In this article, I will cover every topic from G495 including the (in chronological order) Rutherford Scattering, electron and deep inelastic scattering, quark combinations, energy levels of electrons, evidence for the size of a hydrogen atom, nuclear radioactivity (fission and fusion), alpha, beta and gamma radiation, curve of stability analysis, design of a pressurized water reactor, shielding and half thickness, radiation dose, how risk is calculated, eddy currents, transformers: how they work and design of them, the catapult field, motor design, electromagnetic equations, flux linkage, Faraday’s and Lenz’ laws, moving charged particles and relativity, the electric field, the linear accelerator, particle physics, particle identification, the forces, and, finally, electromagnetic field equations with graphs. Please feel free to skip to the parts most relevant to you – I do not expect you to read it all in one go!
- Target – atom (Gold)
- Particle fired in – alpha particle (also known as the nucleus of a helium atom)
- Conclusion -Mass and charge are concentrated in a tiny nucleus and that atoms are mainly made up of empty space.
Distance of Closest Approach
For an electric field:
- F (force between two charged particles) = kQq / r2. k is a constant which is around 9×109 (or 8.85×109 for more accuracy).
- E (electric field strength when the charge is at a point) = kq / r2. The charge ‘q’ is whatever charge is causing the electric field you are entering.
- V (electric potential) = kq / r.
- PE (electrical potential energy) = kQq / r.
At d, the kinetic energy of the alpha particle is 0. Therefore, initial kinetic energy = potential energy at d. From rearranging the formulas, we can come up with the following formula for distant at closest approach:
d (distance at closest approach) = Ze^2 / (2πE x KE)
Where k = 1/4πE = 8.85×10^-9
- Target – Nucleus
- Projectile – Electron (at high energy)
- Conclusion – Scattering electrons can be used to determine the size of the nucleus.
What evidence is there that the nucleus contains more than one type of particle?
The charge of the nucleus can stay the same but the mass can change as a neutron has a zero charge and a mass. Therefore, adding/removing neutrons will change the mass but not the charge. If there was just one particle in a nucleus, being the proton, as you take or add protons, the mass will change along with the charge.
What is an isotope, include an example
An isotope is when there is a difference in the number of neutrons in the nucleus which makes the atom unstable and radioactive. Examples include carbon-14 and oxygen-15 (which is detected during positron emission tomography).
Why do different isotopes of the same element have the same chemical properties?
The only thing that is changing is the mass and not the lepton number, charge etc.
Electron Scattering Measures the Nucleus
Why can’t alpha particles be used to probe inside a nucleus?
The electromagnetic charge of alpha particles stops them from getting close to the nucleus.
What can be used instead and why?
Electrons can be used instead of alpha particles because they are not affected by the force that holds the nuclei together.
Sketch a diagram to show what path an electron scattered by the nucleus might take.
Why do electrons scattering off a nucleus create a diffraction pattern?
Just like alpha particles, they are scattered off. However, instead of repelling away from the nucleus, the electron and nucleus are attracted to each other.
What can this diffraction pattern tell us?
The size of a nucleus. sinθ = 1.22λ / d where λ = the wavelength of the electrons, d = diameter of the nucleus and angle θ = angle of first minimum. We can then use E = pc and λ = h / p to work out the momentum and energy of the electrons.
Why do electrons need to be accelerated to such high energies for this to work?
So that the wavelength is about the same dimensions as of a nucleus.
With electron scattering, you should get a graph such as the one below:
The shape of the graph comes from:
- Rutherford scattering which is an exponential graph.
- Diffraction curve.
Deep Inelastic Scattering
- Target – Proton (quarks inside proton/neutron)
- Projectile – Electron (at high energy).
- Conclusion – there is the existence of three particles inside a proton and neutron being quarks.
Combinations of Quarks
The following combination of quarks are baryons. Baryons are always three quarks. Mesons are always two quarks consisting of a quark/anti-quark pairing. A U (up quark) = 2/3 e charge. A D (down quark) is work -1/3 e charge. There are anti-quarks too which have the opposite charges. U = up quark.
- A proton is made up of UUD = +1.
- An anti-proton is made from anti-quarks of UUD = -1.
- A neutron is made from UDD = 0.
- An anti-neutron is made from anti-quarks of UDD = 0.
Energy Levels of Electrons
This is a potential well for an atom of positive charged. It falls using 1/r because the equation for electric potential is kq / r. This can be approximated to a box because it has straight sides and a box is 1D (where atoms are 3D):
As with the waves on a string, d = nλ / 2. But, we know that λ = h/p and, p = mv and KE = 1/2 x mv². So, KE = p² / 2m and p² = h² / λ². Therefore…
KE = h² / 2mλ²
If we substitute in λ = 2d/n (which came from d = nλ/2…
KE = n²h² / 2m(2d)²
En = n²E
However, this is only an approximation.
Evidence for the Size of a Hydrogen Atom
We know that the potential energy of a charged particle is kQq/r when the charged particles are distance ‘r’ apart from each other.
For a hydrogen atom, it has one proton and one electron. Therefore, PE = – ke² / r. It is negative as one of the charged is positive and the other is negative.
We know that KE = h² / 2mλ². If we take the ground state of the electron. λ/2 = 2r. Therefore, λ = 4r:
From this, we know that the Kinetic energy of an electron in a hydrogen atom, to escape, = h² / 2m(4r)². This is a graph of KE against r.
If we combine the two graphs together, we can find the point when the PE of the electron = it’s KE. At this point, this is the minimum r length a hydrogen atom can be.
There is a minimum size to a hydrogen atom. Below this size, the electron’s KE will be more than it’s PE. Therefore, the electron has enough KE to escape the atom’s ‘potential well’ and the hydrogen atom will no longer exist.
For a hydrogen atom, the energy levels are calculated using:
En = -13.6eV/n²
To find evidence for:
- An atom containing a tiny nucleus, we can use Rutherford’s scattering which fires alpha particles at an atom and notes the deflection caused by the positively charged nucleus. Only a few out of thousands of fire alpha particles are deflected 180 degrees making clear the nucleus is extremely small.
- The atoms in the nucleus are positively charged, we can fire a positively charged alpha particle which will be repelled by the nucleus proving that it is positive (or fire an electron at it which will become attracted to it).
- Radius of the nucleus, we can use an electron scattering which creates a graph combining the Rutherford scattering and direction curve. The ‘first dip’ is the size of the nucleus and is of the order of 10^-15m.
- Protons and neutrons contain smaller particles called quarks, we can use deep inelastic scattering which fired electrons at high energies at protons and neutrons.
Nuclear Fission – What is it?
It is when an unstable atom splits into two smaller and more stable atoms releasing energy. An example of an atom that would undergo fission is Uranium.
A neutron hits the nucleus of e.g. Uranium.this makes the nucleus become very unstable (gets into an excited state). This is because the binding energy of Uranium is not very negative- the added neutron brings it closer to 0. The nucleus breaks in two with two of three neutrons set free at high velocities.
Why does Nucleus Fission happen?
It happens due to atoms have too many electrons or protons making them become unstable. The atom wants to get into a more stable state by making the binding energy per nucleon ‘more negative’. This means it will require more energy for the nucleon to escape the atom making it more stable.
Nuclear Fusion – What is it?
This is when two atoms ‘fuse’ together to make a more stable atom. An example of this is what happens in the sun. two hydrogen atoms form to make a helium.
Comparing Nuclear Radiation
- Structure – Beta + is a positron / Beta – is an electron.
- Charge – Beta + has a charge of +e / Beta – has a charge of -e.
- Penetrating power – Medium: it is stopped by 3mm of aluminium or 10-20cm of air.
- Ionizing power (how easily does it lose or gain an electron) – Medium.
- Deflected by electric and magnetic fields? – Yes, out of all the radiation, it has the highest charge:mass ratio.
- Structure – two protons and two neutrons (nucleus of a helium).
- Charge – +2e.
- Penetrating power – Low: it is stopped by 1cm of air or skin or a sheet of paper.
- Ionizing power (how easily does it lose or gain an electron) – High because it has a large charge.
- Deflected by electric and magnetic fields? – Yes, second most because the charge to mass ratio of alpha particle is less than that of a beta.
- Structure – high energy and frequency gamma ray.
- Charge – No charge.
- Penetrating power – High: need lead to stop it (but it can never completely stop it but reduce it exponentially.
- Ionizing power (how easily does it lose or gain an electron) – Low because it cannot attract atoms with charges.
- Deflected by electric and magnetic fields? – No as the gamma ray has no charge.
The Curve of Stability
The curve of stability can show us a number of things:
What is meant by ‘stable’?
Stable nuclei will not undergo spontaneous nuclear activities. It will not decay.
Why does N = Z for small stable nuclei but N > Z for large ones?
At N = Z, the strong interaction force of a nucleus is greater than the electrostatic force between protons. As the proton number increases, the electrostatic force repelling the protons from each other will increase. Therefore, more neutrons are needed to decrease the electromagnetic force (by increasing the gap between the protons).
What is special about Z = 82?
This is the highest proton number that naturally occurs stable.
When the nuclei have too many neutrons to be stable, they emit electrons (beta – decay). The decay equation would look like n –> e- + p + anti-electron-neutrino + energy (such as gamma radiation). The neutron changes to a proton. When the nuclei do not have enough neutrons, it undergoes beta + decay where it emits a positron: p –> e+ + n + electron-neutrino + energy (such as gamma radiation). When the nuclei has too many neutrons and protons, it emits alpha particles (helium nucleus which consists of two protons and neutrons).
Design of a Pressurized Water Reactor
What is the role of the moderator?
The moderator slows down neutrons to keep the reaction going at a steady rate. The moderator is usually water or graphite.
What is the role of the control rods?
The control rods (made from carbon) absorb neutrons so that the fission reaction is maintained at a steady rate at a critical mass. You want there to be a chain reaction that for every fission decay, there is only one neutron that can go on to cause another fission decay. If there is more than one neutron that goes on to cause another decay. The reaction can quickly expand in decay causing a meltdown to happen. The control rods stop this from happening.
Half thickness – Thickness needed to reduce the number of photons to half.
We will be applying mostly to X-ray and gamma-ray photons that have enough energy to ionize.
By using logarithms and exponential equations, we can come to the final equation of:
I = I(0) / 2
Absorbed Dose = Energy deposited per kilogram (measured in grays)
Dose Equivalent = Absorbed Dose x Quality Factor (measured in Sievert)
The quality factor depends on the type of radiation and the type of tissue being ionized:
- Gamma radiation has a factor of 20.
- Beta radiation, gamma and X-ray have a factor of 1.
- Neutron radiation has a factor of 10.
How is Risk Calculated?
The risk of something happening (e.g. cancer) is often expressed as a percentage per Sievert per person per year.
There are 62.6 million people in the UK and the average level of background radiation is 2000uSv per year. The risk of cancer from this is 5% per Sievert per person per year. Calculate how many people are likely to get cancer from background radiation over a 70 year lifespan.
We can break this question down into three steps:
1) Calculate the percentage risk for a 2000uSv dose – what does this tell you?
Percentage risk from 2000uSv = 2000×10^-6 x 0.05 = 0.0001% (it is very small).
2) Multiply by the number of people – what does this tell you?
0.0001 x 62.6×106 = 6260 (people will get cancer from background radiation each year).
3) Calculate how many people are likely to get cancer over a lifetime.
6260 x 70 = 438,200 people.
How Do Smoke Detectors Work?
Smoke detectors work by having a substance that emits alpha particles. When smoke is present, it blocks the alpha particles from reacting with the receiver causing the alarm to go off.
Eddy currents are more generally associated with turbulence in an electromagnetic field. In electromagnetism, eddy currents are caused when a conducting media interacts with a magnetic field. For example, in a transformer, an alternating current is in the primary coil. This creates alternating flux lines through the core which creates a magnetic field in the core. The secondary coil therefore has alternating flux running in the middle of the secondary coil. This creates an alternating voltage in the secondary coil which, if there is a complete electrical circuit, creates a current in the secondary coil.
The problem is that through creating alternating flux in the core, an EMF is also produced in the core too because the core is a complete electrical circuit too. The EMF produced is perpendicular to the direction of the magnetic flux lines. As there is a complete electrical circuit, currents (being the eddy currents) are induced in the core which have their own magnetic flux lines. These flux lines are opposing the direction of the original flux lines. Therefore, the change in flux in the core will reduce causing the EMF in the secondary coil to decrease. Eddy currents are a resistance to the transformer.
A transformer is a device that changes the voltage (and current output) with little lost of power. This is important especially to the national grid. Current produces heat which is inefficient in sending electricity around the UK. Therefore, the national grid uses transformers to make the voltage extremely high and current as low as possible (step up transformer). As the electricity approaches houses, a step down transformer reduces the voltage increasing the current (as P = IV) so that the electricity can be used in homes.
For a transformer, the following equation can be used:
N (secondary) / N (primary) = V (secondary) / V (primary)
The ratio of the number of turns on the secondary coil to that on the primary coil is the same as the ratio of the voltage on the secondary coil to that on the primary.
How Does a Transformer Work?
The A.C current flowing in the primary coil produces an alternating magnetic field which magnetizes and demagnetizes the core. The core’s changing magnetic field is directed through the secondary coil linking the flux with it. The changing flux in the secondary coil induces an alternating voltage in it which, if connected to a circuit, produces an alternating current.
EMF = – N x dΦ/dt
The above equation is Lenz’s law. The negative sign means that the EMF induced always opposed the field that is causing it.
The sinusoidal variation in EMF is π/2 radians behind the sinusoidal variation in flux (Φ).
Transformer Design of the Core
In an un-laminated (solid iron) core, eddy currents are induced by the primary coils changing magnetic field. These eddy currents reduce the overall flux in the core. This makes the core less efficient.
Lamination in the core reduces the size of the eddy currents and also provide a loop path for the flux. This is because a thin layers of insulation are placed in between layers of iron so that the flux can still move around the core but, the currents cannot move perpendicular to the flux direction because the insulating layers prevents this from happening. In essence, lamination stops the possibility of large eddy currents. The magnetic field circuit is still complete. But, the electrical circuit perpendicularly is not. However, even though large eddy currents cannot be formed, small ones that are individual to each thin layer of iron still form. However, the resistance of these eddy current’s flux lines are much less than before lamination.
Move the magnet further away from the induction coil reduces the max value of Φ. Therefore, dΦ/dt decreases.
dΦ/dt increases as frequency, f, increases (dt falls). therefore, the peak EMF increases.
The Catapult Field
The combined (aggregated) field above the conductor produced a strong field with a force downwards. It’s deformation looks like a catapult elastic being stretched, hence the name – catapult field.
Lenz’s law states that any magnetic effect caused by electromagnetic induction opposes the change that is causing it. For a generator, this means that as the conductor is moved through the magnetic field, the current produced causes its own magnetic field. This field interacts with the original field and opposes the change that is causing it.
As a result, Fleming’s left-hand (motor) and right-hand (generator) rules are mirror images of each other. Lenz’s law is required as it is a direct consequence of the law of conservation of energy.
Additional work is done against the force resulting in the transfer of electrical energy during electromagnetic induction.
What Factors Affect the Power of a Motor?
Power is the rate of transfer of energy which is the rate of doing work.
P = dW/dt
But, W = F x d (where d is the distance moved in the same direction of the force).
Mom (turning effect: torque) = F x d (where d is the distance moved perpendicular to the direction of the force).
W = F x d indicated that greater power goes into the motor if the coil (rotor) is turning faster. But, a fast turning coil has large eddy currents so these oppose the motion of the coil so it reaches an upper limit where torque is zero. This is because all the energy is going into moving the rotor. At lower speeds, the rotor will produce a more useful power output.
A good design of a motor includes the following elements:
- Using an electromagnet for the stater as well as a coil for the motor.
- Good magnetic circuit. This requires the use of materials of relatively high permeability and few and small air gaps (so the magnetic circuit has a high permeance).
- Use more than one pair of poles (N+S) to make torque smoother and more consistent. If there is more than one poles, a multi-part commutator is needed.
- There will be sparking at the commutator/brushes. The brushes are made from carbon which are a good conductor. Metals produce an oxide layer which is not a good conductor. When carbon oxides, it turns to carbon dioxide. Therefore, the oxidation turns some of the carbon atoms to gas maintain the conductance of the carbon brushes.
- Noise and vibrations which could lead to resonance.
Flux density is the flux per unit cross-sectional area. Flux density = Magnetic field strength.
Flux density (B) = Flux (Φ) / Area (A)
This can be re-arranged as:
Flux (Φ) = Flux density (B) x Area (A)
Flux is measured in Weber (Wb), Area in m² and flux density in Wbm^-2. 1 Tesla = 1 Wbm^-2. Another equation for the flux is:
Flux (Φ) = Permeance (Λ) x Number of turns in a coil (N) x Current (I)
The equation for Permeance is:
Permeance (Λ) = ( Permeability (µ) x Area (m²) ) / Length (L)
The units for permeance is WbA^-1turn^-1. An increase in the current increases the field around the wire/conductor. Increasing the number of turns of the coil adds the field lines from each turn of the wire to produce a larger field. The definition of permeance is how much flux that can pass through the magnetizable medium. It is like the conductance for electrical flow but for magnetism.
Calcuate the flux density of a magnetic field where 1700mWb passes through a rectangular area of dimensions 70cm x 1.2m.
- Φ = BA
- B = Φ / A
- A = 0.7 x 1.2 = 0.84m²
- B = 1700×10^-3 / 0.84 = 2.02T
Calculate the permeance of a magnetizable core if a 1500 turn coil which as a 3.2A current flowing through it produces flux of 27Wb.
- Φ = ΛNI
- 27 = 3.2 x 1500 x Λ
- Λ = 27/4800
- Λ = 5.6×10^-3 WbA^-1turn^-1
Lines of flux are analogous to the path followed by an electric current in an
electric circuit. In electromagnetic machines, the loops formed by flux are ‘linked’
with the loops formed by an electric current.
All electric circuits that carry currents induce magnetic circuits that carry flux (these circuits are perpendicular).
Flux Linkage (Φ) = Flux (Φ) x Number of turns (N)
A change in flux linkage in a magnetic circuit will induce an EMF in a linked electric circuit.
EMF = -N x dΦ/dt
The induced EMF opposes the change of flux causing it (hence the minus sign). It is often useful to link the flux density and flux linkage equations to get the following:
Flux Linkage Φ = NAB
Faraday’s law states that a change to a magnetic field and coil will cause an EMF. This can be understand with the following diagram which shows the EMF induced into a coil through a magnet falling through it.
The point D will always have a greater EMF than B is the magnet is falling (it’s speed will be faster than that at B. Therefore, rate of change of flux will increase as dt decreases). Notice how the coil always oppose the motion which is inducing a voltage, current and magnetic field in it. If the magnet is moving down, the coil’s magnetic field wants to move the coil up and vice versa.
Moving Charged Particles and Relativity
We can represent all the movement information of a particle by knowing its momentum
and its initial position – Classical physics (Newton’s law)
We all know that momentum = mass x velocity and that
Kinetic Energy = 1/2 x mass x velocity squared.
At speeds near to the speed of light, the momentum of a particle changes:
momentum = γ x mass x velocity
At the speeds near to the speed of light, the total energy of a particles changes too:
Total Energy = γmc²
This makes clear that to work out γ, it is:
γ = Total Energy / Rest energy
Where the rest energy can be worked out from E = mc² and the total energy = Rest Energy + Kinetic Energy.
The Electric Field
An electric field is a type of force field (not in the sci-fi sense!). Consequently, it exerts a force on an object with property that the field influences. For electric fields, this property is charge.
For a uniform electric field, E = F / Q = V / D. Electrical field strength can be measured in V/m or N/C. For a uniform field, the value of E is constant.
The equipotential lines on the above diagram will also have the same distance between them (and perpendicular to the field lines) because the Electrical field is uniform.
An electron is stationary in an uniform electric field of strength 10N/C. If the charge of the electron is 1.6×10^-19, calculate the force on the electron and it’s acceleration.
- E = F/Q
- F = E x Q
- F = 10 x 1.6×10^-19 = 1.6×10^-18 N
- F = ma
- a = F/m
- a = 1.6×10^-18 / 9.1×10^-31 = 1.8×10^12 m/s²
An alternative expression for the electric field strength is V / d where V = potential difference and d = distance potential differences apart.
The Linear Accelerator
The electrons spend the same time period in each cylinder. An A.C. supply is used to alternate the voltage on the cylinder from positive to negative as the electron passes through (so the cylinders accelerate the electron).
What quantities must be conserved according to the law of physics?
Charge, total energy, linear and angular momentum, lepton, boson and nucleon number.
What is the is the significance of the conservation of momentum in particle-antiparticle annihilation?
The momentum before = momentum after. Therefore, all collisions must have a momentum of 0 if the particles collide at the same speed against each other head on.
Bubble and Cloud Chambers
How are tracks made in a bubble chamber?
Charged particles create an ionization track which liquid vaporizes forming bubbles.
How are tracks made in a cloud chamber?
Instead of water, the chamber has vapor that condenses the air into mist (like a cloud).
What is the significance of the direction of curvature of the track?
If it is an electron or positron, the magnetic field across the chamber attracts/repels the particle causing it to change which why the particle is deflected.
What is the significance of the radius of curvature of the track?
The larger the radius, the faster the particles are moving.
What type of particles do not make any tracks and why?
Photons, neutrons and neutrinos because they don’t ionize and create any bubbles. They are have zero charges.
Fermions and Bosons
What is a fermion? Give examples.
A fermion is any particle with an odd half integer spin (1/2, 3/2 etc.). Protons, neutrons and electrons are all fermions.
What is a boson? Give examples.
A boson is any particle with a whole integer spin. All the forces carries such as photons, gluons etc. are all bosons.
How do they behave differently?
- Bosons want and like to be together (which is a reason why we have lasers).
- Fermions don’t like to be together and prefer to be split up from each other.
What is a neutrino?
A neutrino is a subatomic particle produced by radioactive decay. It belongs to the fermion family and is a lepton. There are six neutrinos: electron, muon and tau neutrinos with each having an antiparticle pair.
Where do neutrinos come from?
Neutrinos were ‘born’ around 15 billion years ago where they were produced at the big bang. They can also be formed at nuclear power plants or anywhere where radioactive decay is (like the sun which features nuclear fusion).
How was the neutrino discovered?
- During experimentation with beta + and – decay, scientists found that the conservationist equations were unbalanced.
- Therefore, to make the equations balanced and conserved, another particle must have been created.
- This particle’s properties, to balance the equation, must have 0 mass, 0 charge, and a lepton number of 1.
Particle Identification Diagram
The weak force, but responsible for the force between astronomical objects. The graviton has not yet been observed. Gravity is felt by all particles with a mass.
- Boson – Graviton.
- Source – Mass.
- Relative Strength – 10^-39.
- Range – Infinite.
Responsible for radioactive Beta decay. The force carriers (W+, W- and Z bosons) have mass and were discovered at CERN in 1983-4. Felt by all particles.
- Boson – W+, W- and Z.
- Source – Weak Charge.
- Relative Strength – 10^-5.
- Range – 10^-18 m.
Holds atoms together and plays a major role in everyday life. The force carrier is the familiar photon. Electricity and magnetism are simply different manifestations of this force. Felt by all particles except neutrinos, which are uncharged.
- Boson – Photon.
- Source – Charge.
- Relative Strength – 10^-2.
- Range – Infinite.
Felt by quarks only, this force also holds the nuclei together. There are eight different types of gluon carrying different combinations of colour.
- Boson – Gluons.
- Source – Colour.
- Relative Strength – 1.
- Range – 10^-15 m.
Electromagnetic Field Equations and Graphs
Below is a complete look at the electromagnetic equations with graphs to go with the equations. This is only looking at a point charge as I have already looked at the uniform electric field.
This is a diagram of what a point charge’s electric field lines with equipotential lines would look like (with a charge of +Q). Remember that field lines are always positive to negative. When you are drawing field lines, think of what direction a positive charge would move if placed in the field. This is the direction of the field lines. Notice in the below diagram that the equipotential lines distance increases the further away from the charge. This makes clear that the field density decreases the further away from the charge.
The electrical field strength at a distance of r from Q is:
E = kQ / r²
The area under this graph is the electric potential.
The electric potential at a distance of r from Q is:
V = kQ / r
The gradient of this graph represents the electric field strength at distance r.
The force on a particle of charge q placed at distance r from particle Q is:
F = kQq / r²
The area under this graph is the potential energy of the particle with charge q at a distance of r from Q.
The potential energy of a particle with charge q placed at a distance r from particle with charge Q is:
PE = kQq / r
The gradient of this graph is the force on particle with charge q from being distance r from particle with charge Q.
I hope this article has proved useful to you…it took me literally a week to write it and do all the diagrams!