Friday Feb. 26, 2016

Eva Cassidy "American Tune" (4:09), Crooked Still "American Tune" (3:25), Black Prairie "Nowhere Massachusetts" (3:20), Punch Brothers "Sometimes" (4:57)

An In-class Optional Assignment was handed out today and collected at the end of class.  If you weren't in class and would like to do the assignment you can download using the link at left.  If you turn in the assignment at the beginning of class next Monday you will receive at least partial credit (as with all Optional Assignments you should have the assignment done before coming to class).


Energy transport by electromagnetic radiation
It's time to tackle electromagnetic (EM) radiation, the 4th and most important of the energy transport processes (it's the most important because it can transport energy through empty space (outer space)).



Many introductory textbooks depict EM radiation with a wavy line like shown above.  They don't usually explain what the wavy line represents.


The wavy line just connects the tips of a bunch of "electric field arrows". But what exactly are electric field arrows?

Static electricity and electric fields



To understand electric fields we need to first step back and review a couple of rules concerning static electricity.
That won't take too long, static electricity is something you're most likely already familiar with. 
Believe it or not there is even a National Static Electricity Day (Jan. 9). 

The static electricity rules are found at the top of p. 59 in the photocopied ClassNotes


Two electrical charges with the same polarity (two positive charges or two negative charges) push each other apart.  Opposite charges are attracted to each other.   Here are some pictures I found online.




This girl became charged with static electricity while jumping on a trampoline and illustrates the repulsive force of like charges.  Her hair and body are all charged up with charge of the same polarity.  The individual hairs are trying to get as far away from each other as they can.  This photo was a  National Geographic Magazine 2013 Photo Contest winner (source)

People's hair will sometimes stand on end under a thunderstorm.  That is a very dangerous situation to be in.

 A cat covered in Styrofoam "peanuts".  Here the cat and the "peanuts" have opposite charges and are attracted to each other. (source)

Being a cat owner I would worry about the cat swallowing one of the peanuts and possibly choking.





An electric field arrow (vector)
 just shows the direction and
gives you an idea of the strength
of the electrical force
that would be exerted on a positive charge at that position.
It's just like an arrow painted on a drive showing you what direction to drive.
Here are a couple of questions to test your understanding. 



First what polarity of charge must be on ground to cause the charges in the figure below to move as they are doing. 
Would the electric field arrow in the air just above the ground point UPWARD, point DOWNWARD, or would the electric field arrow be ZERO?

Here's a second perhaps somewhat harder question


What is the direction of the electric field arrow at Point X halfway between a + and a - charge.

You'll find answers to both questions at the end of today's notes.

Electromagnetic (EM) radiation
Now we'll use what we know about electric field arrows (electric field for short)  to start to understand electromagnetic radiation.  How is it able to carry energy from one place to another.  You'll find most of the following on p. 60 in the photocopied ClassNotes. 




We imagine turning on a source of EM radiation and then a very short time later we take a snapshot.  In that time the EM radiation has traveled to the right (at the speed of light).  The EM radiation is a wavy pattern of electric and magnetic field arrows.  We'll ignore the magnetic field arrows.  The E field arrows sometimes point up, sometimes down.  The pattern of electric field arrows repeats itself.
 
Note the + charge near the right side of the picture.  At the time this picture was taken the EM radiation exerts a fairly strong upward force on the + charge (we use the E field arrow at the location of the + charge to determine the direction and strength of the force exerted on the + charge).



This picture above was taken a short time after the first snapshot after the radiation had traveled a little further to the right.  The EM radiation now exerts a somewhat weaker downward force on the + charge.


A 3rd snapshot taken a short time later.  The + charge is now being pushed upward again. 
A movie of the + charge, rather than just a series of snapshots, would show the charge bobbing up and down much like a swimmer in the ocean would do as waves passed by.





Wavelength and frequency
The wavy pattern used to depict EM radiation can be described spatially (what you would see in a snapshot) in terms of its wavelength, the distance between identical points on the pattern. 


Or you can describe the radiation temporally using the frequency of oscillation (number of up and down cycles completed by an oscillating charge per second).  By temporally we mean you look at one particular fixed point and look at how things change with time. 

Wavelength, frequency, and energy




 EM radiation can be created when you cause a charge to move up and down. If you move a charge up and down slowly (upper left in the figure above) you would produce long wavelength radiation that would propagate out to the right at the speed of light.  If you move the charge up and down more rapidly you produce short wavelength radiation that propagates at the same speed.

Once the EM radiation encounters the charges at the right side of the figure above the EM radiation causes those charges to oscillate up and down.  In the case of the long wavelength radiation the charge at right oscillates slowly.  This is low frequency and low energy motion.  The short wavelength causes the charge at right to oscillate more rapidly - high frequency and high energy.

These three characteristics: long wavelength / low frequency / low energy go together. So do short wavelength / high frequency / high energy.  Note that the two different types of radiation both propagate at the same speed.

The following figure illustrates how energy can be transported from one place to another (even through empty space) in the form of electromagnetic (EM) radiation.




You add energy when you cause an electrical charge to move up and down and create the EM radiation (top left).

In the middle figure, the EM radiation that is produced then travels out to the right (it could be through empty space or through something like the atmosphere). 

Once the EM radiation encounters an electrical charge at another location (bottom right), the energy reappears as the radiation causes the charge to move.  Energy has been transported from left to right.


The electromagnetic spectrum
The EM spectrum is just a list of the different kinds of EM radiation.  A partial list is shown below.





In the top list, shortwave wavelength/high energy forms of EM radiation are on the left (gamma rays and X-rays for example).  Microwaves and radiowaves are longer wavelength/lower energy forms of EM radiation.

We will mostly be concerned with just ultraviolet light (UV), visible light (VIS), and infrared light (IR).  These are shown on an expanded scale below.  Note the micrometer (millionths of a meter) units used for wavelength for these kinds of light.  The visible portion of the spectrum falls between 0.4 and 0.7 micrometers.   UV and IR light are both invisible.  All of the vivid colors shown above are just EM radiation with slightly different wavelengths.  When you see all of these colors mixed together, you see white light.

I've tried to demonstrate colors mixing together to make white light using laser pointers.



But it's too hard to get them adjusted so that the small spots of colored light all fall on top of each other on the screen at the front of the room.  And even if you do the small spot of light is so small that it's hard to see clearly in a large classroom (you need to do the experiment on a piece of paper a few feet away).

Here's the basic idea, you mix red green and blue light together.  You see white light were the three colors overlap and mix in the center of the picture above.

Rules governing the emission of EM radiation
We spent most of the rest of the class learning about some rules governing the emission of electromagnetic radiation.  Here they are:

1.
Everything warmer than 0 K will emit EM radiation.  Everything in the classroom: the people, the furniture, the walls and the floor, even the air, are emitting EM radiation.  Often this radiation will be invisible so that we can't see it and weak enough that we can't feel it (or perhaps because it is always there we've grown accustomed to it and ignore it).  Both the amount and kind (wavelength) of the emitted radiation depend on the object's temperature.  In the classroom most everything has a temperature of around 300 K and we will see that means everything is emitting infrared (IR) radiation with a wavelength of about 10µm.

2.
The second rule allows you to determine the amount of EM radiation (radiant energy) an object will emit.  Don't worry about the units (though they're given in the figure below), you can think of this as amount, or rate, or intensity.  Don't worry about σ (the Greek character rho) either, it is just a constant.  The amount depends on temperature to the fourth power.  If the temperature of an object doubles the amount of energy emitted will increase by a factor of 2 to the 4th power (that's 2 x 2 x 2 x 2 = 16).  A hot object just doesn't emit a little more energy than a cold object it emits a lot more energy than a cold object.  This is illustrated in the following figure:



The cool object is emitting 2 arrows worth of energy.  This could be the earth at 300 K.  The warmer object is 2 times warmer, the earth heated to 600 K.  The earth then would emit 32 arrows (16 times more energy).

The earth has a temperature of 300 K.  The sun is 20 times hotter (6000 K).  Every square foot of the sun's surface will emit 204 (160,000) times more energy per second than a square foot of the earth's surface.

3.
The third rule tells you something about the kind of radiation emitted by an object.  We will see that objects usually emit radiation at many different wavelengths but not in equal amounts.  Objects emit more of one particular wavelength than any of the others.  This is called  λmax ("lambda max", lambda is the Greek character used to represent wavelength) and is the wavelength of maximum emission.  The third rule allows you to calculate  λmax. The tendency for warm objects to emit radiation at shorter wavelengths is shown below.




The cool object could be emitting infrared light (that would be the case for the earth at 300 K).  It might be emitting a little bit of red light that we could see.  That's the 2 arrows of energy that are colored red.  The warmer object will also emit IR light but also shorter wavelengths such as yellow, green, blue, and violet (maybe even some UV if it's hot enough).   Remember though when you start mixing different colors of visible light you get something that starts to look white.  The cool object might appear to glow red, the hotter object would be much brighter and would appear white.

Here's another way of understanding Stefan Boltzmann's law and Wien's Law (the graph below is on the bottom of p. 65 in the ClassNotes).


1.
Notice first that both and warm and the cold objects emit radiation over a range of wavelengths (the curves above are like quiz scores, not everyone gets the same score, there is a distribution of grades).  The warm object emits all the wavelengths the cooler object does plus lots of additional shorter wavelengths.

2.
The peak of each curve is λmax  the wavelength of peak emission (the object emits more of that particular wavelength than any other wavelength).  Note that λmax has shifted toward shorter wavelengths for the warmer object.  That is Wien's law in action.  The warmer object is emitting lots of types of short wavelength radiation that the colder object doesn't emit.

3.
The area under the curve is the total radiant energy emitted by the object.  The area under the warm object curve is much bigger than the area under the cold object curve.    This illustrates the fact that the warmer object emits a lot more radiant energy than the colder object.


It is relatively easy to see Stefan-Boltzmann's law and Wien's Law in action.  The class demonstration consisted of an ordinary 200 W tungsten bulb is connected to a dimmer switch (see p. 66 in the photocopied ClassNotes).  We'll be looking at the EM radiation emitted by the bulb filament.



The graph at the bottom of p. 66 has been split up into 3 parts and redrawn for improved clarity.
















We start with the bulb turned off (Setting 0).  The filament will be at room temperature which we will assume is around 300 K (remember that is a reasonable and easy to remember value for the average temperature of the earth's surface).  The bulb will be emitting radiation, it's shown on the top graph above.  The radiation is very weak so we can't feel it.  We can use Wien's Law to calculate the wavelength of peak emission,  λmax .  The wavelength of peak emission is 10 micrometers which is long wavelength, far IR radiation so we can't see it. 

Next we use the dimmer switch to just barely turn the bulb on (the temperature of the filament is now about 900 K).  The bulb wasn't very bright at all and had an orange color.  This is curve 1, the middle figure.  Note the far left end of the emission curve has moved left of the 0.7 micrometer mark - into the visible portion of the spectrum.  That is what you were able to see, just the small fraction of the radiation emitted by the bulb that is visible light (but just long wavelength red and orange light).  Most of the radiation emitted by the bulb is to the right of the 0.7 micrometer mark and is invisible IR radiation (it is strong enough now that you could feel it if you put your hand next to the bulb).

Finally we turn on the bulb completely (it is a 200 Watt bulb so it got pretty bright).  The filament temperature is now about 3000K.  The bulb is emitting a lot more visible light, all the colors, though not all in equal amounts.  The mixture of the colors produces a "warm white" light.  It is warm because it is a mixture that contains a lot more red, orange, and yellow than blue, green, and violet light.  It is interesting that most of the radiation emitted by the bulb is still in the IR portion of the spectrum (lambda max is 1 micrometer).  This is invisible light.  A tungsten bulb like this is not especially efficient, at least not as a source of visible light.

You were able to use one of the diffraction gratings handed out in class to separate the white light produced by the bulb into its separate colors.

When you looked at the bright white bulb filament through one of the diffraction gratings the colors were smeared out to the right and left as shown at left below.


You may need to rotate the slide 90 degrees to see the spectrum as shown above.


Here are the answers to the two electric field questions embedded earlier in the notes.

#1.  The ground can be either negatively or positively charged.  If the ground were negatively charged the positive charge would be attracted to the ground and the  negative  charge repelled and pushed upward.  That's not what is happening.  So the ground must be positively charged.


The positive charge is creating the force that causes the positive charge to move upward.  So that too must be direction that the electric field arrow is pointing.


#2.  To begin to answer the question we imagine placing a + charge at Point X.


The center charge will be repelled by the charge on the left and attracted to the charge on the right.  The center charge would move toward the right. 

The electric field arrow shows the direction of the force on the center charge.  Since we've determined the + charge will move to the right, that's the direction the electric field arrow should point.  The electric field arrow will point toward the right.