Critical angle and total internal reflection

We have already seen that when a ray of light travels from air into glass its direction is changed (have a look at this post for a refresher): The angle between it and the normal line ( the line at right angles to the boundary between the two substances) gets smaller in the denser medium. When the light goes back, the opposite happens: The angle gets bigger. 

Above: A light ray entering something denser.

Above: A light ray going from something dense to something less dense.

That leads to something slightly odd: If the ray's angle to the normal line is already pretty big inside the glass, it's possible for it to become bigger than 90 degrees as it moves into the air.
Which looks like this:

 Above: Video courtesy fo QuantumBoffin.

Even though the surface isn't a mirror, the light cannot get out of the glass! In fact you might well have seen this if you've ever swum underwater:

Above: Look at the surface of the water. It's reflective, like an upside down mirror. But the world above hasn't been mirrored off, the camera is just being held at a viewing angle where total internal refraction occurs.

This effect is called 'total internal refraction', and the maximum angle to the normal line that an incoming ray can have, before it undergoes total internal refraction, is called the 'critical angle'. To find the critical angle we use this equation:

Where n1 is the refractive index of the material the ray is entering, and n2 is the refractive index of the material the ray is leaving.


Uses:
The use for total internal reflection you're probably most familiar with is the 'cat's eyes' on the road*. Have you ever noticed how real cats eyes reflect light, and seem to glow from the inside?

Like this: Cute, yet ever so creepy. You just know they're planning on eating you.

This effect inspired Percy Shaw to build an artificial version, for marking the edges and middle of roads. An actual cat has a layer of reflective cells at the back of the eyeball, that let it 're-use' incoming light to see better, but a lot of modern versions use a transparent block with lots of tiny cube shaped facets embedded into one face. In this type of reflector, light enters the smooth face, and then bounces around the back surface as a result of a total internal reflection - like this:

Above: Video courtesy of Thorlabs.

Another use is in tracking criminals, by their footprints!
 
Revision questions:

1:
What is the critical angle for a ray of light passing from inside a body of water to the empty air above? Water has a refractive index of 1.3, and we'll assume air has an index of 1.

2:
How does the critical angle change if a sheet of glass with refractive index 1.5 is placed on top of the water?

Answers here.

*If I were a bad man I'd make a roadkill joke there, but I'm not so... whoops.

Things that waves do, part 2: Diffraction and interference

We've looked at reflection and refraction.... so what other things do waves do? Well there are two important ones.

Interference:
We covered many of the ideas behind this before, in this post, but to recap: Remember how every wave is a moving oscillation in whatever medium it's moving through? When two waves meet, or pass through each other, they add together. So, at every point along the two waves where they meet, the amplitude is the amplitude of one wave plus the other wave.

That's nice and simple - just remember that where the wave drops below the X axis it has a negative sign - and a negative added to a positive actually subtracts from it. This means that we can get waves interfereing destructively (decreasing each other's amplitude) like this.... 

Above: Destructive interference.

 

....or we can get waves interfering constructively (increasing each other's amplitude), like this....

Above: Constructive interference.

When two waves are coherent (see this post for an explanation of coherence and phase), and have zero phase difference, they have the same sign along their whole length so you get the most increase in amplitude. When two waves are 180 degrees out of phase (so their peaks align with their troughs) you get the most reduction in amplitude.
Once we extend this idea to waves on a 2D surface, or  a 3D volume, we will start to see more complicated patterns arising - and we'll look at that after the diffraction section below - but it all stems from these basic principles.


Diffraction:
The formal definition of diffraction is:

"The process by which a beam of light or other system of waves is spread out as a result of passing through a narrow aperture or across an edge."

This is a fairly counter intuitive property of waves, and a fairly important one. Imagine we've got an ocean wave passing through a small gap. If the gap is bigger than the length of the wave then the wave passes through fairly unaffected...

But if the gap is smaller than the wavelength than the waves doesn't pass through cleanly, it sploshes through....

... and on the far side of the barrier the 'splosh' acts like water falling straight down: It produces a new set of circular waves. The effect is called diffraction, and it actually happens in all types of waves. The 'how' of a light wave going 'splosh' is a bit complicated - in fact for non-ocean waves the 'splosh' is just a good metaphor, but the diffraction effect still happens anyway! 
In fact the incoming waves don't need to 'splosh' very hard to produce this effect, they just need to have a wavelength that is big compared to the size of the gap they're trying to get through -  so even a big gap can cause diffraction if the waves passing through are also big. The Open University has a nice demonstration that uses a water tank:

 

The smaller the gap is compared to the wavelength, the closer to a semicircle the diffracted waves get. But some diffraction happens even when the slit is very large, as shown below.... 

 

 

 

That's important to get, because it will help with the next part about diffraction: Diffraction can happen even when the 'slit' only has one edge. If we go back to ocean waves we can actually see how this isn't ridiculous - watch a few seconds of this video, showing storm waves hitting a section of sea wall:

 

 

See how the waves 'splosh' off the edge of the sea wall, creating circular patterns of backwash?  Something similar happens to any wave hitting an object with an edge: The wave diffracts off the edge, now going in a different direction, and this lets the wave reach areas it couldn't get to just travelling in a straight line. And, just like all kinds of diffraction, longer wavelengths diffract more than shorter ones.

 

A good example is a person with  two radios, one shortwave and one long wave, trying to send a signal to the other side of a very tall wall. The short wavelength radio waves only diffract slightly, and so cannot reach the receiver:

 

 



 

...But longer wavelengths diffract much more, and do reach the receiver....

 

 

 

...which is why you can usually get your favourite radio station, even when there're houses or even hills in the way.

Just before we move on, it's important to point out (because it's sometimes an exam question) that no property of a diffracted wave is changed by the process, other than it's direction.

The last thing we need to know, that makes diffraction very interesting (and ultimately leads to the weirdest area of physics, quantum mechanics), is that it only occurs with waves. That makes it a good test for whether or not something is a wave, and early scientists used an experiment that combined diffraction and interference to test whether light is wave. The experiment was set up like this:

 

 

 ....if you look at the white screen you can see that at certain points the light waves interfere constructively, and at others they'll interfere destructively - so on the screen, if light is a wave, we should see some areas that are dark, and some areas that are light. And, if you actually do the experiment, you'll see something like this:

 

 

That's definitely a pattern of light and dark, which the old times scientists took as proof that light is a wave.for a hit more on exactly how their ingenious experiment was done with their simple technology, check out the video below. 

 

Radiation and half life crib sheet (and practice questions).

Half-life and radiation factsheet:

General information:

Radioactivity happens because:

1.       Some atoms are unstable, and break apart or loose smaller particles to become more stable atoms – this called the ‘activity’.

2.       Activity is measured Becquerel’s.

3.       When atoms decay they give off radiation.

4.       The amount of time for half the radioactive atoms to decay into stable atoms is called the half life.

5.       Radiation ionises atoms ((knocks off an electron from them) which is damaging to DNA.

Types of radiation are:

Radiation	How penetrating?	What is it?	What’s its charge?	What’s a use?	Weighting factor
Alpha particle	Will be stopped by a few cm of air, or skin. Least penetrating.	Two protons and two neutrons bound together	+2	Used in smoke detectors	20
Beta  particle	Will go through air and thin metal, like aluminium, but not thick or very dense metal like lead.	An electron	-1	Measuring  the thickness of paper in industry	1
Gamma ray	Will go through thick metal sheets, concrete, or rock. The most penetrating.	A very high energy photon (an electromagnetic wave, like light or radio waves)	Neutral	Radiotherapy, to kill cancer cells	1

Natural sources of radiation are everywhere, most are natural radioactive atoms in the air (like radon gas), the rocks and soil (like Uranium ore), or dissolved in water. Some radiation comes from stars or other objects in space, and is called cosmic radiation.

Equations:

Activity:

Activity is defined as ‘The number of atoms to decay per second): the equation for this is:

 
Where A is activity in Becquerels, N is the number of atoms that decayed, and  t is the time in seconds their decay took.

Doses of radiation:

Radiation is measured in two ways:

Absorbed dose, which is measured in Greys (or more often milligreys). The equation for Absorbed dose :

Where D is absorbed dose in Greys, E is the amount of energy the radiation carried in Joules, and M is the mass of the thing absorbing the radiation in kg

Equivalent dose takes into account how damaging  each kind of radiation is, by multiplying the absorbed dose by a weighting factor (a unitless number picked by observing how much damage each type does). It’s equation is:

Where H is the equivalent dose in, W is the weighting factor, D is the absorbed  dose in Greys. Equivalent dose’s  unit is Sieverts.

Half-life and radioactivity practice questions:

 

1: Three kinds of radioactive source give off three different kinds of radiation.

·         Source A gives off radiation that goes through  aluminium sheet but is stopped by  a lead brick

·         Source B gives off radiation that is stopped  by aluminium sheet, but travels through several centimetres of air

·         Source C gives off radiation that is stopped by several centimetre of air.

Which source is giving of which type of radiation?

2: A chunk of radioactive material has a half-life of twenty three seconds. 

a)    How many whole half-lives does it go through before its activity has dropped to 6.25% of the original value? Hint:  1 whole half-life takes the activity to 50%, 2 half-lives takes the activity to  25%, and so on – you can draw a table of  half-lives vs % of activity if it helps.

b)    How many seconds before the activity level drops to 25%?

3: A 4 Kg cat gets hit with 50 joules of radiation.

a)    What is its absorbed dose?

b)    The radiation is alpha particles, with a weighting factor of 20. What is the cat’s equivalent dose?

4:  A radioactive source is made of 10,000,000 radioactive atoms. After 2 seconds 1,000,000 have decayed. What’s its activity in Becquerel’s?

Answers here

Things that waves do, part 1: Reflection and refraction

Right now we're looking at waves, and how studying waves ties into, well, everything. Before we move onto things like resonance, which will let us collapse bridges* (and sometimes resonance does collapse them, just by accident).....


 
.....lets have  quick recap on the main behaviours waves have. In this post we'll look at the two best known:

Reflection:
Waves bounce off things, and we call it reflection... but there's a little bit more to it than that: In two dimensions, or three, waves follow a law of "angle of incidence equals angle of reflection" (the law of reflection). To make it easier to see the law of reflection in diagrams we often use ray diagrams (explained in this post). Drawn with a ray diagram the law of reflection looks like this:

 
When doing calculations we will almost always use ray diagrams, but it's important to remember that light actually moves in waves. A good way of visualising reflection with waves is using a ripple tank with a light behind - so the shadows of the waves are projected onto a screen. Sainnt Mary's University show this nicely: 

When we combine reflection with phase and interference (see this post) we can get some unexpected phenomena, such as standing waves.

Refraction:

Let's go back to our equation that relates wave speed, frequency and wavelength (from this post):

...where V is wavespeed, f is frequency, lambada is wavelength.

One of the things that happens when a relationship binds quantities like these together is that when one of them changes, at least one of the others needs to change to maintain balance in the relationship.
And guess what? In refraction they change: When a wave passes from one medium to a different medium its speed changes - like a cats speed would change if swam from water into treacle.

You're planning me to do what?!

The speed changes but the frequency does not (otherwise the light would change colour).  Because it's speed is different but not its frequency the wavelength must change.That doesn't cause any complications when the wave hits the edge of the new medium  straight on....

 

...but when it comes in at an angle this means that the wave fronts would be broken.....

...unless the angle of the waves changes. And, because the wave fronts don't like to be broken, the angle of them does change: 

To make things a bit simpler we'll draw this out as a ray diagram (see here for a quick run down on ray diagrams)

There is a law that predicts how the angle of the rays change, called Snells' law. It uses a property that every medium has, called 'refractive index'. Refractive index determines how light moves through a medium. For example, empty space has a refractive index of 1, so light moves through it easiest, air as an index of 1.000277 and most glass has an index of around 1.5.
To use Snell's law we need to draw a line (called a 'normal' line) that is at right angles to the surface the rays are passing through, and measure the angles the incoming and refracted rays make to it. When we've done that, Snell's law is this:

 

Where:

Angle 1 = angle between the incoming rays and the line that is at right angle to the surface
Angle 2 is the angle between the refracted rays and the line that is at right angle to the surface
n₁ = index of refraction for the material the light starts in
n₂ = index of refraction of the medium the light ends up in

So, as well as direction, refraction changes wavelength and speed. But by how much? There're related equations that let us work out the change in wavelength and wave speed, using the angles the waves make to the normal line:

...Where:

V = wavespeed

Lambada = wavelength

Angle 1 = angle between the incoming rays and the line that is at right angle to the surface
Angle 2 is the angle between the refracted rays and the line that is at right angle to the surface.

Practice Questions:

 

1: A ray strikes a reflecting surface at a 40 degree angle to that surface. What angle does the reflected ray make with the surface?

 

2: A ray passes through vacuum, then into a block of glass with a refractive index of 1.4. It strikes the glass at an angle of 45 degrees to the normal. What angle does it make with the normal beneath the surface of the glass?

 

3: The light from question 2 has a wavelength of 560 nanometers in the vacuum.

     a) What is its wavelength inside the glass?

     b)What is its speed inside the glass?

* I've mentioned before that a Physicist should have a touch of supervillian.

Introducing ray diagrams....

Before we go any further we need to introduce the concept of ray diagrams. Don't worry, this is fairly painless!
In other posts we've just run with the idea that light travels in waves. And that's true.... but if we're trying to calculate the path light takes, especially when we get to things like diffraction and refraction, accurately drawing waves themselves quickly gets difficult. To simplify this we use ray diagrams: These aren't meant to indicate that light is made up of rays, it's just a convenient way of showing the direction of the waves. 

To turn a wave diagram into a ray diagram is simple: Draw a line at right angles from the line of the wave, pointing in the direction the waves is going. Like so:

This is the 'ray' associated with that wave. If the the wave front is very broad, add some more. If the wave front is curved draw your ray at right angles to the tangent of the wave, and do it at several points, like so:

...which gives you a ray diagram like this:

...and that's all a ray diagram is, a way of showing the direction of motion of the waves!

Phase and Coherence...and a seasick cat.

Up to now we've established that waves are oscillations moving through a medium (this post), that they have properties we can measure, like wave speed and frequency (this post), and that when two waves meet at a point they can interact constructively (reinforcing each othe), or destructively, (cancelling each other out, this post). We've also established that cats are integral to physics, although not always in ways cats would like.

That's all true, but it's a bit of an oversimplification: Two waves don't have to be in exactly the right position to reinforce each other (all the troughs lined up), or in exactly the right position to cancel each other out completely (troughs lined up with peaks). In fact it's unlikely that they will exactly match up either way, in a real situation.
To get to grips with waves interacting in more realistic ways like this (by realistic I mean awkward and complicated) we need to introduce a new wave property, phase, and a new property of two interacting waves coherence.

Lets look at coherence first: Imagine we have  two long trains of waves that are superimposed over each other:

Above: Two superimposed waves -  we'll come back to them in a bit.

Both trains of waves have similar amplitudes, the same frequencies, and the same wavelengths. Yet they are clearly "out-of-step" with each other - the peaks and troughs don't line up. When waves with the same frequency and wavelength (and similar amplitudes) don't 'line up' we say they are coherent - they are the same except that their peaks and troughs do not coincide.

The difference between the waves in the above situation is termed a phase difference between the two waves. Each wave, A and B, has a 'phase' at every point on the graph. Because they're out of step, their phases are different. 

But what is phase? Phase is basically a measure of how far through it's cycle the wave is at any given point.... well, it's easier to show you. Here's a wave:

Now we pick a point on the wave, and to make it stand out we put a cat on it. Because, well, why not?

As the wave moves, that cat moves up and down, in a cycle:

The point the cat is riding moves up and down in a cycle. The phase of the wave at that point can be thought of as: 'How far through it's cycle the cat is'. 

OK, fine (aside from a sea sick moggy). But to use this new quantity we need to be able to put a number to it. And that turns out to be a bit tricky: How far up or down the cat is won't work, because two cats on different waves could be at the exact same height, but one moving up and the other down - so at very different points on their cycles. 

Above: These two cats are at the same 'height' (displacemnt from the X-axis technically), but at different points of their up/down cycle, hence one is moving up and one is moving down - so we can't use X-axis displacemnet to measure phase.

Velocity doesn't work well either, or acceleration: It's hard to put a simple number to how far through it's up-down cycle the cat is, because the speed that a point is propelled up then down by a wave isn't constant, nor is it's acceleration either: Any point with a wave passing through it moves fast, slows down near the peak of the wave, then reverses direction and moves the other way while speeding up. 

But there is a way we can get a number that changes at a steady rate out of this, and so have useful way of measuring phase: For a simple sine wave the variation in the cats vertical velocity actually matches the variation in vertical component of the velocity of a point on a spinning wheel - like this:

So, using this imaginary wheel, we can attach an angle  to each position of the point the cat is riding on it's cycle. So, as the whole wave is made of points going up and down in this way, every point on a wave has a phase that can be expressed as an angle:

We can go a step further with this idea, and plot the angles onto the x - axis, so now the wave is just defined by it's amplitude (maximum height off the X axis) and it's phase:

With this way of measuring the phase of a wave (it's not very intuitive I know, but it makes the maths simpler in the long run and trust me that makes this worth the effort) we can also measure phase difference.

Phase difference:
What if a point has two waves passing through it? If you have two waves passing through a point then the 'phase difference ' is the difference in phases of the two waves at that point. So if there were just wave 'A' present the cat would be here.....

Above: Wave A.

....and if there were just wave B present the cat would be here.

Above: Wave B

Now we can add our imaginary phase wheel in. At any point where they meet two out of phase waves will be at different angles around the rim of the imaginary wheel. The difference between the two angles is the phase difference.


Let's go back to the two waves we saw earlier:

Above: The two wavetrains form the start, because I'm too lazy to draw them again.

Now lets add in a point where we'll measure both their phases, and the wheel for measuring phase:

What's the phase difference between the two waves? It's the angle between the two angles on the wheel. Or, if we were drawing the angles onto the X-axis directly, it's the separation in degrees between a point on wave 1, and its corresponding point on wave 2 (a peak on wave one and the nearest peak in wave 2, for example). If the waves are coherent (so they are identical apart from their phase difference) then their imaginary wheels spin at the same speeds, and the phase difference between them is constant along their entire lengths.

Two more terms you'll need: If two waves have no phase difference (so the peaks align) then they are said to be 'in phase'. If they're exactly 180 degrees out of phase (so each peak perfectly aligns with a trough) they're said to be in 'anti-phase'.

So, to sum up:

• If two waves are coherent they are of identical frequency, similar amplitude, but have a phase difference between them.

• Phase is a measure of how far through it's cycle of oscillation a point on the wave is.

• Phase is measured as an angle, by comparing the oscillation of the wave to a point on a wheel that spins once for every complete wave oscillation. 

• Where two waves meet the difference between their phases is called the phase difference between them.

• Coherent waves will have the same phase difference along their whole length. 

Revision questions:

Answers here



1) Just by looking:

a) Which of these waves are likely to be coherent with each other?

b) Which of these waves are likely to be in phase (no phase difference) with each other?


2)

a) These two coherent waves are out of phase, but by how much?

b) How much will they be out of phase by 100 wavelengths further down the graph?

3)

How many degrees of phase are between the points A and B on this wave?

Next time, we'll look at what happens when we combine the ideas of phase and interference....