person feel special and to allow others in the room to be included in the celebration.
Ceremonial Speaking Wheel
All ceremonial speeches should include the trio –narration, magnification, and identification.
Identification
You are not talking
to
an audience; you are sharing
with
an audience.  Saying “we” are here to honor the recipient and reminding the audience of shared values helps the audience to be a part of the process. Say things that invite the audience in several times throughout the speech. This speech is for everyone present.
Wedding Toast
Instead of saying:
Amy and I played Barbie as children and always made sure that Barbie found her perfect soulmate.
Say
:
As many of you know,
Amy and I played Barbie for hours as children. What you may not know is we always made sure that Barbie found her perfect soulmate.
———-
Graduation Toast
Instead of saying:
I wish you the best in your journey ahead as you graduate and start your new career.
Say:
We here, your family and friends,
all wish you the best in your journey ahead as you graduate and start your new career
———-
Retirement Toast
Instead of saying:
Dad, I was so happy you let me come and play in your office even if I ran your stapler out of staples by shooting them at the invisible villains that were hiding under your desk.
Say:
As many of you here now,
dad would sometimes let me come to work with him. And while
most of you
thought that I was the perfect little bosses’ kids, you may not have known that I was really a super hero executing the invisible villain that I found in dads desk. I killed it with all the staples from dad’s stapler.
Narration
The use of story helps the audience to be drawn in and want to listen. In short toasts, you can tell a series of one-sentence stories. In larger toasts, you can tell one long story or numerous little stories.
Magnification
Take a trait of the individual and magnify it. I’m not talking about superfluous embellishment; I’m talking about honest elaboration. For example, you might talk about
-
Triumph over obstacles
-
Unusual accomplishment
-
Superior performance
-
Unselfish motive
-
Benefit to society
-
The greatness of a simple thing
Now you know of the key elements that should be in your speech, let’s talk about the process of building a toast, writing a toast, and delivering a toast.
Gather the Details
What type of toast is it?
Always adapt your toast to the occasion: Wedding, graduation, retirement, business celebration, award celebration, birthday.
Who will attend?
You will give a different speech to your old college roommate on his birthday if it is just you and the gang versus if his conservative parents are in the room. It will be an even different speech if his new employer is present. Remember, a toast is not just about the person being celebrated but it is about others in the room. As you write your toast, keep imagining the key people present at the event.
Are there expectations about how long the toast will be? It is always good to ask. It can be awkward if there are several people toasting and each gives a five-minute toast and you have prepared only a two-minute toast. Typically wedding-type toasts, retirement toasts, and graduation toasts are longer speeches and people speak from notes. On the other, shorter toasts are usually memorized
Brainstorm
Sit down and just start jotting ideas. Start brainstorming ideas and memories one day and then give it a day or two to think about more memories. The next day after brainstorming, your brain will keep cranking out ideas …write them down immediately, put them on a notepad on your phone, or even call yourself and leave yourself a voicemail of the new ideas. Don’t lose them or let them get away.
Let’s think of your brain as a water faucet. You are not going to get any water until you turn the thing on. Many people complain they can’t think of ideas, but they never actually turn on the idea faucet. Think of brainstorming as turning those ideas on and getting your creative juices flowing.
When I turn off my faucet at home, there are usually a few drips that happen as the rest of the water gets out of the line. Think of your brain that way. Give it time to drip out the last ideas and be there to catch them. They just might be the best ideas of all. Some people find if they brainstorm before bed that they wake up with the best ideas. Always assume that you will forget your ideas, so write them down as soon as you think of them.
Sit Down and Write
At this point, just get your ideas down. Put down more ideas than you will use. Allow yourself to even write down some cheesy things—maybe you will use this somehow but probably not. Most importantly, the creative ideas are hiding in your brain behind the cheesy ones and if you don’t get the cheesy ideas out of the way, you will never get to the next level.
Now. Let’s think about that faucet again. Imagine there is something clogging it. You have to put some pressure on it so you can get it working. I like to think of cheesy ideas as that clog in the line. I have to get the cheesy clog out of the way by giving my mind permission to go there and once that line is clear. I need to be ready for all the great ideas that will come out in force after the clog comes out.
Taking It to The Professional Level
If you want to give your toast the professional edge, try using a theme and then adding in some parallel construction.
Use a Theme
Pick a theme for your toast and carry it throughout the whole toast. I like to think of the theme as the container that holds the speech together and the vehicle that helps ideas move from place to place.
For example, Drew gave a best man speech and used the theme of a road trip. He told stories of him and his best friend and their adventures (at least the adventures you the ones that you can tell when mom is in the room). He talked about fighting over the music, having to take bathroom breaks, and his friend always getting lost. These were fun stories that we could all imagine. In his speech, he told us that whoever was in the passenger seat was called the “roundtrip captain.” He gave the new bride advice on how to navigate the relationship since he knew so much about his friend from all these road trips. The speech wrapped up with Drew telling the bride that she was the map reader now and he passed to her the title of “road trip captain.”
Erin gave a graduation toast to her family members who supported her through college. She used the theme of climbing hills. It was the perfect theme since the University of Arkansas campus seems to be arranged so everything is uphill–both ways. She talked about walking the hills both physically and symbolically. In her speech, she talked about how beautiful the view is from the top. She wrapped up her speech by saying now that she is graduated how she has new hills to climb.
Keep in mind the theme is just the container. If you have only a container, it will seem cliché and sound like a cheap greeting card. The theme is not the speech, it is the container that holds your many stories. Make sure to fully develop those stories.
Use Parallel Construction
Parallel construction is where you take one sentence and you repeat it. It lets you tell a lot of little stories in a short period. Always have at least 3 sentences, keep them together, and try to give them rhythm.
Consider this format and how it might be used for the different toast types.
At a Wedding Toast
I remember when ___
I remember when ___
I remember when ___
I knew you were the “one” because…
I knew you were the “one” because…
I knew you were the “one” because…
At a Retirement Toast
Many of you know dad
for selling insurance, but I know dad for selling me on the idea to go to college
Many of you know dad
for working long hours at the office, but I know dad for working long hours on school projects that I only mentioned to him the night before they were due.
Many of you know dad
as a business profession, but I know dad as my super hero.
Many of you know dad for …., but I know dad for ….
Many of you know dad for …., but I know dad for ….
Many of you know dad for …., but I know dad for ….
TOAST PATTERNS
When picking a speech pattern for the toast, consider the audience and occasion. Many toasts are just one to two minutes while others can be five to seven minutes.
Short Toasts Pattern
-
Let us raise our glass to… Today, I would like to recognize…
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Make a list of attributes/accomplishments of the person. With each statement there is a building sensation –each statement you make is better than the next.
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Build to a climax. Create a building sensation using parallel construction.
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Raise your glass near the end.
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End with a poem, quote, or wish them well. (I really think a well-chosen quote makes these speeches special…it is worth the time to look for one to make it really special)
Note: Typically short toasts are memorized and longer toasts are written as a manuscript.
Wedding Toasts Possible Patterns
Give Their New Spouse the Manual of How to Get Along with Them
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Tell a story or two about your friendship with your friend
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Tell a story about when your friend met the person they are marrying. What did they say about this person? What were your first impressions? Tell anything about the two of them together? What are they like as a couple? How do you know they are right for each other? Tell something you observe about them when they are together?
-
Give the person advice on how to navigate life with your friend since you know them so well. Think of this as passing on the manual.
Three Stories
Tell three stories about the couple.
-
Maybe, how they met, the engagement, and a story they tell on themselves.
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Maybe, three ways you know they are in love
-
Maybe, three things you noticed about how they care for each other.
I Knew It Was Love…
-
Tell five ways that you knew it was love. Tell a story attached to each of the ways
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Tell what your friend was like before and how they are better now than they are with the one they love.
Note: Never mention old girlfriend/boyfriends, never give away secrets to clueless family members (they are living together, that you and your friend snuck out in high school), and keep it positive. You likely have inside information about the person you are toasting–their wedding is not the time to reveal it.
Retirement Toast Possible Patterns
If you are a work colleague:
-
These are the three things that set this person apart.
-
These are the three things that I learned from this person.
If you are the child or the spouse:
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Talk about what their career looked like from your position. How were they dedicated?
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How did it help you benefit from their job and friends? What did you learn from watching them work?
Note: It can be fun to use their job as the theme. For example, if the person is a coach make the whole speech in the theme of a game. If the person is in sales, use marketing terms to hold it together. If they collected something interesting that was displayed in the office, it lends itself to the theme.
Graduation Toast
Looking back, looking ahead
-
Tell a few stories about them before graduation and tell what you expect to see in the future.
We’ve come through a lot together, Here’s some advice for your next step.
-
Tell stories of your relationship, how that relationship talks about character attributes that led them to graduate.
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Give advice on how to navigate the next step.
Top Ten List
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Ten things you need to know about (surviving college, having your first real job, etc.).
-
This can be especially fun coming from the sibling that is in college to the one headed to college.
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Top ten things you learned in school that you can use in life.
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Top ten things about you that will make you successful.
Let Us Not Forget
-
A series of memories that you and most of the group present share. Great for a group of friends
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Find some great meaning to it all. What did it all mean? what did you learn?
Note: It can be fun to use their major, their future job, or their hobbies as the theme.
Anniversary Toast
Three Stories
-
Tell three stories about the couple.
-
Maybe how they met, the engagement, and a story they tell on themselves.
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Maybe, three ways you know they are in love.
I Knew It Was Love…
-
Tell five ways that you knew it was love and tell a story attached to each of the ways.
Interview five of their friends and tell the five stories from their friends about love.
How They Taught You What Love Is
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What have you learned by watching them love each other well?
A Biography of Love
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Tell a story about their love when they met when they overcame when they did things together.
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Tell a timeline of love events and stories about every major milestone.
Ending a Toast Speech
Typically, a toast speech ends with a thoughtful saying, witty quote, or poem. There are many toast websites with great one-liners or short poems you can use to end a toast. If it is a longer toast, consider using a theme and connecting the ending with the theme.
“May misfortunes chase you all of your life and never catch up.”
“As you slide down the banister of life may the splinters never point the wrong way.”
Short, Short Toasts
There may be times you are called to give a quick, one to three-sentence toast. It is a good idea to have a few memorized quotes ready that can be woven into a short, short toast.
Here is to those who’ve seen us at our best and seen us at our worst and can’t tell the difference.
May you live for as long as you want, and never want for as long as you live!
May we get what we want, but never what we deserve.
May the best of your past be the worse of your future.
May you always lie, cheat, and steal. Lie beside the one you love, cheat the devil, and steal away from bad company.
Here’s to friends and family who know us well but love us just the same.
Let us drink to bread, for without bread, there would be no toast.
“Here’s to that long straight piece in Tetris.”
Toast Etiquette
How Do You Hold the Glass?
Both wine and champagne glasses should be held by the stem, not by the bowl. “You should always hold the glass by the stem, no matter what the shape or size of the glass or the type of wine,” says Michael Greenlee, the sommelier and wine director of Gotham Bar and Grill, in New York City. “The most common mistake I see in restaurants and at dinner parties is people holding a wineglass by the bowl.”
Three reasons to hold the glass by the stem:
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The glass is pretty; you don’t want your fingerprints to mess it up.
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Putting your hand on the glass would block seeing the color of the wine or block your ability to watch the bubbles dance in the champagne.
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White wine and champagne are chilled, and red wine is served a room temperature. The temperature influences the taste and your hands on the glass will warm up the beverage and change it from its optimal temperature.
What Do You Do When Listening to a Toast?
Those who are listening to someone being toasted should hold their glass when the toast is being made. At the end of the toast, listeners should raise their glasses, tilt the glass towards the person being toasted, and then take a sip.

When you are being toasted, you should sit and listen. Smile and nod to let the person toasting know you are listening. You should not hold or lift your glass. Once the toast is over and everyone has had a sip, then you may nod and/or say, “Thank You.” One of the more common questions I get is, “Do I drink to myself?” and the answer is no, you do not drink to yourself.
Should We Clink our Glasses?
Check five different etiquette specialists and you will get five different answers. Some groups clink and some groups don’t clink. It is best to take your lead from those with the highest power of those in the center of the event. For example, if you have dinner with your co-workers and your boss is present, see what the boss does. If you are at a wedding, see what the head table does and copy.
The larger the group, the less likely there will be glass touching because it is difficult to touch everyone’s glasses. If you are in a clinking group, when you touch the bowl of the glasses together, you should always look them in the eye and smile.
“As with many of our food traditions, the clinking of glasses traces its root to the health and safety of the drinker. In this case, it goes back to the tendency of nobles to kill each other off by poisoning their food! Wine was very commonly drunk during medieval days because it was one of the only safe liquids available. Water was often polluted, and milk was both useful for other things and thought to be for children only. As the wine was often full of sediment, a poison was easily introduced into it.
To prove that his wine was safe, the host would pour a bit of his guest’s wine into his own glass and drink it first, to prove it was safe. If the guest trusted his host, however, he would merely clink his flagon against that of his host’s when his host offered his cup for the sample. The ‘clink’ (or perhaps ‘clunk’ back then, since wood or metal was more common for drinking vessels) was a sign of trust and honesty.”
Text from Wineintro. Clinking of Wine Glasses and Toasts. Retrieved March 28, 2008, from
http://www.wineintro.com/champagne/clinkglass.html
Toasting Rules
Hold up your glass while giving a toast.
Never toast with an empty glass, it is considered bad luck.
Always hold the glass by the stem.
Avoid getting the beautiful glasses smudged. That means, not touching the bowl. It also means sipping from the same place on the glass.
Always drink to the person being toasted. If you don’t take a sip then it means that you disagree with the nice things that were said about the person.
If you oversee the filling of the glasses, hold the wine bottle near the bottom and only fill glasses halfway up to give the wine room to breathe. Always offer wine to others before pouring your own wine.
In some regions, people will gently use a knife to tap the glass to bring the group to attention.
At some weddings, the quests may tap their glasses to mean the bride and groom should kiss.
Brainstorming About Your Person
Read through these questions and try to answer them about your person. Not all the questions will relate, and that is OK. That question may trigger an unrelated memory that you can use.
Story
What is the story of the two of you as children?
What is the first memory you have of this person?
How did you meet?

What is the story of the two of you as adults?
What is an adventure you had together?
When did you do something involving this person that didn’t work out as planned?
What is a story that is often told about this person?
What is a struggle this person has overcome of you overcame together?
What story will history remember about this person?
What story will you tell your kids one day?
Why is this person special to you in a way that no one else knows?
When did you and this person make a difficult situation fun?
How have you seen them grow?
Stuff
What do they collect?
What are their hobbies?
What types of things are important to them? (Pets, cars, shoes)
What are they known for having? (Name brand clothes, a great boat)
What does their house/car/backpack say about them?
What do you have of theirs that means a lot to you?
How is their favorite movie, favorite superhero, favorite song a reflection of them?
The Person
What are the physical characteristics that become part of their persona (Tall, big feet, curly hair)?
What are they known for emotionally (being kind, being passionate, being stubborn)?
What are the phrases they say often?
What are they famous for? (Cooking, getting lost)
Relationship
How did you meet?
What has kept your relationship with this person going?
What has been the highlight of the relationship?
What do their friends say about them?
Wedding Toast
When and how did they meet?
What were your first impressions of them as a couple?
What did your friend say about the first time they met?
If you were given three wishes for them as a couple, what would they be?
Toast Speech Samples from College Students Toasting Other College Students
To Blake who has taught us the importance of enjoying life to the fullest, whether fly fishing or spending time with loved ones.
You’re a man of great humor.
You’re a man who has brought us all to tears with your Tommy Boy impression.
You’re a man who embodies the word “chill”
Every time I would look over in class, you would be just sitting back and smiling.
You were always a great person to make a speech to, because even if the speech was bad, we could look over and you’d still be smiling.
Today, I would like to raise my glass (raise your glass) to by the far the most gifted athlete in our class. She showed us what it takes to be a student athlete at the University of Arkansas. She opened up her heart and shared the stories about her mother with us. I know we were all touched.
She taught us how to improve our puts and she taught us that golf can be fun.
We learned that even the best athletes get nervous sometimes.
In the words of Roy, tin cup, McEvoy, “I hit it again because that shot was a defining moment, and when a defining moment comes along, you define the moment… or the moment defines you.”
To Ana, may you always define the moment and not let it define you!
I ask students to toast their classmates and to give reference to things that student has said or done in class, consider the following speech ending that references three of the student’s former speeches.
Sean here this is to you for all you have given us.
May your bike never fail you (reference to speech on changing a tire),
May your sense of nature keep you wandering (reference to speech on camping)
and may your lessons live on, in this lifetime and the next (reference to speech on missionary work).
To someone who
Has opened her heart and mind to this class
With hopes that they too will join in her efforts to serve God and humanity
To someone who
Has made us all feel as if her papaw was our own
Pitching and swinging the bat
To someone who
Has made each person in this class believe
As if we can make a difference in the world
One Krispy Kreme doughnut at a time
Anne Frank said,
“How wonderful it is that nobody need wait a single moment
Before starting to improve the world. “
To Haley, we toast to you (raise glass)
For you have already begun improving the world.
Best Man Toasts the Couple by Joe Burlingame
Thank you all for gathering here tonight on the beautiful occasion to celebrate the love of Korey and Paola.
My name is Joe Burlingame and I know many of you have traveled from long distances to be here and we are so grateful that y’all came.
Since the lovely maid of honor spoke so highly of Paola I am going to try and make Korey seem like he is worthy.
When Korey told me that things were getting serious and that he was going to propose,
My initial reaction was woah Korey
I am flattered but I am perfectly content with being fraternity brothers, but everybody knew he was talking about Paola.
Korey and I’s relationship started about 5 years ago
And to this day I remember my first impression–it was like any first impression when we first meet an accounting major/// Aw—wk–kward.
Our friendship though took off like a rocket ship, and before we knew it we were roommates up to the time he betrayed me and found a permanent roommate.
I won’t say any names but she is in this room.
In the 4 years, I lived with Korey, I have found 3 things that make him the perfect roommate.
First, Korey loves spontaneous trips.
Korey and I went to 6-7 concerts one semester. We only bought tickets in advance for one and that was because it was the one band I wanted to go see. The others were concerts that Korey had caught wind of last-minute and persuaded me to go. One concert, in particular, was the night of Korey and Paola’s first date. They had gone and got coffee and macaroons I think it was maybe an hour-long date. When he got back Korey talked about it as if they had spent a year with each other. Paola I knew then that you had to be special if you had him talking about you for that long.
Second, Korey is a humble servant.
 I would complain about how hot it was mowing our yard, but you would not only mow ours but our elderly neighbor’s yard as well. You thought you were invisible– I saw it all. Whether it be cleaning dishes, cooking, or constantly going to our neighbors to see if they needed anything done, you were a servant and I was thankful for you.
Paola I am jealous, I am now going to have to clean and cook for myself now. Nik Birchfield said it best “Korey doesn’t just talk the talk but walks the walk”
Lastly, Korey is someone who is thoughtful.
I remember when we first moved in Korey brought in this huge water jug of change and a little wooden chest, I asked what the heck is that for. He responded with “Man I plan on getting married and in order to get married you need a ring.” I was blown away– I never would have thought of that.
Paola I regret that we never really got to hang out. When I saw Korey take that jug of bills and came back with a ring and he started describing why he picked this specific ring out because it matched your characteristics I knew we would be friends.
Korey and I were roommates the night before he was going to pop the question.
He practiced a lot with what he was going to say. I know because he practiced on me.

To Paola and Korey!
Hilarious Father of the Bride Toast
This toast is a good example of how to tell a funny story in a speech. This father’s story is brilliant, and his long pauses are perfect for this speech. He goes from having you crying from laughter to tearing up with sentiment.


5.E: Current and Resistance (Exercises)
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Conceptual Questions
9.2 Electrical Current
1. Can a wire carry a current and still be neutral—that is, have a total charge of zero? Explain.
2. Car batteries are rated in ampere-hours ( A⋅h ). To what physical quantity do ampere-hours correspond (voltage, current, charge, energy, power,…)?
3. When working with high-power electric circuits, it is advised that whenever possible, you work “one-handed” or “keep one hand in your pocket.” Why is this a sensible suggestion?
9.3 Model of Conduction in Metals
4. Incandescent light bulbs are being replaced with more efficient LED and CFL light bulbs. Is there any obvious evidence that incandescent light bulbs might not be that energy efficient? Is energy converted into anything but visible light?
5. It was stated that the motion of an electron appears nearly random when an electrical field is applied to the conductor. What makes the motion nearly random and differentiates it from the random motion of molecules in a gas?
6. Electric circuits are sometimes explained using a conceptual model of water flowing through a pipe. In this conceptual model, the voltage source is represented as a pump that pumps water through pipes and the pipes connect components in the circuit. Is a conceptual model of water flowing through a pipe an adequate representation of the circuit? How are electrons and wires similar to water molecules and pipes? How are they different?
7. An incandescent light bulb is partially evacuated. Why do you suppose that is?
9.4 Resistivity and Resistance
8. The IR drop across a resistor means that there is a change in potential or voltage across the resistor. Is there any change in current as it passes through a resistor? Explain.
9. Do impurities in semiconducting materials listed in Table 9.1 supply free charges? (Hint: Examine the range of resistivity for each and determine whether the pure semiconductor has the higher or lower conductivity.)
10. Does the resistance of an object depend on the path current takes through it? Consider, for example, a rectangular bar—is its resistance the same along its length as across its width?
11. If aluminum and copper wires of the same length have the same resistance, which has the larger diameter? Why?
9.5 Ohm's Law
12. In Determining Field from Potential, resistance was defined as \(R≡\frac{V}{I}\). In this section, we presented Ohm’s law, which is commonly expressed as V= IR. The equations look exactly alike. What is the difference between Ohm’s law and the definition of resistance?
13. Shown below are the results of an experiment where four devices were connected across a variable voltage source. The voltage is increased and the current is measured. Which device, if any, is an ohmic device?
14. The current I is measured through a sample of an ohmic material as a voltage V is applied. (a) What is the current when the voltage is doubled to 2V (assume the change in temperature of the material is negligible)? (b) What is the voltage applied is the current measured is 0.2I (assume the change in temperature of the material is negligible)? What will happen to the current if the material if the voltage remains constant, but the temperature of the material increases significantly?
9.6 Electrical Energy and Power
15. Common household appliances are rated at 110 V, but power companies deliver voltage in the kilovolt range and then step the voltage down using transformers to 110 V to be used in homes. You will learn in later chapters that transformers consist of many turns of wire, which warm up as current flows through them, wasting some of the energy that is given off as heat. This sounds inefficient. Why do the power companies transport electric power using this method?
16. Your electric bill gives your consumption in units of kilowatt-hour (kW ⋅· h). Does this unit represent the amount of charge, current, voltage, power, or energy you buy?
17. Resistors are commonly rated at \(\frac{1}{8}W, \frac{1}{4}W, \frac{1}{2}W\), 1 W and 2 W for use in electrical circuits. If a current of I=2.00A is accidentally passed through a R=1.00Ω resistor rated at 1 W, what would be the most probable outcome? Is there anything that can be done to prevent such an accident?
18. An immersion heater is a small appliance used to heat a cup of water for tea by passing current through a resistor. If the voltage applied to the appliance is doubled, will the time required to heat the water change? By how much? Is this a good idea?
9.7 Superconductors
19. What requirement for superconductivity makes current superconducting devices expensive to operate?
20. Name two applications for superconductivity listed in this section and explain how superconductivity is used in the application. Can you think of a use for superconductivity that is not listed?
Problems
9.2 Electrical Current
21. A Van de Graaff generator is one of the original particle accelerators and can be used to accelerate charged particles like protons or electrons. You may have seen it used to make human hair stand on end or produce large sparks. One application of the Van de Graaff generator is to create X-rays by bombarding a hard metal target with the beam. Consider a beam of protons at 1.00 keV and a current of 5.00 mA produced by the generator.
(a) What is the speed of the protons?
(b) How many protons are produced each second?
22. A cathode ray tube (CRT) is a device that produces a focused beam of electrons in a vacuum. The electrons strike a phosphor-coated glass screen at the end of the tube, which produces a bright spot of light. The position of the bright spot of light on the screen can be adjusted by deflecting the electrons with electrical fields, magnetic fields, or both. Although the CRT tube was once commonly found in televisions, computer displays, and oscilloscopes, newer appliances use a liquid crystal display (LCD) or plasma screen. You still may come across a CRT in your study of science. Consider a CRT with an electron beam average current of 25.00μA25.00μA. How many electrons strike the screen every minute?
23. How many electrons flow through a point in a wire in 3.00 s if there is a constant current of I=4.00A ?
24. A conductor carries a current that is decreasing exponentially with time. The current is modeled as \(I=I_0e^{−t/τ}\), where \(I_0=3.00A\) is the current at time t=0.00s and τ=0.50s is the time constant. How much charge flows through the conductor between t=0.00s and t= 3τ ?
25. The quantity of charge through a conductor is modeled as \(Q=4.00\frac{C}{s^4}t^4−1.00\frac{C}{s}t+6.00mC\). What is the current at time t=3.00s ?
26. The current through a conductor is modeled as \(I(t)=I_msin(2π[60Hz]t)\). Write an equation for the charge as a function of time.
27. The charge on a capacitor in a circuit is modeled as \(Q(t)=Q_{max}cos(ωt+ϕ)\). What is the current through the circuit as a function of time?
9.3 Model of Conduction in Metals
28. An aluminum wire 1.628 mm in diameter (14-gauge) carries a current of 3.00 amps.
(a) What is the absolute value of the charge density in the wire?
(b) What is the drift velocity of the electrons?
(c) What would be the drift velocity if the same gauge copper were used instead of aluminum? The density of copper is \(8.96g/cm^3\) and the density of aluminum is \(2.70g/cm^3\). The molar mass of aluminum is 26.98 g/mol and the molar mass of copper is 63.5 g/mol. Assume each atom of metal contributes one free electron.
29. The current of an electron beam has a measured current of I=50.00μA with a radius of \(1.00\) mm. What is the magnitude of the current density of the beam?
30. A high-energy proton accelerator produces a proton beam with a radius of r=0.90mm . The beam current is I=9.00μA and is constant. The charge density of the beam is \(n=6.00×10^{11}\) protons per cubic meter.
(a) What is the current density of the beam?
(b) What is the drift velocity of the beam?
(c) How much time does it take for \(1.00×10^{10}\) protons to be emitted by the accelerator?
31. Consider a wire of a circular cross-section with a radius of R=3.00mm . The magnitude of the current density is modeled as \(J=cr^2=5.00×10^6\frac{A}{m^4}r^2\). What is the current through the inner section of the wire from the center to r=0.5R ?
32. A cylindrical wire has a current density from the center of the wire’s cross section as \(J(r) = Cr^2\) where \(r\) is in meters, \(J\) is in amps per square meter, and \(C=10^3\) A/m 4 . This current density continues to the end of the wire at a radius of 1.0 mm. Calculate the current just outside of this wire.
33. The current supplied to an air conditioner unit is 4.00 amps. The air conditioner is wired using a 10-gauge (diameter 2.588 mm) wire. The charge density is \(n=8.48×10^{28}\frac{electrons}{m^3}\). Find the magnitude of
(a) current density and
(b) the drift velocity.
9.4 Resistivity and Resistance
34. What current flows through the bulb of a 3.00-V flashlight when its hot resistance is 3.60Ω ?
35. Calculate the effective resistance of a pocket calculator that has a 1.35-V battery and through which 0.200 mA flows.
36. How many volts are supplied to operate an indicator light on a DVD player that has a resistance of 140Ω , given that 25.0 mA passes through it?
37. What is the resistance of a 20.0-m-long piece of 12-gauge copper wire having a 2.053-mm diameter?
38. The diameter of 0-gauge copper wire is 8.252 mm. Find the resistance of a 1.00-km length of such wire used for power transmission.
39. If the 0.100-mm-diameter tungsten filament in a light bulb is to have a resistance of 0.200Ω at 20.0°C , how long should it be?
40. A lead rod has a length of 30.00 cm and a resistance of 5.00μΩ . What is the radius of the rod?
41. Find the ratio of the diameter of aluminum to copper wire, if they have the same resistance per unit length (as they might in household wiring).
42. What current flows through a 2.54-cm-diameter rod of pure silicon that is 20.0 cm long, when \(1.00×10^3\) is applied to it? (Such a rod may be used to make nuclear-particle detectors, for example.)
43. (a) To what temperature must you raise a copper wire, originally at 20.0°C , to double its resistance, neglecting any changes in dimensions? (b) Does this happen in household wiring under ordinary circumstances?
44. A resistor made of nichrome wire is used in an application where its resistance cannot change more than 1.00% from its value at 20.0°C . Over what temperature range can it be used?
45. Of what material is a resistor made if its resistance is 40.0% greater at 100.0°C than at 20.0°C ?
46. An electronic device designed to operate at any temperature in the range from −10.0°C to 55.0°C contains pure carbon resistors. By what factor does their resistance increase over this range? 47.
(a) Of what material is a wire made, if it is 25.0 m long with a diameter of 0.100 mm and has a resistance of 77.7Ω at 20.0°C ? (b) What is its resistance at 150.0°C ?
48. Assuming a constant temperature coefficient of resistivity, what is the maximum percent decrease in the resistance of a constantan wire starting at 20.0°C ?
49. A copper wire has a resistance of 0.500Ω at 20.0°C , and an iron wire has a resistance of 0.525Ω at the same temperature. At what temperature are their resistances equal?
9.5 Ohm's Law
50. A 2.2-kΩ resistor is connected across a D cell battery (1.5 V). What is the current through the resistor?
51. A resistor rated at 250kΩ is connected across two D cell batteries (each 1.50 V) in series, with a total voltage of 3.00 V. The manufacturer advertises that their resistors are within 5% of the rated value. What are the possible minimum current and maximum current through the resistor?
52. A resistor is connected in series with a power supply of 20.00 V. The current measure is 0.50 A. What is the resistance of the resistor?
53. A resistor is placed in a circuit with an adjustable voltage source. The voltage across and the current through the resistor and the measurements are shown below. Estimate the resistance of the resistor.
54. The following table show the measurements of a current through and the voltage across a sample of material. Plot the data, and assuming the object is an ohmic device, estimate the resistance.
| I(A) | V(V) |
|---|---|
| 0 | 3 |
| 2 | 23 |
| 4 | 39 |
| 6 | 58 |
| 8 | 77 |
| 10 | 100 |
| 12 | 119 |
| 14 | 142 |
| 16 | 162 |
9.6 Electrical Energy and Power
55. A 20.00-V battery is used to supply current to a 10-kΩ resistor. Assume the voltage drop across any wires used for connections is negligible.
(a) What is the current through the resistor?
(b) What is the power dissipated by the resistor?
(c) What is the power input from the battery, assuming all the electrical power is dissipated by the resistor?
(d) What happens to the energy dissipated by the resistor?
56. What is the maximum voltage that can be applied to a 20-kΩ resistor rated at \(\frac{1}{4}W\)?
57. A heater is being designed that uses a coil of 14-gauge nichrome wire to generate 300 W using a voltage of V= 110V. How long should the engineer make the wire?
58. An alternative to CFL bulbs and incandescent bulbs are light-emitting diode (LED) bulbs. A 100-W incandescent bulb can be replaced by a 16-W LED bulb. Both produce 1600 lumens of light. Assuming the cost of electricity is $0.10 per kilowatt-hour, how much does it cost to run the bulb for one year if it runs for four hours a day?
59. The power dissipated by a resistor with a resistance of R= 100Ω is P=2.0W . What are the current through and the voltage drop across the resistor?
60. Running late to catch a plane, a driver accidentally leaves the headlights on after parking the car in the airport parking lot. During takeoff, the driver realizes the mistake. Having just replaced the battery, the driver knows that the battery is a 12-V automobile battery, rated at 100 A⋅h . The driver, knowing there is nothing that can be done, estimates how long the lights will shine, assuming there are two 12-V headlights, each rated at 40 W. What did the driver conclude?
61. A physics student has a single-occupancy dorm room. The student has a small refrigerator that runs with a current of 3.00 A and a voltage of 110 V, a lamp that contains a 100-W bulb, an overhead light with a 60-W bulb, and various other small devices adding up to 3.00 W.
(a) Assuming the power plant that supplies 110 V electricity to the dorm is 10 km away and the two aluminum transmission cables use 0-gauge wire with a diameter of 8.252 mm, estimate the percentage of the total power supplied by the power company that is lost in the transmission.
(b) What would be the result is the power company delivered the electric power at 110 kV?
62. A 0.50-W, 220-Ω resistor carries the maximum current possible without damaging the resistor. If the current were reduced to half the value, what would be the power consumed?
9.7 Superconductors
63. Consider a power plant is located 60 km away from a residential area uses 0-gauge \((A=42.40mm^2)\) wire of copper to transmit power at a current of I=100.00A . How much more power is dissipated in the copper wires than it would be in superconducting wires?
64. A wire is drawn through a die, stretching it to four times its original length. By what factor does its resistance increase?
65. Digital medical thermometers determine temperature by measuring the resistance of a semiconductor device called a thermistor (which has α=−0.06/°C ) when it is at the same temperature as the patient. What is a patient’s temperature if the thermistor’s resistance at that temperature is 82.0% of its value at 37°C (normal body temperature)?
66. Electrical power generators are sometimes “load tested” by passing current through a large vat of water. A similar method can be used to test the heat output of a resistor. A R= 30Ω resistor is connected to a 9.0-V battery and the resistor leads are waterproofed and the resistor is placed in 1.0 kg of room temperature water ( T=20°C ). Current runs through the resistor for 20 minutes. Assuming all the electrical energy dissipated by the resistor is converted to heat, what is the final temperature of the water?
67. A 12-guage gold wire has a length of 1 meter.
(a) What would be the length of a silver 12-gauge wire with the same resistance?
(b) What are their respective resistances at the temperature of boiling water?
68. What is the change in temperature required to decrease the resistance for a carbon resistor by 10%?
Additional Problems
69. A coaxial cable consists of an inner conductor with radius \(r_i=0.25cm\) and an outer radius of \(r_o=0.5cm\) and has a length of 10 meters. Plastic, with a resistivity of \(ρ=2.00×10^{13}Ω⋅m\), separates the two conductors. What is the resistance of the cable?
70. A 10.00-meter long wire cable that is made of copper has a resistance of 0.051 ohms.
(a) What is the weight if the wire was made of copper?
(b) What is the weight of a 10.00-meter-long wire of the same gauge made of aluminum?
(c)What is the resistance of the aluminum wire? The density of copper is \(8960kg/m^3\) and the density of aluminum is \(2760kg/m^3\).
71. A nichrome rod that is 3.00 mm long with a cross-sectional area of \(1.00mm^2\) is used for a digital thermometer.
(a) What is the resistance at room temperature?
(b) What is the resistance at body temperature?
72. The temperature in Philadelphia, PA can vary between 68.00°F and 100.00°F in one summer day. By what percentage will an aluminum wire’s resistance change during the day?
73. When 100.0 V is applied across a 5-gauge (diameter 4.621 mm) wire that is 10 m long, the magnitude of the current density is \(2.0×10^8A/m^2\). What is the resistivity of the wire?
74. A wire with a resistance of 5.0Ω is drawn out through a die so that its new length is twice times its original length. Find the resistance of the longer wire. You may assume that the resistivity and density of the material are unchanged.
75. What is the resistivity of a wire of 5-gauge wire (\(A=16.8×10^{−6}m^2\)), 5.00 m length, and 5.10mΩ resistance?
76. Coils are often used in electrical and electronic circuits. Consider a coil which is formed by winding 1000 turns of insulated 20-gauge copper wire (area \(0.52mm^2)\) in a single layer on a cylindrical non-conducting core of radius 2.0 mm. What is the resistance of the coil? Neglect the thickness of the insulation.
77. Currents of approximately 0.06 A can be potentially fatal. Currents in that range can make the heart fibrillate (beat in an uncontrolled manner). The resistance of a dry human body can be approximately 100kΩ .
(a) What voltage can cause 0.2 A through a dry human body?
(b) When a human body is wet, the resistance can fall to 100Ω . What voltage can cause harm to a wet body?
78. A 20.00-ohm, 5.00-watt resistor is placed in series with a power supply.
(a) What is the maximum voltage that can be applied to the resistor without harming the resistor?
(b) What would be the current through the resistor?
79. A battery with an emf of 24.00 V delivers a constant current of 2.00 mA to an appliance. How much work does the battery do in three minutes?
80.
(a) What is the current if the battery terminals are momentarily shorted together?
(b) What is the terminal voltage if the battery delivers 0.25 amps to a circuit?
Challenge Problems
81. A 10-gauge copper wire has a cross-sectional area \(A=5.26mm^2\) and carries a current of I=5.00A . The density of copper is \(ρ=89.50g/cm^3\). One mole of copper atoms (\(6.02×10^{23}atoms\)) has a mass of approximately 63.50 g. What is the magnitude of the drift velocity of the electrons, assuming that each copper atom contributes one free electron to the current?
82. The current through a 12-gauge wire is given as I(t)=(5.00A)sin(2π60Hzt) . What is the current density at time 15.00 ms?
83. A particle accelerator produces a beam with a radius of 1.25 mm with a current of 2.00 mA. Each proton has a kinetic energy of 10.00 MeV.
(a) What is the velocity of the protons?
(b) What is the number (n) of protons per unit volume?
(c) How many electrons pass a cross sectional area each second?
84. In this chapter, most examples and problems involved direct current (DC). DC circuits have the current flowing in one direction, from positive to negative. When the current was changing, it was changed linearly from \(I=−I_{max}\) to \(I=+I_{max}\) and the voltage changed linearly from \(V=−V_{max}\) to \(V=+V_{max}\), where \(V_{max}=I_{max}R\). Suppose a voltage source is placed in series with a resistor of R= 10Ω that supplied a current that alternated as a sine wave, for example, \(I(t)=(3.00A)sin(\frac{2π}{4.00s}t)\). (a) What would a graph of the voltage drop across the resistor V(t) versus time look like? (b) What would a plot of V(t) versus I(t) for one period look like? (Hint: If you are not sure, try plotting V(t) versus I(t) using a spreadsheet.)
85. A current of I= 25A is drawn from a 100-V battery for 30 seconds. By how much is the chemical energy reduced?
86. Consider a square rod of material with sides of length L=3.00cm with a current density of \({J} =J_0e^{αx}k̂=(0.35\frac{A}{m^2})^{e(2.1×10^{−3}m^{−1})x}k̂\) as shown below. Find the current that passes through the face of
87. A resistor of an unknown resistance is placed in an insulated container filled with 0.75 kg of water. A voltage source is connected in series with the resistor and a current of 1.2 amps flows through the resistor for 10 minutes. During this time, the temperature of the water is measured and the temperature change during this time is ΔT=10.00°C .
(a) What is the resistance of the resistor?
(b) What is the voltage supplied by the power supply?
88. The charge that flows through a point in a wire as a function of time is modeled as \(q(t)=q_0e^{−t/T}=10.0Ce^{−t/5s}\).
(a) What is the initial current through the wire at time t=0.00s ?
(b) Find the current at time \(t=\frac{1}{2}T\).
(c) At what time t will the current be reduced by one-half \(I=\frac{1}{2}I_0\)?
89. Consider a resistor made from a hollow cylinder of carbon as shown below. The inner radius of the cylinder is Ri=0.20mmRi=0.20mm and the outer radius is \(R_0=0.30mm\). The length of the resistor is L=0.90mm . The resistivity of the carbon is \(ρ=3.5×10^{−5}Ω⋅m\). (a) Prove that the resistance perpendicular from the axis is \(R=\frac{ρ}{2πL}ln(\frac{R_0}{R_i})\). (b) What is the resistance?
90. What is the current through a cylindrical wire of radius R=0.1mm if the current density is \(J=\frac{J_0}{R}r\), where \(J_0=32000\frac{A}{m^2}\)?
91. A student uses a 100.00-W, 115.00-V radiant heater to heat the student’s dorm room, during the hours between sunset and sunrise, 6:00 p.m. to 7:00 a.m.
(a) What current does the heater operate at?
(b) How many electrons move through the heater?
(c) What is the resistance of the heater?
(d) How much heat was added to the dorm room?
92. A 12-V car battery is used to power a 20.00-W, 12.00-V lamp during the physics club camping trip/star party. The cable to the lamp is 2.00 meters long, 14-gauge copper wire with a charge density of \(n=9.50×10^{28}m^{−3}\).
(a) What is the current draw by the lamp?
(b) How long would it take an electron to get from the battery to the lamp?
93. A physics student uses a 115.00-V immersion heater to heat 400.00 grams (almost two cups) of water for herbal tea. During the two minutes it takes the water to heat, the physics student becomes bored and decides to figure out the resistance of the heater. The student starts with the assumption that the water is initially at the temperature of the room \(T_i=25.00°C\) and reaches \(T_f=100.00°C\). The specific heat of the water is \(c=4180\frac{J}{kg}\). What is the resistance of the heater?


1.4.2: Ionization energy
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- Kathryn Haas, Swetha Ramireddy, Bingyao Zheng, & Emily Nguyen
- Duke University
Ionization energy (IE) is the energy required to remove an electron from a neutral atom or cation in its gaseous phase . IE is also known as ionization potential .
\[A^{n+}_{(g)} \longrightarrow A^{(n+1)+}_{(g)} + e^- \hspace{1cm} IE = \Delta U \nonumber \]
Conceptually, ionization energy is the affinity of an element for its outermost electron (an electron it already has in its valence shell).
1 st , 2 nd , and 3 rd Ionization Energies
The symbol \(I_1\) stands for the first ionization energy (energy required to take away an electron from a neutral atom, where \(n=0\)). The symbol \(I_2\) stands for the second ionization energy (energy required to take away an electron from an atom with a +1 charge, \(n=2\).)
- First Ionization Energy, \(I_1\) (general element, A): \[ A_{(g)} \rightarrow A^{1+}_{(g)} + e^- \nonumber\]
- Second Ionization Energy, \(I_2\) (general element, A): \[ A^{1+}_{(g)} \rightarrow A^{2+}_{(g)} + e^- \nonumber\]
- Third Ionization Energy, \(I_3\) (general element, A): \[ A^{2+}_{(g)} \rightarrow A^{3+}_{(g)} + e^- \nonumber\]
Each succeeding ionization energy is larger than the preceding energy. This means that \(I_1 < I_2 < I_3 < ... < I_n\) will always be true. For example, ionization energy increases as succeeding electrons are taken away from \(\ce{Mg}\).
\[ Mg \,_{(g)} \rightarrow Mg^+\,_{(g)} + e^- \;\;\; I_1= 738\, kJ/mol \nonumber \]
\[Mg^+ \,_{(g)} \rightarrow Mg^{2+}\,_{(g)} + e^- \;\;\; I_2= 1451\, kJ/mol \nonumber \]
Ionization energy is correlated with the strength of attraction between the positively-charged nucleus and the negatively-charged valence electrons. The higher the ionization energy, the stronger the attractive force between nucleus and valence electrons, and the more energy is required to remove a valence electron. The lower the ionization energy, the weaker the attractive force between nucleus and valence electrons, and the less energy required to remove a valence electron.
General periodic trends in electron affinity
In general, ionization energies increase from left to right and decrease down a group; however, there are variations in these trends that would be expected from the effects of penetration and shielding. The trends in first ionization energy are shown in Figure \(\PageIndex{1}\) and are summarized below.
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Across a period:
As Z* increases across a period, the ionization energy of the elements generally increases from left to right. However there are breaks or variation in the trends in the following cases:
- IE is especially low when removal of an electron creates a newly empty p subshell (examples include \(I_1\) of B, Al, Sc)
- IE energy is especially low where removal of an electron results in a half-filled p or d subshell (examples include \(I_1\) of O, S)
- IE increases more gradually across the d- and f-subshells compared to s- and p- subshells. This is because d- and f- electrons are weakly penetrating and experience especially low Z*.
- From one period to the next: There is an especially large decrease in IE with the start of every new period (from He to Li or from Ne to Na for example). This is consistent with the idea that IE is especially low when removal of an electron creates a newly empty s-subshell.
- Noble gases: The noble gases posses very high ionization energies. Note that helium has the highest ionization energy of all the elements.
- Down a group: Although Z* increases going down a group, there is no reliable trend in IE going down any group; in some cases IE increases going down a group, while in other cases IE decreases going down a group.
Plots of the \(I_1, I_2,\) and \(I_3\) of elements from hydrogen to krypton (first four periods) are shown in Figure \(\PageIndex{2}\). Notice that \(I_3>I_2>I_1\). Also notice that trends mentioned above for \(I_1\) hold true for subsequent ionizations when electron configurations are considered!


4.5: Experimental Tests
Universality of Tensor Behavior
The techniques developed in this chapter allow us to make a variety of new predictions that can be tested by experiment. In general, the mathematical treatment of all observables in relativity as tensors means that all observables must obey the same transformation laws. This is an extremely strict statement, because it requires that a wide variety of physical systems show identical behavior. For example, we already mentioned in section 2.5 the 2007 Gravity Probe B experiment (discussed in detail on in sections 5.5 and 6.2 ), in which four gyroscopes aboard a satellite were observed to precess due to special- and general-relativistic effects. The gyroscopes were complicated electromechanical systems, but the predicted precession was entirely independent of these complications. We argued that if two different types of gyroscopes displayed different behaviors, then the resulting discrepancy would allow us to map out some mysterious vector field. This field would be a built-in characteristic of spacetime (not produced by any physical objects nearby), and since all observables in general relativity are supposed to be tensors, the field would have to transform as a tensor. Let’s say that this tensor was of rank 1. Since the tensor transformation law is linear, a nonzero tensor can never be transformed into a vanishing tensor in another coordinate system. But by the equivalence principle, any special, local property of spacetime can be made to vanish by transforming into a free-falling frame of reference, in which the spacetime is has a generic Lorentzian geometry. The mysterious new field should therefore vanish in such a frame. This is a contradiction, so we conclude that different types of gyroscopes cannot differ in their behavior.
This is an example of a new way of stating the equivalence principle: there is no way to associate a preferred tensor field with spacetime. 12
Note
This statement of the equivalence principle, along with the others we have encountered, is summarized in Appendix C .
Speed of Light Differing from c
In a Lorentz invariant theory, we interpret c as a property of the underlying spacetime, not of the particles that inhabit it. One way in which Lorentz invariance could be violated would be if different types of particles had different maximum velocities. In 1997, Coleman and Glashow suggested a sensitive test for such an effect. 13
Assuming Lorentz invariance, a photon cannot decay into an electron and a positron ( example 12 ):
\[ \gamma → e^{+} + e^{−}\]
Suppose, however, that material particles have a maximum speed c m = 1, while photons have a maximum speed c p > 1. Then the photon’s momentum four-vector, (E, \(\frac{E}{c_{p}}\)) is timelike, so a frame does exist in which its three-momentum is zero. The detection of cosmicray gammas from distant sources with energies on the order of 10 TeV puts an upper limit on the decay rate, implying
\[c_p −1 \lesssim 10^{−15}.\]
An even more stringent limit can be put on the possibility of c p < 1. When a charged particle moves through a medium at a speed higher than the speed of light in the medium, Cerenkov radiation results. If \(c_p\) is less than 1, then Cerenkov radiation could be emitted by high-energy charged particles in a vacuum, and the particles would rapidly lose energy. The observation of cosmic-ray protons with energies ∼ 10 8 TeV requires
\[c_p − 1 \lesssim −10^{−23}.\]
Degenerate Matter
The straightforward properties of the momentum four-vector have surprisingly far-reaching implications for matter subject to extreme pressure, as in a star that uses up all its fuel for nuclear fusion and collapses. These implications were initially considered too exotic to be taken seriously by astronomers. For historical perspective, consider that in 1916, when Einstein published the theory of general relativity, the Milky Way was believed to constitute the entire universe; the “spiral nebulae” were believed to be inside it, rather than being similar objects exterior to it. The only types of stars whose structure was understood even vaguely were those that were roughly analogous to our own sun. (It was not known that nuclear fusion was their source of energy.) The term “white dwarf” had not been invented, and neutron stars were unknown.
An ordinary, smallish star such as our own sun has enough hydrogen to sustain fusion reactions for billions of years, maintaining an equilibrium between its gravity and the pressure of its gases. When the hydrogen is used up, it has to begin fusing heavier elements. This leads to a period of relatively rapid fluctuations in structure. Nuclear fusion proceeds up until the formation of elements as heavy as oxygen (Z = 8), but the temperatures are not high enough to overcome the strong electrical repulsion of these nuclei to create even heavier ones. Some matter is blown off, but finally nuclear reactions cease and the star collapses under the pull of its own gravity.
To understand what happens in such a collapse, we have to understand the behavior of gases under very high pressures. In general, a surface area A within a gas is subject to collisions in a time t from the n particles occupying the volume V = Avt, where v is the typical velocity of the particles. The resulting pressure is given by P ∼ \(\frac{npv}{V}\), where p is the typical momentum.
Definition
- Nondegenerate gas : In an ordinary gas such as air, the particles are nonrelativistic, so \(v = \frac{p}{m}\), and the thermal energy per particle is \(\frac{p^{2}}{2m} ∼ kT\), so the pressure is \(P ∼ \frac{nkT}{V}\).
- Nonrelativistic, degenerate gas : When a fermionic gas is subject to extreme pressure, the dominant effects creating pressure are quantum-mechanical. Because of the Pauli exclusion principle, the volume available to each particle is \(∼ \frac{V}{n}\), so its wavelength is no more than \(∼ (\frac{V}{n})^{1/3}\), leading to \(p = \frac{h}{\lambda} ∼ h(\frac{n}{V})^{1/3}\). If the speeds of the particles are still nonrelativistic, then \(v = \frac{p}{m}\) still holds, so the pressure becomes \(P ∼ (\frac{h^{2}}{m})(\frac{n}{V})^{5/3}\).
- Relativistic, degenerate gas : If the compression is strong enough to cause highly relativistic motion for the particles, then v ≈ c, and the result is \(P ∼ hc(\frac{n}{V})^{4/3}\).
As a star with the mass of our sun collapses, it reaches a point at which the electrons begin to behave as a degenerate gas, and the collapse stops. The resulting object is called a white dwarf. A white dwarf should be an extremely compact body, about the size of the Earth. Because of its small surface area, it should emit very little light. In 1910, before the theoretical predictions had been made, Russell, Pickering, and Fleming discovered that 40 Eridani B had these characteristics. Russell recalled: “I knew enough about it, even in these paleozoic days, to realize at once that there was an extreme inconsistency between what we would then have called ‘possible’ values of the surface brightness and density. I must have shown that I was not only puzzled but crestfallen, at this exception to what looked like a very pretty rule of stellar characteristics; but Pickering smiled upon me, and said: ‘It is just these exceptions that lead to an advance in our knowledge,’ and so the white dwarfs entered the realm of study!”
S. Chandrasekhar showed in that 1930’s that there was an upper limit to the mass of a white dwarf. We will recapitulate his calculation briefly in condensed order-of-magnitude form. The pressure at the core of the star is P ∼ ρgr ∼ GM2/r4 , where M is the total mass of the star. The star contains roughly equal numbers of neutrons, protons, and electrons, so M = Knm, where m is the mass of the electron, n is the number of electrons, and K ≈ 4000. For stars near the limit, the electrons are relativistic. Setting the pressure at the core equal to the degeneracy pressure of a relativistic gas, we find that the Chandrasekhar limit is \(∼ (\frac{hc}{G})^{3/2} (Km)^{−2} = 6M_{\odot}\). A less sloppy calculation gives something more like 1.4M. The selfconsistency of this solution is investigated in homework problem 15 .
What happens to a star whose mass is above the Chandrasekhar limit? As nuclear fusion reactions flicker out, the core of the star becomes a white dwarf, but once fusion ceases completely this cannot be an equilibrium state. Now consider the nuclear reactions
\[n \rightarrow p + e^{-} + \bar{\nu}\]
\[p + e^{-} \rightarrow n + \nu,\]
which happen due to the weak nuclear force. The first of these releases 0.8 MeV, and has a half-life of 14 minutes. This explains why free neutrons are not observed in significant numbers in our universe, e.g., in cosmic rays. The second reaction requires an input of 0.8 MeV of energy, so a free hydrogen atom is stable. The white dwarf contains fairly heavy nuclei, not individual protons, but similar considerations would seem to apply. A nucleus can absorb an electron and convert a proton into a neutron, and in this context the process is called electron capture. Ordinarily this process will only occur if the nucleus is neutron-deficient; once it reaches a neutronto-proton ratio that optimizes its binding energy, neutron capture cannot proceed without a source of energy to make the reaction go. In the environment of a white dwarf, however, there is such a source. The annihilation of an electron opens up a hole in the “Fermi sea.” There is now an state into which another electron is allowed to drop without violating the exclusion principle, and the effect cascades upward. In a star with a mass above the Chandrasekhar limit, this process runs to completion, with every proton being converted into a neutron. The result is a neutron star , which is essentially an atomic nucleus (with Z = 0) with the mass of a star!
Observational evidence for the existence of neutron stars came in 1967 with the detection by Bell and Hewish at Cambridge of a mysterious radio signal with a period of 1.3373011 seconds. The signal’s observability was synchronized with the rotation of the earth relative to the stars, rather than with legal clock time or the earth’s rotation relative to the sun. This led to the conclusion that its origin was in space rather than on earth, and Bell and Hewish originally dubbed it LGM-1 for “little green men.” The discovery of a second signal, from a different direction in the sky, convinced them that it was not actually an artificial signal being generated by aliens. Bell published the observation as an appendix to her PhD thesis, and it was soon interpreted as a signal from a neutron star. Neutron stars can be highly magnetized, and because of this magnetization they may emit a directional beam of electromagnetic radiation that sweeps across the sky once per rotational period — the “lighthouse effect.” If the earth lies in the plane of the beam, a periodic signal can be detected, and the star is referred to as a pulsar. It is fairly easy to see that the short period of rotation makes it difficult to explain a pulsar as any kind of less exotic rotating object. In the approximation of Newtonian mechanics, a spherical body of density \(\rho\), rotating with a period \(T = \sqrt{\frac{3 \pi}{G \rho}}\), has zero apparent gravity at its equator, since gravity is just strong enough to accelerate an object so that it follows a circular trajectory above a fixed point on the surface ( problem 14 ). In reality, astronomical bodies of planetary size and greater are held together by their own gravity, so we have \(T \gtrsim \frac{1}{\sqrt{G \rho}}\) for any body that does not fly apart spontaneously due to its own rotation. In the case of the Bell-Hewish pulsar, this implies \(\rho \gtrsim 10^{10}\) kg/m 3 , which is far larger than the density of normal matter, and also 10-100 times greater than the typical density of a white dwarf near the Chandrasekhar limit.
An upper limit on the mass of a neutron star can be found in a manner entirely analogous to the calculation of the Chandrasekhar limit. The only difference is that the mass of a neutron is much greater than the mass of an electron, and the neutrons are the only particles present, so there is no factor of K. Assuming the more precise result of 1.4M \(\odot\) for the Chandrasekhar limit rather than our sloppy one, and ignoring the interaction of the neutrons via the strong nuclear force, we can infer an upper limit on the mass of a neutron star:
\[1.4 M_{\odot} \left(\dfrac{Km_{e}}{m_{n}}\right)^{2} \approx 5 M_{\odot}\]
The theoretical uncertainties in such an estimate are fairly large. Tolman, Oppenheimer, and Volkoff originally estimated it in 1939 as 0.7M \(\odot\) , whereas modern estimates are more in the range of 1.5 to 3M \(\odot\) . These are significantly lower than our crude estimate of 5M \(\odot\) , mainly because the attractive nature of the strong nuclear force tends to pull the star toward collapse. Unambiguous results are presently impossible because of uncertainties in extrapolating the behavior of the strong force from the regime of ordinary nuclei, where it has been relatively well parametrized, into the exotic environment of a neutron star, where the density is significantly different and no protons are present. There are a variety of effects that may be difficult to anticipate or to calculate. For example, Brown and Bethe found in 1994 14 that it might be possible for the mass limit to be drastically revised because of the process e − → K − +\(\nu_{e}\), which is impossible in free space due to conservation of energy, but might be possible in a neutron star. Observationally, nearly all neutron stars seem to lie in a surprisingly small range of mass, between 1.3 and 1.45M \(\odot\) , but in 2010 a neutron star with a mass of 1.97 ± .04 M \(\odot\) was discovered, ruling out most neutron-star models that included exotic matter. 15
For stars with masses above the Tolman-Oppenheimer-Volkoff limit, theoretical predictions become even more speculative. A variety of bizarre objects has been proposed, including black stars, gravastars, quark stars, boson stars, Q-balls, and electroweak stars.
It seems likely, however, both on theoretical and observational grounds, that objects with masses of about 3 to 20 solar masses end up as black holes; see section 6.3 .
References
13 arxiv.org/abs/hep-ph/9703240
14 H.A. Bethe and G.E. Brown, “Observational constraints on the maximum neutron star mass,” Astrophys. J. 445 (1995) L129. G.E. Brown and H.A. Bethe, “A Scenario for a Large Number of Low-Mass Black Holes in the Galaxy,” Astrophys. J. 423 (1994) 659. Both papers are available at adsabs.harvard.edu.
15 Demorest et al., arxiv.org/abs/1010.5788v1.