Muons, Pions and Eminem

The Muon Decay Experiment

What you need to know about Muons:

Muons are leptons - a type of subatomic particle. They are created created in the atmosphere, 10 km from the surface of the Earth. The cosmic rays from the sun hit the atmosphere and can create muons that travel up to 0.99c.

Muons have an average lifetime of 2.2 x 10^-6 seconds. This is measured in the frame of reference of the muon (so it is proper time)

So the maximum distance a muon can travel is 2.2 x 10^-6 x 3 x 10⁸ = 660m

Awwwww, isn't it cute?

The thing is, eventhough the muons are not supposed to reach anywhere near Earth, we are still able to detect them. (using the cute-o-meter...joke)

So, remember that anything moving at the speed of light (in this case it s 0.99c) will experience the effects of special relativity.

A muon moving at 0.99c has a Lorentz factor of 7.1.

In terms of time dilation:

From the muon's frame of reference: the muon measures proper time. 2.2 x 10^-6 seconds.

From the Earth observer's frame of reference: the muons move towards the stationary observers at 0.99c and is thus time dilated to 7.1 x 2.2 x 10^-6 seconds = 1.562 x 10^-7 seconds

Remember that time dilation is concered with longer time periods while length contraction is concerned with shorter distances.

 In terms of length contraction:

From the muon's frame of reference: everything around the muon will now appear shorter (but not the muon - imagine sitting on the muon like a train, everything on the train stays the same) due to length contraction by a factor of 7.1.



Mr. Lorentz likes to dilate time

Firstly, watch my video on deriving the time dilation equation 

(Thanks for the birthday coloured pens Rynda, Rita, Manini and Tanya! I'm putting them to good use =D)

Speaking of stationary and moving bulbs...this is what happens when you move a bulb into a microwave.

Anyway, what is important is the Lorentz function. Which is linked to the time dilation equation.

If you graph it, the Lorentz equation looks like this.

And some solutions from today's work.

Time Dilation

Makes me think of pupils.

Anyway, learn that a light clock is a device which uses a beam of light reflected between two mirrors to measure time. This is what you see below:



Now lets tackle Time Dilation.

So Jack is at rest.

Set Jill to move at 0.3 the speed of light c and click "play".

Notice that for Jack, the photon travels up and down in a straight line...nothing complicated.

Jill is moving away from Jack from Jack's perspective (we would say from Jack's frame of reference). Notice that Jill's photon travels a further distance as seen by the traingle shape path.

The description on the top right hand side is wrong...this is the correct interpretation

From Jack's frame of reference: Jack measures 10.0 seconds on the clock he is holding. When he looks at Jill's clock (so he takes out some binoculars to see Jill's clock which is very far away), he sees the clock read 10.4 seconds. This is time dilation.

From Jill's frame of reference: the ship, the floor, the walls and everything else around her are stationary while Jack appears to be moving away. From Jill's frame of reference, she will see 10 seconds on her clock and when peering through binoculars to Jack's clock, will see 10.4 seconds.

This is like the man and boat example. For the man on the boat, a ball dropped from the mast of the ship will follow a straight path downwards. For the external observer, the ball appears to follow a parabolic path as the boat is moving. Identical situation but different conclusions from different frames of reference. (This is Simultaniety...the same 

Time stretches and contracts because it is the speed of light which ios constant.

Inertial Frames of Reference and very Special Relativity

People always tell you to consider things from different perspectives because the same thing can look different to different observers.

Einstein proved this mathematically with Special Relativity! Isn't physics awesome?

Okay so first, learn that an inertial frame of reference is one in which Newton's Law of Motion apply. The two conditions neccesary for an intertial referece are constant speed or zero speed. An inertial frame of reference has no acceleration.

Dropping a ball to measure g in a static lab yields the same result as doing the experiment on a train moving at constant speed.

If speed is changing - this means there is accelration. Accelerated frames of reference are non inertial frames. In circular motion, velocity is changing and the body is accelerated towards the center, so a rotating frame is a good example for a non inertial frame of reference. Although the speed is the same, velocity is changing (due to its vector nature).

Lets build on that. The Two Postulates of Relativity state that:

1.) The Law of Physics are the same in all inertial frames of reference. If you are on a car at 30km/h, you can assert that you are travelling at 30km/h relative to the ground (so the ground is the inertial frame of reference). However you would be stationary relative to the car (this time the car is the inertial frame of reference).

2.) The speed of light is the same to all observers. The speed of light is measured relative to the observer and is a constant defined by Maxwell's equation.

Simultaneity can be explained using Special Relativity. If two events happen simultaneously, they are called simultaneous events.

Watch video:



The video is trying to say that two event CANNOT be simulateous in the two inertial frames of reference. The lady saw the flashes happen one before the other while the external observer saw both happen at the same time, the two observations are correct according to special relativity.

IB questions have the same style to simultaneous questions. You get marks for saying:

1.) No simultaneity can occur between the two inertial frames of reference. So if A sees something happening at the same time, then B will see them happening one before the other and vice versa.
2.) Speed of light is constant in all frames of reference
3.) The tricky part is determining which event happens first.

Stuff I have forgotten!

Capacitance relates specifically to CCD's. The CCD relies on the photoelectric effect to liberate electrons.

To get the equation for Escape Speed. Equate KE with Work done using Newton's Universal law of gravitation and solve.

Albedo is the ratio of reflected radiation to incident radiation. It is REFLECTED radiation, questions will try to catch you out with emitted radiation. A body can either absorb or reflect light. High albedo (90%) means a lot of the light is reflected (e.g Snow).

Emissivity is the ratio of the of the energy radiated by a particular body/material to the energy radiated by a black body at the same temperature. 

Why Relativity is relatively hard

The important thing to realize is that the point of view of the oberver is called the frame of reference.

In the animation above, the ball would be falling horizontally from the frame of reference of the man on the boat.

If the frame of reference is the lady, the ball would follow a parabolic path.

Its an identical situation - but the motions are different depending on the frame of reference.

Another example: If we had a mosquito flying in a car, from the driver's frame of reference, the mosquito flies at "mosquito speed"

From a person on the sidewalk, the mosquito travels at "car speed + mosquito speed". Again, same situation but different speeds depending on the frame of reference.

So...Study Guide says "When a frame of reference is adopted,the position of the observer in that frame becomes the zero of position. If the frame is a moving on, the zero of position moves with observer"

To understand this situation consider this scenario:

The last point is crucial -> V (A relative to B) = V(A) - V(B)

Do not confuse this with HL Math vectors where AB = b - a

Musical Quasars

When house music forms a binary star system with cosmic lyrics...the result is a song of intergalactic proportions.

Cosmic Girl (Tom Belton Remix) by Jamiroquai. One of the best songs I have in my music library.

Listen to the lyrics...the physics makes sense!




Anyway the lyrics talk about Quasars:

Quasars or Quasistellar objects appear to be point like sources of light and radio waves that are very far away. We know this due to very large red shifts - they are at the limits of our observable universe. Because we still see/detect these stars, they must be emitting a load of energy for their size (10^40 W!).

We still have yet to understand how this energy is made, but some theoretical models rely on super massive blackholes. They think the energy results from whole stars "falling" into these black holes.

Cool! Would love to see one!

Weird galactic names and the fate of a star

Before we start, check out this link https://spreadsheets.google.com/ccc?hl=en&key=tALs8n-NiKGxCs3_FROKSkg&authkey=CICK0YwN&hl=en#gid=1

which contains explanations of the fate of stars depending on their masses.

What is interesting though is the markscheme way of explaining what happens:

What will get you the marks with questions involving Supernovas
1.) After star leaves the main sequence the fusion of hydrogen stops. The fusion of helium begins and heavier elements are fused in concentric shells around the core
2.) Fusion stops at Iron (Highest BE per nucleon) and the mass exceeds the Chandrasekhar limit. Therefore it does not form a white dwarf but instead...
3.) Forms a supernova
4.) The core then collapses into a neutron star

Some Astro-Boy definitions

1.) Chadrasekhar Limit = mass after which a white dwarf does not form. Equivalent to 1.4 solar masses.
(The heavy mass causes the star collapse into itself to form a neutron star)

2.) Openheimer-Volkoff Limit = a neutron star is stable up to this mass which is equivalent to 3 solar masses, beyond this point and it forms a blackhole.

Breakdancers and Pulsars

Puslars: All pulsars are neutron stars. A pulsar can be distinguished because the Earth is in its beam of radiation that sweeps across as the neutron star (its a pulsar now) rotates.

Why does this happen?
When the star was formed (we are talking about right at the beginning of the star's life), there was rotation in the dust cloud as it compressed together. This rotation is conserved due to the conservation of momentum.

As a star collapses, it spins faster and faster - in the same way a breakdancer does a faster headspin by pushing his legs up higher and bringing his arms together.

But neutron stars are way cooler because their magnetic field increases as they spin faster. Squish the Earth into a pea and the magnetic field lines would be very close. This accelerates charged particles which emit EM radiation (Radio to X ray) in a fine beam along the lines of the poles.

If these "beams" are not in line with the axis of rotation of the neutron star, the radation pulsates (aha! Puls-ar) like a lighhouse! If the light house sweeps across the Earth we can detect these radio waves or X rays!

What will you get you the marks in the Exam:

1.) Neutron star rotates rapidly and has a large magnetic field

2.) Charged particles accelerated due to large magnetic field

3.) Produces directed beams of EM radiation 

Astro-Examination Techniques

No I won't be talking about how Astro-Boy takes the IB:

Can't you see he has better things to do like fly on super cool rocket boosters? Gee. Obvious.

Anyway back on Earth, After doing some past paper questions...there are important "Exam style answers" that are worth remembering as they crop up multiple times in our IB Galaxy.

Remember these word for word...

1.) Luminosity: the total energy emitted by the star per second.

Alternatively: the power emitted by the star

2.) 2 factors affecting Luminosity: 1. Temperature 2. Surface Area

3.) Cepheid Variables have a periodically changing luminosity because:

-The outer surface expands and contracts periodically

-As luminosity is directionally proportional to surface area

-This means luminosity increases and decreases periodically

4.) Apparent Brightness: The incident power per unit area received on the surface of the Earth

5.) Apparent Magnitude: A measure of how bright an object appears from Earth 

(Good to know but not needed: it is a comparative scale)

6.) Absolute Magnitude: The apparent magnitude when observed at a distance of 10 parsecs

Alternatively: a measure of how bright the object appears from 10 parsecs away. (I recommend the first definition using apparent magnitude...easier to remember)

The fate of our Universe! And what you can do to help...

There are 3 things that can happen to our universe. We'll probably never live to see it, but it is still worth knowing.



Lets designate a few symbols here:
- The density of our universe is r

- The critical density is r0  
The critical density is the density of the universe that would lead to a flat universe. This is where the the size (or radius) of the universe increases towards one value. It would theoretically take an infinite amount of time to reach this value. (Hey Further Maths people...its like a converging series!)

- Omega is the ratio of these two values  W = r/r0
 a closed Universe has W > 1, 

In other words the density of our universe is greater than the critical density. Think of it like "because there is loads more mass...there will be greater gravitational attraction so the universe will collapse on itself"

 a flat Universe has W = 1 

This is the converging universe discussed above. It is 1 because we are using the critical density as a reference vaue.

 an open Universe has W < 1.

This is where the density of the universe is less than the critical density. Think of it like "because there is way less mass...the galaxies in the universe will just drift and float away because the attraction of gravity is too weak"

So if you are anxious over the fate of the universe, please visit us on the link http://www.stoptheexpansionoftheuniverse.org/. We are a non-governmental organization dedicated to stopping the expansion of the universe by all means. Donations are welcome.

Paypal is preferred.

Obler's Paradox

You know when you come with an awesome theory only to realize that something is not quite right, but then you don't really want to make it too obvious because then your work will be ignored. Yup, we've all done it before...and so did Newton.

So Newton - in a flash of inspiration - decides to assume:

1.) The universe is static
2.) The universe is infinite in size

Great. I might as well assume there was a cheeseburger at the center of the universe.

The Hertzsprung Russel Diagram

Scientists love to plot to graphs, yay...because its so much fun! So one day Mr. Hertzsprung and Mr. Russel decided to meet up for some tea. 

Mr. Hertzsprung: "Mr. Russel old friend, are you up for a game of putting?"

Mr. Russel: "Actually I was thinking about plotting luminosity and against surface temperature today"

Mr. Hertzsprung: "..."

So anyway, this is what they come up with.

Important things to notice:

1. On the bottom x axis: This can be surface temperature in Kelvins or Spectral Class. 

The surface temperature in Kelvins increases from right to left...opposite to normal convention. This is most likely because we are used to having blue on the left and right on the right to show the visible spectrum. 

If you look at the black body radiation graph 

We see that peak wavelength is inversely proportional to the temperature. Remember that the black body is emitting all wavelenghts, but there is only one peak wavelength at the specific temperature. 

Notice that blue is on the left, red is the on the right and that the temperature increases from right to left.

Spectral class is the classification of stars according to its temperature and chemical composition. If they have the same chemical composition = they will have similar absorption or emission spectra. If they have the same temperature = they will appear to the same colour because they have the same peak wavelength.

2. On the vertical axis we have luminosity or absolute magnitude.

Luminosity is defined by the total amount of energy emitted by the star per second (in watts as it is joules/sec).

Luminosity depends on the temperature of the star and its radius of surface area (in otherwords...its size). Here size and hotness does matter...lol.

Absolute magnitude is the luminosity the star would have if it was observed from 10 parsecs away. Why make this movement of the star? 

The stars in the observable sky are different distances from the sun. We need to move all the stars to the same distance to in order to make comparisons between them. 

3. The line running from the top left to the bottom right is the main sequence. Our sun is right now towards the bottom right of the main sequence.

White dwarfs are located towards the bottom left.

Giants are towards the top right.

Red Giants are located above the Giants.

There is always a 3 marker question in the exam on simply..."label the location of the white dwarfs...etc"

Astrophysics Notes

Stellar Clusters and Constellations


A stellar or globular cluster are a group of stars that are physically close to one another in space. They are created by the collapse of the same gas cloud. Stellar clusters can contain 100,000 to 1,000,000 in a region 30-100 light years across. The stellar clusters form as a result of gravitational attraction.

Remember that a Light Year (ly) is 9.4 x 10^15m. This is the distance travelled by light in one year.


The Astronomical unit is 1 AU =1.5 x 10^11m. This is the distance between the Sun and Earth.

A constellation is a pattern of starts formed as a result of human imagination (and boredom...lol).

Constellations are usually featured in films as part of a guy's attempt to impress a girl he is attracted to. First, the guy will point towards the sky, tracing out the stars forming one of the 88 constellations. He will then proceed to hold the girl's hand, helping her point towards the sky while moving closer to her. At this point, the two look into each other eyes, perhaps making a few romantic comments about how he or she like a shining star, while the camera pans towards the starry night sky.

Exactly, good call cameraman, we don't care about what happens to the couple, we're more interested in the astrophysics of the starry night.

Oh look! Its a shooting star!

Oh no wait its an asteroid.


Asteroids exist between the orbits of Jupiter and Mars and can range from dust particles to chunks of rock hundred of kilometers in radius.

Its usually featured in sci-fi films when the protagonist is escaping a fleet of evil ships:

Star Wars Episode 1

Well look what we have here: it goes up to like Star Wars Episode 6

ZOMG its even in Star Trek!

Okay so now the girl starts arguing with you and says that it can't be an asteroid because its too far from Earth (its between Jupiter and Mars...Many Virgins Eat Monkey - asteriods located here - Jelly So Uncles Nibble) 

She now explains that it is actually a comet, a small orbiting body that is made up of loose particles of ice and rock that are blown off by solar wind, forming a tail. 

Holy, she's smart. Is your arm still around her. Yes. Lol. 

Astrophysics connects people =D

Nuclear Energy Levels and Exponential Decay

Waving hello to the wave function

Okay, so I was making notes on Schrodingers model and looked at a book I read last summer about Quantum Mechanics called "The Quantum World" by Kenneth Ford.


After reading this book (again...lol), the IB textbook and the study guide...here is how I would explain the wave function. (My head really hurt after that lesson btw)


If it clears things up...awesome...this blog is doing its job!


If this makes it more confusing, close your web browser! But I recommend reading this very slowly...skimming will make everything more confusing.


Lets begin...


1.) First, please don't be scared with the next equation...


All you need to notice is the fact that Ψ exists in the equation and will have a value. Its like having d = s x t....where t has a value when you need to solve something.

Blah Blah Blah...All you need to take away is that Ψ is a variable quantity that can take up different values.

Okay. Move on.

2.) The textbook says (Read slowly) "The probability of finding the particle at any point in space within the atom is given by the square of the amplitude of the wave function at that point".

Right. The important thing to realize here is whenever the word "wave function" pops up or the symbol Ψ appears...you can visualize the wave function as a graph.

If I said a quantity (I'll call it y) is related to the sine function, all of you can immediately picture the sine graph and the "value"  that the quantity y can therefore take. So the peak of the sine graph is the maximum value of the quantity y.

However, the wave function is way more complicated than the sin graph because...

3.) The wave function looks different depending on the situation. The sine graph always looks the same. How the wave function looks like can change. 

What do I mean by this? Look at this graph:

The "wave function" looks like this for the lowest energy state of an electron in the hydrogen atom. 

However, in a higher energy level of the hydrogen atom...the "wave function" looks like this:

Get it get it? The "wave function" looks different depending on the situation. 

When you say "wave function"...it is a collection of graphs where each one applies to a different situation.

Look at how the "value" of the wave function (ie what you would read on the y axis at a certain distance from the atom) changes according to the distance from the atom.

4.) Read slowly. "The probability of finding the particle at any point in space within the atom is given by the square of the amplitude of the wave function at that point".

Now this statement makes more sense. 

If I want to find the probability of locating an electron say at 10^-11 m from the hydrogen atom, I would look for the value on the axis, go up the graph, read the y value (which is Ψ don't forget) and then square it.

There you have it...the number you have (say 0.11) is the probability of locating the electron at that point which is 10^-11 m from the hydrogen atom.

4.) Because the value of Ψ changes according to where you are from the atom...Ψ is a variable value as stated in the beginning.

In the text book you will read "At any instant in time, the wave function has different values at different points in space".

Booyah, just look at the 2 graphs...thats what the textbook is saying. 

-------------Now we will link the standing waves we learnt in class with the wave function.

5.) So why are there loads of different graphs for this one symbol Ψ ?

Notice that at higher energy levels, the wave function completes more oscillations.

VS

Guess what...a standing wave in a string has more oscillations at higher harmonics.

Ahhhhh...there must is some link between the two.

This is the link:

6.) For a standing wave to set up on a string, the boundary conditions are 2 nodes on either side. 

The reason why only certain frequencies can exist is because these specific frequencies satisfy the conditions necessary for a standing wave to form...2 nodes at either end. 

We all agree that more than 1 frequency can thus form a standing wave as long as 2 nodes can form at either end.

7.) Notice how the graph for the wave function (the hydrogen graphs) all reach zero a long distance from the atom. This makes sense, the electron is less likely to be found very very far away from the atom.

Well, in order for Schrodinger's equation to work properly (the long one above), the wave function must rapidly tend to zero (go towards zero) at very far distances from the atom. 

This is like a boundary condition. A valid wave function can only "exist" if Ψ approaches zero further away from the atom. There are only specific energies where the wave function approaches zero. These energies (or energy levels) are the ones we see in the textbook:

Other energies would make the wave function graph not reach zero, which is why that graph doesn't exist. Other energies actually make the wave function infinite as opposed to zero.

Think of this diagram:

as having a node at either end when it is reaching zero, thus a standing wave forms. If it was going to infinity (going up and and up), its like having an antinode...which will not form a standing wave.

See the link? 

In the same way that certain standing waves exist on a string because only certain frequencies satisfy the boundary conditions (and there are many of these frequencies known as harmonics)...only specific energies can produce a wave function that tends to zero and thus make the Schrodinger Equation work (and there are many of these energies and hence many valid wave function graphs).

Thats why we can use the electron in a box model to describe the nature of electrons under  the Schrodinger model. There are connections between standing waves and Schrodinger's model. The energy levels are still quantized like Bohr's model but are now described in terms of probability.

8.) So lets return back to "The probability of finding the particle at any point in space within the atom is given by the square of the amplitude of the wave function at that point".

We now know why there are many valid wave function graphs (in the same way there are many possible frequencies to form a standing wave on a string)

Related to the wave function is the probability (the statement above). This probability changes according to the wave function (because it is the wave function squared. duh) and exists in 3 dimensions. So the dips and peaks on the 2D graph can be mapped into a 3D electron cloud.

Aha!...look at the study guide.

A peak is the denser part of the cloud, a dip does is the less dense part of the cloud.

The shape of the cloud changes according to the wave function (think about the value of Ψ on the y axis as you move left and right), and the wave function in turn only exist for specific energies. So there are specific shapes to the electron clouds because there are specific wave functions related to them! 

That should make sense...read it again =)

The shape of the cloud are the shapes we have come across in HL Chemistry when we talk about s, p and d orbitals. 

s orbitals

p ortibal

d orbital

They look the way they are because they are the only possible shapes that can exist and satisfy the conditions of the wave function and Schrodinger's equation...they are solutions to Schrodinger's Equation!

COOL!

Thats why its not some random hexagon. 

I love physics =D