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archived as http://www.stealthskater.com/Documents/Standard_02.doc

(also …Standard_02.pdf) => doc pdf URL-doc URL-pdf

more Physics-related articles are on the /Science.htm page at doc pdf URL

note: because important websites are frequently "here today but gone tomorrow", the following was

archived from http://www.particleadventure.org/standard-model.html on March 21, 2019. This is

NOT an attempt to divert readers from the aforementioned website. Indeed, the reader should

only read this back-up copy if it cannot be found at the original author's site.

The Standard Model of Particle Physics

(for dummies)

Answer: The Greek thinker Empedocles first classified the fundamental elements as fire, air, earth, and

water (although our particular diagram reflects Aristotle's classification).

Did you know?

The ancient Chinese believed that the five basic components (in Pinyin, Wu Xing) of the physical

Universe were earth, wood, metal, fire, and water. And in India, the Samkhya-karikas by Ishvarakrsna

(c. 3rd

century AD) proclaims the 5 gross elements to be space, air, fire, water, and earth.

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Answer: The Greek root for the word atom ("atomon") means "that which cannot be

divided." But the entities that we call atoms are made from even more fundamental particles!

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Are protons and neutrons fundamental?

Physicists have discovered that protons and neutrons are composed of even smaller particles called

quarks. As far as we know, quarks are like points in geometry. They're not made up of anything else.

After extensively testing this theory, scientists now suspect that quarks and the electron (and a few

other things that we'll see in a minute) are fundamental.

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Quiz - Standard Model

Question: It is known that the 100s of particles are all made from how many fundamental particles?

Answer: 6 quarks, 6 leptons, 6 antiquarks, 6 antileptons, and the force carriers.

Trivia Question: In what year did physicists know for sure that there were more than just protons,

neutrons, electrons, and photons? 1901? 1936? 1976?

Answer: 1936! In the 1930s, physicists found muons. But hundreds more were found with high energy

accelerators in the 1960s and 1970s.

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Matter and Antimatter

For every type of matter particle we've found, there also exists a corresponding antimatter particle or

anti-particle.

Antiparticles look and behave just like their corresponding matter particles except that they have

opposite charges. For instance, a proton is electrically positive whereas an antiproton is electrically

negative. Gravity affects matter and antimatter the same way because gravity is not a charged property

and a matter particle has the same mass as its antiparticle.

When a matter particle and antimatter particle meet, they annihilate into pure energy!

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Bubble chambers were an important kind of particle detector from 1953 well into the 1970s. The

idea behind a bubble chamber is that when you shoot charged particles into a superheated liquid, the

particles will leave behind a track of bubbles. This makes it easy to track the particles and figure out

important things like their charge and mass. (A superheated liquid is made by lowering the pressure in

the chamber when the liquid is just below the boiling point.)

In order to find out things about particles, physicists measure their charge and momentum. To do this

they observe particle collisions in strong magnetic fields, because different kinds of particles behave

very differently in a magnetic field depending on their charge and their momentum.

For one thing, the signs of charged particles can easily be read from their paths

since they curve in opposite directions in the same magnetic field.

For another, the momenta of particles can be calculated easily because the path

of a particle with greater momentum bends less than that of one with lesser

momentum.

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Cork model/bad pun by Don Groom, Particle Data Group, LBNL.

Up

Down

Charm

Strange

Top

Bottom

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On March 2, 1995, Fermilab announced the discovery of the 'top' quark, the last of

the 6 predicted quarks. The search began in 1977 when physicists found the 5th

quark

('bottom') at Fermilab. It took this long because the 'top' quark was much more massive

than was originally imagined. So it required a more powerful accelerator to create it.

Although the 'top' quark decays too fast to be observed, it does leave behind particles

that give evidence of its existence - a 'top' quark "signature". The 'top' quark can decay

in more than one way. Since a 'top' quark appears only once in several billion collisions, it was

necessary to perform trillions of collisions.

Physicists still do not understand why the 'top' is so massive. It is 40 times heavier than the next

heaviest quark and about 35,000 times heavier than the 'up' and 'down' quarks that make up most of the

matter we see around us. In fact, the question still remains why particles have such different masses.

A weird thing about hadrons is that only a very very very small part of the mass of a hadron is due to

the quarks in it. For example, a proton (uud) has more mass than the sum of the masses of its quarks:

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Most of the mass we observe in a hadron comes from its kinetic and potential energy. These

energies are converted into the mass of the hadron as described by Einstein's equation that relates energy

and mass, E = mc2.

Answer: Even though "lepton" comes from the Greek for "small mass", the tau lepton is more than 3,000

times as massive as the electron. (Leptons were named after the tau was discovered.)

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(Animation is at => http://www.particleadventure.org/unseen.html )

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(Life is just a neat example of ElectroMagnetic force!)

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Quarks emit gluons

Color charge is always conserved. When a quark emits or absorbs a gluon, that quark's color must

change in order to conserve color charge. For example, suppose a red quark changes into a blue quark

and emits a red/anti-blue gluon (the image below illustrates antiblue as yellow). The net color is still

red. This is because after the emission of the gluon, the blue color of the quark cancels with the antiblue

color of the gluon. The remaining color then is the red color of the gluon.

Quarks emit and absorb gluons very frequently within a hadron. So there is no way to observe the

color of an individual quark. Within a hadron, though, the color of the 2 quarks exchanging a gluon will

change in a way that keeps the bound system in a color-neutral state.

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We know how to calculate gravitational forces. But we do not know how to integrate gravity into

the mathematics of the Quantum Theory of the Standard Model. (The fact that we have not seen the

graviton yet is not a surprise in the Standard Model because the graviton has extremely weak

interactions and so is rarely produced and rarely detected.)

In the same way that Isaac Newton's laws of mechanics were not wrong but needed to be extended

by Einstein to be more accurate about very high velocities, we need to extend the Standard Model with a

new theory that will explain Gravity thoroughly.

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While quarks have a fractional electric charge of 2/3 and 1/3 electron charges, they are only found in

composite particles that have an integral electric charge. You can never observe an isolated quark.

(Spin is the internal angular momentum of a particle, in units of ħ = 1.055 x 10-34

J s. This is Planck's

Constant.)

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Helium has a boson nucleus (2 neutrons and 2 protons). So it does not ever crystallize even when

cooled to almost Absolute Zero. It becomes a "superfluid" which is a liquid with strange properties such

as having zero viscosity and no surface tension. We will probably discover other strange properties of

atoms with boson nuclei in the future.

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JPEG PDF Order Chart in 3 sizes

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Answer: The computer monitor right in front of you (unless you have an LCD monitor)!

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(Animation at => http://www.particleadventure.org/accel_ani.html )

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(Animation at => http://www.particleadventure.org/circle.html )

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Answer: The conservation of momentum appears to be violated. But there were unseen neutrinos.

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(Animation at => http://www.particleadventure.org/measure.html )

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http://www.particleadventure.org/an-example-of-an-actual-event-with-a-possible-higgs-decay.html

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(Animation at: http://www.particleadventure.org/time-evolution-of-higgs-boson-data.html)

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[StealthSkater note: There are other GUTs such as Loop-Quantum Gravity that do not require

extra dimensions to unite General Relativity. In particular, Matti Pitkanen's Topological

GeometroDynamics not only unites GR and QM but also Consciousness as well as proposing a

different model for the 'Big Bang'; replacing zero-point energy with Zero-Energy Oncology;

and showing how dark matter-energy contributes to Quantum-Biology and remote-

viewing/ESP/Near-Death Experiences. The super-membranes in string theory are replaced by

magnetic flux tubes about 104 times bigger. See => doc pdf URL ]

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What is Decay?

The Standard Model explains why some particles decay into other particles.

In nuclear decay, an atomic nucleus can split into smaller nuclei. This makes sense. A bunch of

protons and neutrons divide into smaller bunches of protons and neutrons. But the decay of a

fundamental particle cannot mean splitting into its constituents because "fundamental" means it has no

constituents.

Here, particle decay refers to the transformation of a fundamental particle into other fundamental

particles. This type of decay is strange because the end products are not pieces of the starting particle

but rather totally new particles.

Nuclear Decay Particle Decay

In this section, we will discuss the types of decay, how they happen, and under what circumstances a

decay will or will not happen.

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(Animation at: http://www.particleadventure.org/decay_start.html)

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Confusion about decays

Many heavy elements decay into simpler things. But a close observation of these decays reveals

several confusing problems.

Consider Uranium-238 decay. A lump of Uranium-238 will decay at a constant rate such that in

4,460,000,000 years (give or take a few days), half the Uranium will be gone. But there is no way to tell

when a specific Uranium atom will decay. It could decay 5 minutes from now or in 10 billion years.

Why will an atom decay only according to some probability?

Uranium-238 has a mass of 238.0508 atomic mass units (u). It can decay into Thorium (234.0436 u)

and an alpha particle (4.0026 u). But Uranium's mass minus the mass of its decay products is 0.0046 u.

Why is there missing mass?

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[StealthSkater note: Einstein actually didn't believe in a Supreme Entity. He thought it was a

childish notion. When he made his famous dice statement, Niels Bohr replied: "Stop telling

God what to do!"]

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Animation at: http://www.particleadventure.org/top_pro.html

Answer: We ignored the color force field that develops as the b quark and b antiquark move apart. This

energy is converted into another quark/antiquark pair. Eventually only distinct color-neutral

particles emerge (B mesons). The same is true for the u quark and d antiquark. To see what really

happens, look at an analogous process in the picture of e+ and e

- --> D+ and D-.

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StealthSkater note: another introduction to the Standard Model (including mathematics) is at =>

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http://physicsforidiots.com/

Atoms and Nuclei

Atoms are one of the most stable forms of matter. They consist of a nucleus made up of protons and neutrons, which are

surrounded by a ‘cloud’ of electrons. A normal atom will have the same number of protons and electrons, making it

electrically neutral. It will then have a set number of neutrons than are needed to keep the nucleus stable.

If the atom loses or gains one-or-more electrons then it becomes an ion (i.e., a charged atom). If it gains or loses neutron

it becomes an isotope. Because the neutrons control the stability of the nucleus, it is common for isotopes to be unstable and

hence radioactive.

Building Hydrogen

If you want to build Hydrogen, you need one proton and one electron. The negative electron is bound around the

positive proton and you have an atom. The problem is that everyday experience tells us that the positive-negative attraction

would just pull them together until they were touching, leaving no stable structure to the Universe. But the experience we

have turn out to be just a large scale effect. At the Quantum level, things work differently.

In the Quantum Mechanics of atoms, the energies that the bound electrons can have can only take certain values in an

analogous way to the strings in a piano only being able to play certain notes. Theses energies in a basic atom are given

roughly by

where Z is the charge of the nucleus of the atom and n ranges from - to-infinity and at a basic level counts the different

energy levels. Each of these energies will have a corresponding orbital radius given by

So if we take our proton and add an electron to make Hydrogen, the lowest level it can go into is n=1. Which means the

electron will have to have an energy of -13.6 eV and orbit at a radius of 52.9 picometers. It can’t get any closer as there are

no lower energy levels or radii for it to be at.

This gives us our first step to structure. One hydrogen atom has to have a volume of around 620000 cubic picometers.

You can’t squash it any smaller. But what about other elements that have more than one electron?

What about Carbon? For Carbon, we have Z=6 and 6 electrons. So do they all go in the n=1 level and have energies of -

489 eV and orbit at 8.8 picometers? The answer is no.

Exclusion

The thing that gives atoms and all matter additional structure is the Pauli Exclusion Principle. If we have a system of

two particles A and B in two possible states 1 and 2, then the description of the system will be of the form

We now need to outline 1 important features of particles;

I. All particles of the same type are indistinguishable from each other.

II. Matter particles are antisymmetric.

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From point I, we don’t know which particle is in state 1 and which is in states 2. So the proper description of the system

should take into account both possible combinations. This is done in the following way:

From point II, we have to pick the minus condition. If we were dealing with force particles (which are symmetric), it

would be the plus. But for matter, the anti-symmetry means we have a minus sign. So now we have

This is an expression for a system containing 2 identical particles. What happens if we try to put them in the same state,

say φ1?

Therefore you cannot have 2 identical particles in the same state.

So if we go back to our case of Carbon, all 6 electrons are not allowed to just orbit at the same distance with the same

energy. We can add the first electron to the n=1, -489 eV/8.8 pm state. But to add the second electron, something will have

to be different. In this case it will be spin.

You may or may not know that electrons have a quantum property called 'spin'. It can either be up or down and provides

a way to get two electrons at the same energy and radius. One will be spinning up and the other down meaning the states will

be different. As soon as you’ve got the first two electrons on that state, it is then filled. And so you move on to n=2 with a

new energy and radius.

At this point, things get a lot more complicated and you get more-and-more ways of getting electrons into the same n

level with slightly different states. You have to include things like orbital angular momentum, magnetic effects, and Hund’s

rules. But electron by electron, you can build up the entire Periodic table.

Neutron Glue

For small atoms, you tend to find that the number of protons and neutrons in the nucleus are roughly the same. Once you

start to get past Calcium, you get more-and-more neutrons for each proton. This is because the neutrons feel the Strong

nuclear force but not the Electromagnetic force.

As the number of protons increases, the Electromagnetic force that’s trying to rip apart the nucleus increases. And so in

order to keep the nucleus stable, more neutrons are added so that the strong force is increased in line with the electromagnetic

repulsion.

Thermodynamics

The 4 Laws

There are 4 laws to Thermodynamics. They are some of the most important laws in all of Physics. The laws are as

follows

1. Zeroth law of Thermodynamics – If two thermodynamic systems are each in thermal equilibrium with a third, then

they are in thermal equilibrium with each other.

2. First law of Thermodynamics – Energy can neither be created nor destroyed. It can only change forms. In any

process, the total energy of the Universe remains the same. For a thermodynamic cycle, the net heat supplied to

the system equals the net work done by the system.

3. Second law of Thermodynamics – The entropy of an isolated system not in equilibrium will tend to increase over time

approaching a maximum value at equilibrium.

4. Third law of Thermodynamics – As temperature approaches absolute zero, the entropy of a system approaches a

constant minimum.

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Before I go over these laws in more detail, it will be easier if I first introduce Entropy.

Entropy and Phase Space

Entropy is a very important thing in the realm of Thermodynamics. It’s the core idea behind the Second and Third laws

and shows up all over the place. Essentially, entropy is the measure of disorder and randomness in a system. Here are 2

examples.

● Let’s say you have a container of gas molecules. If all the molecules are in one corner, then this would be a low

entropy state (highly organized). As the particles move out and fill up the rest of the container, then the entropy

(disorder) increases.

● If you have a ball flying through the air, then it will start off with its energy organized (i.e., the kinetic energy of

motion). As it moves through the air however, some of the kinetic energy is distributed to the air particles so that

the total entropy of system has increased (the total energy is conserved, however, due to the First law).

To get a more detailed picture of entropy, we need to look at the concept of Phase Space. Some of the concepts for this

may be a bit confusing. But bear with me. Once you’ve got your head around it, it’s not that bad.

A phase space is just like a graph. But a point on this graph represents the whole state of a system. Let’s use an

example. Imagine I have a box with 4 gas particles inside. Each point in the phase space for this system tells you where all 4

balls are located in the box.

In our example, we are only interested in the positions of the 4 particles. So each point in phase space must contain an x,

y, and z coordinate for each particle. So our phase space is 3N dimensional where N is the number of particles in the system.

In our case, the phase space is 12 dimensional in order that each point can describe the location of 4 bodies.

In all the diagrams, I will depict the phase space as 2D to make it easier to convey what it actually represents. For our

purposes, we will not need to consider the dimensions.

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If we imagine that each of the particles is a different color, we can keep track of their positions easier. If we imagine the

case where all of the particles are located in one corner of the container, then we have the situation

In terms of the system, there are multiple other combinations of the 4 particles that will be as organized as the above

state

and so on. Each of these set-ups will correspond to a different position in phase space as they are all different layouts of

the system of the 4 particles. If we add these to the phase space along with the original, we get something like

These 5 layouts of the 4 particles along with the 11 other combinations make up a set of states that are (apart from the

colors) indistinguishable. So in the phase space, we could put a box around the 16 states that defines all the states inside it as

being Macroscopically indistinguishable.

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The total phase space of a system will have many regions all of different shapes and sizes and could look like the

following:

But how is all this abstract representation linked to entropy? Entropy (given in equations as the symbol S) is defined then

as

where k is the Boltzmann constant (1.38x10-23

JK-1

) and V is the volume of the box in phase space. All the points in a region

of phase space have the same entropy. The value of the entropy is related to the logarithm of the volume (originally,

Boltzmann never put the constant k in the formula as he wasn’t concerned with the units. The insertion of the k seemed to

have come first from Planck).

Entropy can also be defined as the change when energy is transferred at a constant temperature

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where ∆S is the change in entropy, Q is the energy or heat, and T is the constant temperature.

The Zeroth Law

The Zeroth law is so named as it came after the other three. Laws 1, 2, and 3 had been around for a while before the

importance of this law had been fully understood. It turned out that this law was so important and fundamental that it had to

go before the other three. Instead of renaming the already well-known 3 laws, they called the new one the Zeroth law and

stuck it at the front of the list.

But what does it actually mean? The law states

“If two thermodynamic systems are each in thermal equilibrium with a third, then they are in thermal

equilibrium with each other.”

Basically if A=B and C=B, then A=C. This may seem so obvious that is doesn’t need stating. But without this law, we

couldn’t define temperature and we couldn’t build thermometers.

The First Law

The First law of Thermodynamics basically states that energy is conserved. It can neither be created nor destroyed, just

changed from one for to another,

“The total amount of energy in an isolated system is conserved.”

The energy in a system can be converted to heat or work or other things. But you always have the same total that you

started with.

As an analogy, think of energy as indestructible blocks. If you have 30 blocks, then whatever you do to or with the

blocks you will always have 30 of them at the end. You can't destroy them; only move them around or divide them up. But

there will always be 30. Sometimes you may lose one or more. But they still have to be taken account of because Energy is

Conserved.

Fundamental thermodynamic relation

From the Second law, we can write that the change in the internal energy U of a system is equal to heat supplied to the

system Q minus any work done by the system W.

(1)

From the definition of Entropy above, we can replace ∂Q and also make the replacement ∂W=PdV giving us

(2)

Now if we have a system of particles that are different, then we may get chemical reactions occurring. So we need to

add one more term to take this into account.

(3)

The Second Law

This is possibly the most famous (among scientists at least) and important laws of all Science. It states

“The entropy of the Universe tends to a maximum.”

In other words, Entropy either stays the same or gets bigger. The entropy of the Universe can never go down.

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The problem is that this isn’t always true.

If you take our example of 4 atoms in a box, then all of them being in one corner is a highly ordered system and so will

have a low entropy. Then over time, they’ll move around become more disordered and increasing the entropy. But there is

nothing stopping them all randomly moving back to the corner. It’s incredible unlikely. But not actually impossible.

If you look at the problem in terms of phase space, you can see that over time that it’s more likely you’ll move into a

bigger box which means higher entropy. But there’s no actual barrier stopping you moving back into a smaller box.

The Third Law

The Third law provides an absolute reference point for measuring entropy, saying that

“As the temperature of a system approaches absolute zero (−273.15°C/0°K), then the value of the entropy

approaches a minimum.”

The value of the entropy is usually 0 at 0°K. However, there are some cases where there is still a small amount of

residual entropy in the system.

The Basics

When you heat something (depending on what it’s made of), it takes a different about of time to heat up. Assuming that

power remains constant, this must mean that some materials require more energy to raise their temperature by 1°K (1°K is

actually the same as 1°C; they just start at a different place.) than others.

If you think about it, this makes sense. A wooden spoon takes a lot longer to heat up than a metal one. We say that

metal is a good thermal conductor and wood a poor thermal conductor. The energy required to raise 1kg of a substance by

1°K is called its specific heat capacity. The formula we use to find how much energy is required to raise 1 kg of a substance

by 1°K is:

where Q = Energy, m = mass, c = specific heat capacity and ∆T = change in temperature.

1a. Laura is cooking her breakfast before work on a Sunday morning (please send your sympathy messages that I had to

work on a Sunday here). She doesn’t want to have to do any more washing up that is absolutely necessary. So she decides to

stir the spaghetti she is cooking with her fork rather than have to wash a wooden spoon. She leaves the fork in the pan while

she spreads her toast with margarine and grates some cheese. The stove provides 1000J of energy to the fork in the time she

leaves it unattended.

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What would be the temperature increase in the fork assuming half the energy provided will be lost to the surroundings

and the initial temperature of the fork was 20°C,and the mass of the fork is 50g and is made of a material with a specific heat

capacity of 460 Jkg-1

K-1

?

Although I’m fairly sure I read somewhere that trying to work out energy changes in forks first thing in the morning was

a symptom of insanity, it’s something I find myself doing from time-to-time. For this question, we’re going to need the

equation Q=mcΔT. This is an equation that you’ll probably need a lot. So it’s worth trying to memorize it. It also springs up

in Chemistry.

First things first. We need to rearrange the equation to make ΔT the subject. Once you’ve rearranged this question, you

should get ΔT=Q/(mc). Substituting the values given to us in the question, you get:

ΔT= 1000/(50 x 10-3 x 460)

ΔT= 43K

So since the initial temperature of the fork was 20°C, the final temperature of the fork would be 63°C.

Different Measures of Energy

Internal Energy:

Helmholtz free energy: F = U - TS

Enthalpy: H = U + PV

Gibbs Free Energy: G = U + PV - TS

Maxwell’s Relations

(4)

(5)

(6)

(7)

ElectroMagnetism

All of It

For Electromagnetism, all you need to know is what happens when you have + or – charges; what happens when they get

close; and what happens when they move. That’s it!

For all of non-quantum E-M, there are only 5 formulas you need. The 4 Maxwell Equations and the Lorentz equation

describe all of electricity, magnetism, light, sound, radiation, and actually most of Physics:

(1)

(2)

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(3)

(4)

(5)

How bad can a topic be if you can describe it all with just 5 equations! You could probably fit them all on the back of a

beermat.

Now that you’ve seen the conclusion, we can go to the beginning and read the whole story in detail. Unless you’re doing

a university course, you can get away with not knowing exactly what the equation mean or do. Bt this site will explain them

later. First, lets get back to basics.

The Basics

Charge comes in 2 types (positive and negative) and is measured in Coulombs (C). If you have a charge on its own, it

emits a field in all directions. The field from a charge is represented by E as in E-lectricity.

If you put another charge in the field, it experiences a force. Like charges repel and unlike charges attract. The bigger

the charge, the stronger the force. And the further away the charges, the weaker the force (exactly what you’d expect). This

relationship can be represented by Coulombs Law:

and

The q‘s are the two charges and r2 is the distance between them squared. The other bit is just a constant which roughly

equals 9000000000. From these, you can see that the force is just the field times by whatever charge you put in: F = qE.

Using this, you can work out the field or force between particles or atoms or anything with charge provided they’re not

moving. Once you start a charge moving, other things happen.

Stuff Moving

As soon as a charge starts to move, it produces another field. The new field is magnetism and is represented by B (as in

B-magmatism).

The reason it’s B is simply that it was the second thing in an alphabetical list:

● Electromagnetic vector potential: A

● Magnetic induction: B

● Total electric current: C

● Electric displacement: D

● Electromotive force: E

● Mechanical force: F

● Velocity at a point: G

● Magnetic intensity: H

(This also explains where H comes from for those interested).

So now your particle or atom or whatever has 2 fields coming out. The full equation to describe how both fields act on a

particle is

If we expand out the above expression, we have

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But we can already describe one of these bits. qE is just Coulombs Law. Also,at A-level or below, the situation will

probably be simplified so you only have to consider the E and B fields separately. So you will probably only have to use one

of the following 2 formulas

F = qE F = qvB .

Obviously, F is the force and q is charge; E and B are the two fields previously described; and v is the velocity of the

moving charge. The electric field is measured in the SI units of Newtons per coulomb (N/C) or, equivalently, volts-per-meter

(V/m). The magnetic field has the SI units of Teslas(T), equivalent to Webers-per-square-meter (Wb/m2) or volt-seconds per

square meter (V s/m2).

Circuits

Now I’m not a big fan of circuits (never have been. But hopefully I’ll be professional enough that my disliking of them

won’t come across in this section (but if it does, I apologize in advance). If I really start to struggle with my hate, I may have

to call in a second writer.

A circuit is basically just a series of moving charges with the occasional object or device in the way that affects the flow.

Now when I say the electrons are moving around, most people will think that their speeding around at close to the speed-of-

light. But this is wrong. The actual electrons are moving EXTREMELY slowly. It’s the wave that travels fast. As stated

above like charges repel, so put one electron next to another and they will move apart.

With a current in a wire, you basically have a tube of electrons and you’re adding one to one of the ends. This causes the

next electron to move down which in turn pushed the next one and so on. So you have a Mexican wave like effect that

moves quickly. But the electrons themselves are only moving slowly.

Circuits usually contain all sorts of different objects and devices depending on what they’re for. Depending on how you

set them all up in the circuit depends how you do all of you calculations.

Which is Which?

If you set up all your component in a closed loop like

then we say that all the components are in series. If you set them up with branching paths like

then we say that the components are in parallel. You can also make circuits that are a mixture of series and parallel section

like

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Amps, Volts, and Ohms (oh my!)

We call the moving charges a current and it is measured in the SI unit of Amps (A). Amps are equivalent to the amount

of charge passed in a certain time. So 2 coulombs in 6 seconds will be equivalent to 0.3A. This (like most things in physics)

can be expressed in a nice formula for you to learn:

Another important idea in circuits is voltage (or potential difference). Volts are basically the difference in the electric

potential at two different points. The electric potential between 2 points is given as

where l is the distance between a and b. It’s basically field times distance.

Another important idea when it comes to circuits is resistance. Resistance is basically a measure of how much

resistance opposes an electric current. Almost all objects or devices in a circuit cause resistance. To calculate the total

resistance in a circuit, you use one-or-more of these rules:

...

...

One of the most important and fundamental equations in circuits is Ohm’s law. It relates current, voltage and resistance:

V = I R

The Deep End

This is it. Classical E-M goes no deeper than this. These four are the fundamental equation for ALL fields in E-M.

They may take a bit to get your head around. But once you do, it should all make sense (sort of):

Gauss’ Law

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This says that the integral of the electric field E through a closed area ∮.dA is equal to the total charge inside of the area

Qinside divided by ε0. ε0 is a constant called the permittivity of free space and shows up all over Physics along with μ0 which

is the Permeability of free space.

What this equation means is you can take ANY closed surface you like and find the E field going through provided you

can do the maths. Usually you can’t. However, there are a number of cases when it's nice and easy. Cases when the E field

is coming straight out through the surface evenly. The cases are

● A Spherical surface around a point or sphere

● A Cylindrical surface around an infinite wire

● A Regular surface over a section of an infinite plane

I admit these sound vague and abstract. So I will demonstrate with the aid of a diagram.

These are the Gaussian surfaces. Basically with these surfaces, all you’re trying to do is make life easier. You just make

sure that the surface is always the same distance from the charge source and that the field is always going through at 90

degrees. You can then work out the integral with your eyes closed its that easy. The left-hand side of Gauss’ law becomes E

times the surface of the shape you chose.

● A Spherical surface becomes π r2 E where r is the radius of the sphere.

● A Cylindrical surface becomes π rl E where l and r are the length and radius of the cylinder.

● A Regular surface becomes AE where A is the Area above and below the infinite surface (you need the factor of 2 as

the field goes above and below the surface at 90 degree).

So Gauss’ law for a sphere becomes

which was introduced earlier as Coulombs Law (now you know where it came from). Gauss’ Law for an infinite line of

charge is just

Now in this, something new has been introduced (λ). If you have an infinite line of charge, then the total charge on it is

infinite and there is no way of knowing how much of that infinite charge you would have inside your gaussian surface.

That’s where λ comes in. It's a value of charge per unit length. So if λ =4C/m and you have 5 meters, then the charge q is

just 20C. That’s all λ is . J just a value of charge.

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For an infinite surface, Gauss’ Law becomes

Once again, a new symbol has been added. But it's just like the one before. σ is just the charge per unit area. So if σ =

C/m2 and you have a 100m

2 area, the total charge is 500C.

Charged Ring

Let's say you’ve got a charged ring and you need to know the field produced from it. Once again, we’ll be employing

one of the most important tools in Physics making stuff easier.

Firstly, we’ll only look at the field along the axis of the ring. (Otherwise things just get too complicated and it’s not

worth the effort). Now let's just take a very small part of the ring and say that it’s a sphere. This isn’t really true. But the

smaller we make the section, the more we can make it resemble a point charge. So you have something like this

You want to find the field at a point z along the axis from the ring of total charge q and radius a. The little square section

at the top, that’s the bit that you assume is a charged sphere. Now we don’t know how much charge is in that little section as

you can make it any size you want. So we just call the charge dq (a small amount of q). So we now have

Now if you think about it, every bit of the ring above the axis pushing down will have an equal bit below the axis

pushing up. It’ll also be the same for left and right and all other parts of the ring. So all the force from the ring will only be

acting along the axis. To work out only this bit, we need to use some trig. We need to multiply the field by cosθ to get the

axial component.

As you may or may not know, cosθ can also be described (using SOH CAH TOA) by the following relationship for our

situation: cosθ = z / r as z is the adjacent side and r is the hypotenuse. So now we have

However, we may not know what r is. We do know the radius of the disk, a, and the distance we are from the disk z.

Using a bit of the old Pythagoras, we can rewrite r in terms of z and a:

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So now our equation looks like this

Now we want to get rid of that dq. So we integrate

Now we know from the diagram at the start that the total charge on the disk is q. So if we add up all the little bits of dq,

the total should be q. So the integral is just q.

So there you have it. The E field from a charged disk. All you need is the field from a point and some trig knowledge

and you can work it out. I could have just given you the final solution. But this way, you can see where it came from. Then

if you forget it, you may be able to work it out from first principles like above.

Gauss’ Law for Magnetism

This one is nice and easy but has some big implications. Gauss’ Law for Magnetism is

It's like the ordinary gauss’ law in that it describes a field. But this time it's the magnetic field B. It says that the integral

of B over a closed surface dA is zero. Nothing. Every field line that goes out of the surface has an equivalent that goes in.

There is no overall field.

This means that it's impossible to get sources of a magnetic field. Whereas electrons and protons are origins of field

from which field lines diverge from or converge to, there is no magnetic analog. Magnetic field lines are always closed

loops. No start, no end. This of course hasn’t stopped people from preparing in case we do find a magnetic monopole.

This equation may seem nice (and it is). But it is utterly useless on its own. Usually a zero result in Physics is quite

important. It means something special might be happening. Here it shows that magnetic monopoles don’t exist.

Faradays Law

Now things are getting more complex. Here we have Faradays law:

I’ll walk you through each bit to show you what it actually means.

First we have the left hand side which is easy. It's just like Gauss’ law only the integral is over a different thing. Instead

of finding the total E field through a surface dA, we are now finding the total E field around a closed loop dl. That’s all that’s

different with the left-hand side. No more surfaces. Just closed loops.

Now on to the right-hand side. First up, we have a minus. Noting complicated about that. Why it's there will be

explained later.

Next we have another integral and this one looks horrible. The ∂ symbol basically means a small change. So ∂B is a

change in B and ∂ t is a change in t where t is time.

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The whole ∂B/∂t is the rate of change of B. It's how much B is changing (∂B) in a given time (∂t). And that is being

integrated over an area dA (the area inside the closed loop dl). If you draw some random squiggly thing making sure that the

line doesn’t cross itself and that it joins itself, then the length around the line is your dl and the area inside the line is your dA.

Simple yes? So the total E around a loop is just equal to the minus of the changing B through the loop.

What happens if there is no B? Well, there is no ∂B so ∂B/∂t is zero which makes the integral 0. So no E.

What happens if you have a constant B? Well again, ∂B is 0. So ∂B/∂t is zero which makes the integral 0,. So again, no

E. You can only induce an E field from a changing B field.

The importance of the minus sign comes from the fact that E fields create B fields and B fields create E fields (as shown

in Faraday’s and Ampere’s Laws). If the minus wasn’t there, then the fields would just keep building and building eventually

giving infinite energy. And that is not allowed!

Ampère-Maxwell Law

The last of Maxwell’s Equations is the Ampere-Maxwell law. Just like the first 2 laws were similar, so are the last two.

There is a pattern to them in this order that can make them easier to remember. E over an area; B over an area; E around a

loop; and now finally B around a loop. The equation is

Left-hand side, easy. Integral of B around a closed loop. Right-hand side, not so easy.

First let's ignore the μ0I bit (I’ll come back to that). Other than the μ0I, it's very similar to Faradays law. You have

another changing field integrated over an area. But this time it's E.

This time, though, instead of multiplying by minus 1, you’re multiplying by μ0ε0. Once again, these are two very

important values in Physics alone and combined. They are at the very heart of E-M.

So your magnetic field around a loop is just equal to the changing E field going through it times by μ0ε0. But then you

have to add on a bit. This is the \ μ0I bit. This is just the current going round the loop times by μ0. This is because (as said in

"Stuff Moving") if you have a moving charge (i.e,. a current), then you get a magnetic field. So you have to add the 2 bits

together. There, done!

Another Form of the Deep End

As well as writing Maxwell’s equations above in what is called integral form, you can also write them in differential

form like

Yet Another Form of the Deep End

Writing Maxwell’s equations in one of the above 2 forms is really a simplification. Both the integral form and the

differential form are vector equations and they save you having to write out the full 8 Maxwell equations for the E and B

fields in all 3 dimensions.

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[su_spoiler title=”8 ‘Original’ Maxwell Equations” style=”fancy”]

[/su_spoiler]

Well, it turns out you can also compactify the 4 vector Maxwell equations into 2 tensor equations like

Here Jβ is a vector with 4 components (sometimes called the 4-current) and F

αβ is a 4×4 matrix called the

electromagnetic tensor. They are defined as

(6)

(7)

where c is the speed-of-light. The α and β just tell you where in the vector or matrix to look. But confusingly for some start

at 0. So J0 = cρ and J

3 = Jz (not to be confused with J-cubed). Same with F

αβ. So F

02 = Ey / c and F

33 = 0.

Light

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Light is energy that we can see. It is given off (emitted) by matter when it is hot or in some way excited. The colors that

human eyes can see show how much energy is in the light (ignoring how bright it is) ranging from low energy red to high

energy blue. This scale stretches further in both the high energy and low energy directions than the human eye can detect.

Generally the term 'light' refers to the range of energy’s we can see. But across all energy ranges electromagnetic

radiation or just radiation is a better term. That said, I’m going to keep calling it 'light' because it’s easier.

This radiation/light can be thought of as both a wave and a particle. Sometimes it behaves like one and at other times it

behaves like the other. This may seem very odd. And to be honest it is. At least compared to other things in day-to-day life.

I’m not going to go into how it can be both. That can be explained better later in the "Quantum mechanics" section.

Anyway, if we think about light as a wave, how does some light have more energy than other light? Imagine holding

one end of a string, dangling it. If you move your hand back-and-forth, that motion will travel along the string as a wave.

There are 2 ways of putting more energy into this. One is to move your hand farther in the same amount of time

increasing the amplitude (tallness) of the wave (which is like the brightness of the light). The other is to move your hand

back-and-forth the same distance faster causing waves to travel down the string more often.

As you probably know, this will tire you more than moving your hand slowly and this is because you’re putting more

energy into the wave. This decreases the length of the waves travelling along the string and increases the number of waves

that pass a certain point on the length of the string each second. These 2 things are known as wavelength and frequency

Quantum Mechanics

In the Beginning

In the beginning, there was continuous flow. Then Max Planck came along and proposed quantization.

Quantization basically just means that instead of being continuous, things such as E-M radiation can only exist in

multiples of certain values. It’s a little bit like having a tube of smarties. The whole tube represents a beam of light. Inside it

you have the smarties. You can split the tube so you can have less smarties in there. Or you can get another tube and have

smarties. But you have to have a whole number of smarties because they can’t be split (if anyone e-mails me suggesting I

squash/crush/split a smartie, I will hunt them down and make them pay!).

Planck came to this conclusion when working on the “Ultraviolet Catastrophe”. According to Classical

ElectroMagnetism, the number of ways an electromagnetic wave can vibrate a in a 3-dimensional cavity per unit frequency is

proportional to the square of the frequency. This means that the power you would get out per unit frequency should follow

the Rayleigh-Jeans law. Which means that the power would be proportional to the frequency-squared. So if you put the

frequency up higher-and-higher, the power would be unlimited.

Planck said that electromagnetic energy did not follow the Classical description. He said that it could only be emitted in

discrete packets of energy proportional to the frequency E = hf (or sometimes written as E = ħ w) where ħ (pronounced “h-

bar”) is h/(2π).

These equations mean that the radiation eventually goes to zero at infinite frequencies and the total power is finite.

Planck called these packets of energy “quanta”. The value of h is 6.626 x 10-34

J-s and the value of ħ is 1.06 x 10-34

J-s.

Quantum behavior differs from Classical behavior because h is not equal to 0.

Let There be Little Packets of Light

If you shine a light onto a metal surface for long enough, the surface will heat up. This must mean that the light is

transferring energy to the metal. So in theory, it is possible that if you shone a light on a surface for long enough, enough

energy would be transferred to liberate an electron from an orbit. Even with a weak light, you should be able to wait long

enough for the energy to build up and an electron to be emitted.

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So physicists tried the experiment. It failed miserably. For some metals, specific light would cause electron emissions

while for other metals the same light source wouldn’t no matter how long it was left. And it was found that the electrons

came out with higher energies depending on the color of the light, not the intensity.

The problem of the photoelectric effect was solved in 1905 by Einstein and was what he won the Nobel Prize for in

1921. Einstein applied Planck's theory of quantization to light and said that the light is not a continuous stream of energy but

rather loads of little packets of a certain energy value that depended on its wavelength. This explained why no matter how

long you left the light on the surface, there would be no emission unless the individual photons had enough energy.

This also explained why different colors gave the emitted electrons different energy values. The energy was shown to be

related to wavelength by Planck's equation. Einstein also showed that the energy of the emitted electrons would be equal to

E = h f - φ where φ is the energy needed to get the electron from inside the metal to just outside the surface and is called

the “Work Function”.

Schrödinger Equation

We start off with a standard equation for a wave:

(1)

A(x,t) means that what this wave looks like depends on position (x) and time (t). The description is set out in complex

number form and can be displayed with an Argand diagram. This wave is a solution of the Wave Equation. What we want to

see is if the wave equation can be used to describe matter waves. The wave equation is

(2)

What this equation is saying is that if you partially differentiate your wave A with respect to x twice, it will equal the

partial differential of your wave with respect to t twice multiplied by a constant. Which in this case is 1/c2 .

So now we need to see if it will work. First we take our wave (1) and differentiate it twice with respect to x to give

(3)

You may be wondering why I’ve changed the original equation whilst doing the differentiations. Originally we had

A0ei(kx-ωt)

and now we have A0 eikx

e-iωt

. This is just a math trick that you can do to exponential powers and I personally think

it makes the differentiation easier. We now differentiate the wave twice with respect to time to get

We can now substitute these two results into equation (2) to give

Conveniently the minus signs, the A‘s, and the squares cancel to give us

(3)

Now if we take the 2 base quantum formulas from the first section

(4)

(5)

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and try and substitute (3) into them, we get a problem.

However, E ≠ pc for matter. For non-relativistic matter, the relationship between energy and momentum obeys the

following law:

(6)

So it looks like we have a problem. The Wave Equation (2) doesn’t work for matter

One way to try and get it to work is to say that instead of ω2 ≈ k

2, what if we tried to get it so it was ω ≈ k

2? To do this,

we would need a wave equation that was differentiated twice with x and only once with t. Also if we replace the constant, we

can make life easier for ourselves. So let's try

as our new wave equation. We have now changed A to φ as this will be the equation that works and φ is the common

symbol used for quantum mechanical waves. The equation for φ is the same as for A. So if we now do the differentiation

which when we put back in our new wave equation gives

Which is the relationship between ω and k that we were after. So now we can rearrange for ω and substitute into

equation (4) to get

We can now chose our constant α as

so that we get ...

which is correct!! So far so good. Our new wave equation for matter gives us the correct energy. So let’s put the constant

back in.

And now once again rearrange to get it in a nicer looking form ...

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We’re nearly there now. The equation is almost complete. However when we solve it for the energy of a particle, we

get

But sometimes a particle can get energy from its surroundings. For example if it was in a potential. So we have to make

one slight adjustment to account for all of the particle's possible energies.

which means our wave equation becomes

This is called the One-Dimensional Time-Dependent Schrödinger Equation

Time Independence

In most cases, you’ll learn about involving matter waves like electrons, the potentials they’re in don’t really depend on

time, and they don’t suddenly change shape after so many seconds. If this is the case (and most of the time it is), then we can

use the Separation of Variables method on the Schrödinger Equation.

First thing we do is assume that the φ can be split into 2 functions. One that only depends on x and one that only

depends on t like ...

You then get your Schrödinger Equation. Wherever there is a φ , you just replace it with uT. So you get

Now if you divide by u, you get rid of the one on the left as that differential doesn’t depend on u. And if you divide

through by T, you get rid of the T on the right as that differentiation doesn’t depend on T. So you get

Now let's say t changed. That would mean that the left-hand of the equation would now have a different value.

However as u is independent of t, the right-hand side of the equation wouldn’t change. This would cause an error. The 2

sides of the equation were equal before. Now one side has changed and they still have to be equal.

To get around this problem you set both sides equal to a constant. In this case we shall call it E. So now we have 2

separate equations ...

(7)

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(8)

We have already prefaced that we are only interested in cases where time has no effect on the potential. So we can

ignore equation (7) and just use equation (8) which is our One-Dimensional Time-Independent Schrödinger Equation. In the

case of a free particle, V(x)=0. So the solution to the time independent equation (8) becomes

where

Which is of an identical form to equation (1) provided that the constant E is the total energy of the system (which is a

good thing). This means that provided we know the energy of the system, we can work out a solution to the wave equation.

Also, provided the energy remains constant, T has no effect on φ. So |φ|2 is just the same as |u|

2. So then we can write

Particles and Forces

Spin

Before I get into the different types of particle, there’s a bit more back story you need. All particles have a property

called 'spin'. The spin of a particle has a fixed value that depends only on the type of particle. Spin can also have direction

up or down. And the particle carrying the spin can have a handedness left or right. This gives 4 possible combinations

shown below:

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I admit that this looks quite confusing. But it can be possibly simplified with the help of your own left and right hands.

If you get your right hand in a grip position with your thumb sticking straight up, then your fingers represent the handedness

of the particle (or direction of spin) and your thumb represents whether the particle is orientated up or down.

For example, if a clock is thrown with its face directed forwards, it’s left-handed. In order to represent its clock hands

motion with your gripped fingers and the overall clock flying through the air motion with your thumb, you have to use your

left hand.

Types of Particle

Throughout the whole of the known Universe, there are only 2 types of particle. Particles that make up matter and

particle that carry force. They are the only 2 types found so far.

Now you may be thinking that yes, there may only be two categories. But I bet they’re filled with hundreds of different

subgroups and types. Thankfully this isn’t the case. Particles follow specific rules and once you known them, everything

gets a lot easier. The two groups are called Fermions and Bosons.

Fermions

Fermions are all particles that make up matter. The name comes from the fact that all particles of matter follow a

certain set of laws called Fermi-Dirac Statistics developed by Enrico Fermi and Paul Dirac in 1926.

All fermions in existence possess half integer spin (i.e., 1/2, 3/2, 5/2 etc.). For example, every electron in the Universe

possesses a spin of 1/2. Fermions also obey the Pauli Exclusion Principle. This sounds complicated but it’s relatively

simple to describe. What it means is that only certain combinations of matter can exist in the same space. More specifically,

it states that

no 2 identical fermions may occupy the same quantum state simultaneously

For example, take Helium. It’s got a lowest energy shell for the electrons. You can put one electron in easy. However,

the Pauli Exclusion Principle says 2 electrons can’t occupy the same quantum state. So the second one has to have the

opposite spin. This then allows the 2 electrons because spin is part of the quantum state of the electron. So the two electrons

are occupying different quantum states.

The spin, however, can only be one of two things -- up or down (+1/2 or -1/2). If for example you had a Lithium atom

which has 3 electrons, then the third electron can’t fit into the 1st shell. So to fit it in, it has to move up to the next shell. The

entire Periodic Table is built up from this Principle.

There are two different types of fermions -- Leptons and Quarks.

Leptons

There are 6 sub-atomic particles that make up the leptons. The Electron and the Electron Neutrino, the Muon and Muon

Neutrino (which are basically heavier versions of the Electron and the Electron Neutrino), and the Tau and the Tau Neutrino

(which are heavier versions still). The electron, muon, and tau all have charges of -1 whereas all the neutrinos have charges

of 0.

Quarks

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Quarks are the other type of matter particle along with the leptons. Like the leptons, there are 6 quarks grouped in 3 sets

of 2 with each successive group basically just a heavier version of the previous. Like the leptons, the quarks in each set have

a charge difference of 1. But instead of nice whole numbers, the charges of quarks come in fractions of e. The 6 quarks are

named Up, Down, Charmed, Strange, Top, and Bottom.

Hadrons, Mesons, and Baryons

Physicists seem to love their labels and groups. As soon as you put quarks together in groups, then the resultant particles

are called Hadrons. But the names and classes don’t stop there. If you make a Hadron out of 2 quarks, it’s called a Meson.

And if you make a Hadron out of 3 quarks, it called a Baryon.

The reason you get groupings of 2 or 3 quarks is because of their' color'. Quarks can be 'red', 'green', or 'blue'. Anti-

quarks can be anti-red anti-green and anti-blue. The particles aren’t actually colored (for one thing, they’re much smaller

than the wavelength of visible light). The “colors” are just labels. Quarks have a property that can take 3 distinct values. So

physicists called those values 'red', 'green', and 'blue'.

Quarks exist in groups that have no overall color charge. So you can get groups that are red+blue+green=white, anti-

red+anti-blue+anti-green=white, red+anti-red=white, blue+anti-blue=white, or green+anti-green=white. (i.e., either 3 quarks

or anti quarks together or 1 quark and 1 anti quark together; Baryons and Mesons).

Particles like the proton and neutron are examples of Baryons as they are comprised of 3 quarks. While particles like the

π+ and π

- are Mesons as they are made from a quark and an anti-quark. However, all four of them are types of Hadrons.

Bosons

Bosons are the particles that carry force. They are characterized by having whole integer spin (e.g., -1, 0, 1) and don’t

obey the Pauli Exclusion Principle. So you can have loads of them in the same space. Each of the fundamental forces of

Nature has its own Bosons.

For ElectroMagnetism, the force carrier is the photon. They are sometimes called virtual photons as they only exist for

very small intervals of time or space. If an electron gets near another electron, it emits a virtual photon which is absorbed by

the second electron and lets it know that it needs to move away.

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This is a Feynman Diagram named after the amazing physicist Richard Feynman. It's an easy way of describing or

visualizing particle interactions. Particles are represented as lines (either straight or wavy) and interactions are depicted as a

vertex of the lines.

Most of the time, the lines will have arrows to show more specifically how the particles are moving. In the above

example, 2 electrons move towards each other. Then we have the interactions with the boson of the electromagnetic force.

Then they move away from each other.

For the Strong Nuclear Force, the boson is the gluon. It has zero rest mass and zero charge. Despite there only being

one boson for this force, it can come in different colors.

For Gravity, the boson is theorized to be the graviton. It is thought to have zero rest mass and zero charge but has not

been discovered yet.

The Weak Nuclear Force looks like the odd one out. It has 3 bosons -- the W+, W

-, and the Z

0. None of them are

massless like the photon. On average, they’re about half the mass of a caffeine molecule.

Forces

What is stuff made of? Compounds, molecules, atoms, electrons and quarks. At the moment, you can go no further.

Forces are the same. There are loads of them about. But really they’re just combinations of 4 fundamental forces. This isn’t

enough for most physicists (myself included) and so research has been going on for a long time now to try and find just one

description of all of them. James Clerk Maxwell did it with Electricity and Magnetism. So why can't we do it with the rest?

Actually we partially have (but more on that later). For now here are the 4 forces.

Gravity

Gravity is the weakest of all the forces which seems odd at first. It holds planets together and holds them in their orbits.

It is also the longest ranged force mainly because it is always attractive. You can easily overcome gravity just by jumping

(that’s how weak it is). Gravity is felt by anything with mass. If it has mass, Gravity can act on it. Gravity works via the

following law

It’s an inverse square law. So it gets weaker the further you move away, it also gets stronger for objects of larger mass.

ElectroMagnetism

ElectroMagnetism is 1 trillion, trillion, trillion times stronger than Gravity. However unlike Gravity which is always

attractive, ElectroMagnetism can be both attractive and repulsive. This is because there are 2 types of electromagnetic

“matter” -- positive charge and negative charge. ElectroMagnetism follows the following law:

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You can see that its very similar to the law for Gravity. It’s inversely proportional to distance and is stronger for objects

of larger charge. It’s a long-ranged force. However, the mix of positive and negative charge cancel each other so it’s hardly

ever felt on large scale (unlike Gravity).

Weak Nuclear Force

The Weak Nuclear Force 10 trillion trillion times stronger than Gravity. The weak nuclear force is responsible for all

three types of nuclear decay (Alpha, Beta, and Gamma).

Alpha decay is the emission of a helium nucleus from an atom. Beta decay is when an electron or positron is emitted

from an atom. And Gamma decay is the emission of a high energy photon from an atom.

The Weak Nuclear Force is the odd one out of all the forces. Firstly because of its bosons. The weak force has 3 bosons

unlike the others which only have one each. The bosons are also unlike the others as they have charge and mass,. S much

mass in fact that they are heavier that atoms of Rubidium!

This is why the force only acts over small distances. In one type of decay, an Up quark can emits a W-. That’s a particle

emitting something that is 40,000 times heavier!

The Weak Nuclear Force is also different as it only affects left-handed particles or right-handed antiparticles with flavor.

Strong Nuclear Force

Inside a nucleus, you have protons and neutrons. Due to the ElectroMagnetic force, however, all of the protons in the

nucleus are pushing each other apart trying to break free. The thing that holds them together is the Strong Nuclear force. It's

100 times stronger that E-M and affects all particles with color. The Strong Nuclear Force gets stronger with distance.

However, it is a very small-ranged force only acting over a range of 10-15

m.

Are There Really Just 4 Forces?

It turned out that most of the forces seem to be just different aspects of the same thing. ElectroMagnetism and Weak

Nuclear Forces have shown that at high enough energies, the 2 forces are the same (called the Electroweak interaction).

Above the unification energy of about 100 GeV or 1015

Kelvin, they would merge into a single Electroweak force. Work

is currently being done on adding the Strong force and then hopefully Gravity.

Black Holes

What is a Black Hole?

A black hole is a point in space with so much gravity that not even light (the fastest thing around) can escape (hence the

name). To an observer, it would just appear as a sphere of perfect blackness. At the heart of a black hole is an object called a

singularity. A point of zero size and infinite density. Yes, you have read that correctly. Zero Size and Infinite Density.

Any object can become a black hole. But don’t worry. It would take a hell of a lot of work for something like your car

to collapse space (unless of course your car weighed about the same as the Sun).

A Black Hole is an object for which nothing can get a high enough escape velocity to get away from it. Think of a

cannonball being fired straight up in the air. As it goes up, it will be slowed down by gravity and come crashing back down.

If the speed is high enough, however, it will keep going until it escapes the gravitational pull. For the Earth, the escape

velocity is 12km/s or 26,843mph. For something bigger like the Sun, the escape velocity is 618km/s or 1,382,430mph.

When the body is outside of the gravitational pull, its kinetic energy and potential energy will be 0. So if we equate them

...

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and the rearrange for v, we get an expression for the escape velocity:

(1)

where M is the mass of the planet or body and r is the radius you are taking off from.

The formula contains no mass of the escaping object. If you wanted to get a space shuttle off the Earth, you would have

to get it to the same speed as if you wanted to get a pebble off the Earth, the difference being the amount of energy it would

take getting something as heavy as a space shuttle up to the right speed.

Cambridge scientist John Michell argued that if you made the value of M big enough in the escape velocity formula, then

you could get a value for v that was bigger than the speed-of-light. We wouldn’t be able to see these objects as no light

would be able to reach us. And as nothing can travel faster than light, no objects would be able to escape their pull once they

were close enough. This is a Black Hole.

Forming a Black Hole

Naturally occurring black holes form when stars collapse. Stars are massive. Our nearest star the Sun is roughly

1300000000km wide and weighs about 2000,000,000,000,000,000,000,000,000,000kg (2×1030

kg). Due to their enormous

mass, they have a HUGE amount of gravitational force. As you may or may not remember, Gravity is related to mass via the

following equation

(2)

This is Newton’s Law of Gravitation. The value of the constant G is 6.67428×10-11

which is quite small. However when

you put in the mass of the Sun and the Earth and the distance between them, the force that comes out is 3.76×1022

N which is

equivalent a thousand million million (1 followed by 15 zeros) Saturn V rockets.

When the force of gravity from a star becomes bigger than the outwards pressure caused by its temperature, then the

force starts to make the star collapse pulling all its mass inwards to a central point. This point gets smaller-and-smaller and

denser-and-denser as all of the stars mass is squashed into a tiny point.

Not all collapsing stars form black holes, however. In order for an object to form a black hole, it has to be compressed

below a certain radius. This radius is given by

(3)

This radius is known as the Schwarzschild radius after physicist Karl Schwarzschild who discovered it in 1916.

So for example if you wanted to turn the Earth into a black hole, you would have to compress it all down to about the

size of a large mosquito. And if you wanted to turn your car into a black hole, you would have to squash it down to the size

of a neutrino which is pretty small (about 1×10-24

m wide).

Point of No Return

Once an object has been compressed to Schwarzschild radius, it will continue to collapse until it becomes a singularity.

Centered on this singularity will be a sphere of Schwarzschild radius called the Event Horizon. The Event Horizon is the

last distance from which light can escape the pull of the black hole. Inside the Event Horizon, everything including light

must move inward, getting crushed at the center.

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The Event Horizon itself is not some physical barrier in space. It just represents the last distance at which it is possible

to escape the gravitational pull. A person falling into a black hole through the Event Horizon wouldn’t notice anything

different (although they may be preoccupied with the excruciating pain of being crushed and stretched by all that gravity).

Due to the extreme nature of gravity around the Event Horizon, some very weird things can happen. As I just stated,

someone falling into a black hole wouldn’t notice any changes as he went through the Event Horizon. However, for someone

watching at a safe distance, it wouldn’t be that simple. Things moving away from a body get slowed down by the

gravitational pull. The bigger the pull, the more things get slowed down. Also the closer you are, the more you are going to

be slowed down. Both of these come from the fact that F∝M and F∝1/r2(from equation 2).

As we sit at a safe distance and watch the unlucky person get closer-and-closer, they seem to slow down! This can’t be

right, can it? Stuff moving away is meant to slow down. Stuff moving towards the black hole should speed up!

The way we see the person falling in is through photons (particles of light) being reflected off them and into our eye. As

they get closer to the black hole, the photons get slower and slower due to the increase of gravitational force so that they take

longer to reach to reach the observer.

A diagram to show what occurs when something falls into a black hole

The photons given off when the person crosses the Event Horizon will be slowed down to 0 by the gravity and so an

observer will never see them disappear.

A Hot Black Hole?

Despite the fact that black holes just sit there in space sucking things in with their enormous gravity, it is possible for

them to radiate and thus have a temperature.

In the vacuum of Space, particle and antiparticles are continuously created and annihilated randomly. Usually these are

just classed as 'virtual particles' as they don’t really interact with anything and can’t usually be detected or measured.

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But if a virtual particle pair are created outside of the Event Horizon, then it’s possible that one of them falls into the

black hole before they can annihilate. The particle that is left can then fly off into space as a real particle.

A diagram showing particle-antiparticle pairs being separated when forming near a black holw

To someone observing from a safe distance, it would appear that the black hole is radiating and therefore will have a

temperature. This temperature was found by Stephen Hawking as

(4)

where we have the speed of light (c), Planck's constant (ħ), Gravitational Constant (G), Boltzmann's constant (k), and the

mass of the black hole (M). Notice that this equation contains both ħ and G. This indicates that black hole temperature is a

‘quantum-gravitational’ effect.

If we have the temperature of a black hole, then we can also work out the Entropy

(5)

where k, c, G and ħ are the usual constants and A is the surface area of the Event Horizon. Like the temperature equation, the

equation for entropy contains both Planck's constant and the Gravitational constan, showing that black hole entropy is a

strictly ‘quantum-gravitational’ effect.

Strange Evaporation

As a black hole radiates via this virtual particle method, it will lose mass and get smaller. Using Equation (4), we see

that a smaller mass means a bigger temperature. So as the black hole radiates, it gets smaller and hotter.

If it gets hotter, then the rate of radiation will increase and so you’ll have a chain reaction with the black hole getting

ever smaller and hotter until it gets to zero mass. What happens then? Does the black hole just disappear with all the

information that fell into it lost?

A black hole of about 5 solar masses would have a temperature of about 12×10-9

Kelvin. The temperature of Space is

about 2.7K which is quite a lot bigger. So the temperature of Space would be providing loads of energy to the black hole via

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the Second Law of Thermodynamics. This heat energy will add to the mass of the black hole (via E=mc2) which will make

the black hole bigger and colder.

In order to ‘see’ an evaporation, you would need the black hole to have a temperature above the temperature of Space

which equates to starting off as a mass that's lighter than the Moon. If you had managed to squash your car down to a black

hole (squashing it to 1×10-24

m wide), then it would evaporate in about 1×10-9

seconds.

Black Holes have no Hair

This may seem like an odd title. And it is. But trust me when I say that it is a perfectly acceptable phrase in Physics.

It refers to the fact that all you need to perfectly describe any black hole is 3 externally observable numbers. The rest of

the information (hair) about the matter that made the black hole is lost.

The 3 numbers you need to describe a black hole are its Mass (M), Angular Momentum (J) and Charge (Q). All other

information is lost. Which is potentially a huge amount as a black hole with a given M, J, and Q can be formed from many

different collections of particles.

So if you had 2 stars with the same values for M, Q, and J but one was made of matter and the other antimatter, then

when they collapsed to black holes, they would be identical. You wouldn’t be able to tell which was which. Information is

lost behind the event horizon.

The main equations we have so far for a black hole are 3, 4, and 5. As you can see, the only variable in Equations 3 and

4 is mass (M). Equation 5 only depends on the surface are of the Event Horizon A. However, the surface area of a sphere

depends on its radius. From Equation 3, we know that radius depends on M.

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