38 KiB
SL Physics - 2
The course code for this page is SPH4U7.
Magnetism
Magnets are objects with north/south dipoles that create a field around them. Although ferromagnetic substances can repel each other, paramagnetic substances are always attracted to a magnetic field. See HL Chemistry#Physics properties of transition elements for more details.
Magnetic fields
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Kognity)
Similar to electric and gravitational fields, magnetic fields (also known as B-fields) are drawn by their effect on a north pole. Since magnetic poles always appear in equal magnitude pairs, all magnetic field lines for a magnet must form closed loops from north to south outside and south to north inside the magnet. Much like electric field lines, magnetic field lines never touch
(Source:
Kognity)
Atoms in ferromagnetic materials are tiny magnets with dipoles. These dipoles act on neighbouring dipoles and can cause the whole object to align — this is known as an electric domain.
!!! note Nickel, cobalt, or any alloy with nickel, cobalt, or iron can become magnetised this way.
Unmagnetised domains have dipoles pointing in random directions that are aligned when exposed to a magnetic field where they become magnetised domains. As such, bar magnets are always broken into smaller magnets, each with two poles — a monopole is impossible to create.
Straight-line electromagnets
Moving electric charges produce magnetic fields. A circle filled with an “x” indicates that the current is moving away from the viewer in the third dimension while a dotted circle indicates it is moving toward the viewer.
These magnetic fields are centred on the conductor, are in a plane perpendicular to the conductor, and have decreasing magnetic field strength over distance.
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Kognity)
The right-hand rule for straight-line conductors indicates that when the conductor would be grasped by the right hand, the thumb would point in the direction of current and the fingers pointing in the direction of the magnetic field.
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Kognity)
Selenoid electromagnets
A selenoid is a conductor coil in a tight helix. Current passed through a selenoid will generate a uniform magnetic field inside the coil with a pattern identical to that of a bar magnet outside it.
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Kognity)
The right-hand rule can be applied again to a selenoid to identify the direction of the north pole or direction of magnetic field in the coil:
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Kognity)
Properties of moving charges
As only moving electric charges generate magnetic fields, stationary electric charges are unaffected by external magnetic fields. Moving charges are affected by Newton’s third law of motion — equal and opposite forces are exerted on the charge and the magnet. As such, where \(q\) is the charge of the particle and \(\vec{v}\times \vec{B}\) is the cross product (vector multiplication) of the velocity of the particle and the magnetic field strength in Teslas: \[\vec{F_m}=q\vec{v}\times \vec{B}\]
Magnetic field strength (\(B\)) represents the force acting per unit current in a conductor of unit length perpendicular to the field with the unit Tesla (\(\pu{T}\))
The magnetic force is always plane perpendicular to both \(\vec{v}\) and \(\vec{B}\). Just the magnitude of the force can be found by using the angle between the two vectors (\(\theta\)): \[|F_m|=qvB\sin\theta\]
Regardless of \(\theta\), the force vector is always perpendicular to both \(B\) and \(v\),
The above equation can be rearranged to find electromagnetic force in terms of current and wire length in a uniform magnetic field: \[|F_em|=BIL\sin\theta\]
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Kognity
The right-hand-rule can be used to determine the direction of force — the thumb points in the direction of current/velocity, the fingers point in the direction of the magnetic field, and the palm points in the direction of force. Alternatively, just three fingers can be used.
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Kognity)
When two straight-line conductors with current flowing through them are brought together, they either mutually attract or repel. The ampere is defined based on the current required to flow through a scenario involving two parallel straight-line conductors.
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Kognity)
Inside a uniform magnetic field, charges move in uniform circular motion at a constant velocity. If the particle did not enter the field at a perfect right angle, some of the velocity is used to change the path of the particle to be in a spiral — still perfectly circular, but additionally moving in the third dimension perpendicular to the circle. \[\Sigma F_c = F_m\]
Nukes
Atomic structure
!!! definition - A photon is a particle that represents light. - A nucleon is a subatomic particle in an atomic nucleus — that is, a proton or neutron. - A nuclide is a nucleus with a specific number of protons and neutrons.
Please see HL Chemistry#2.1 - Atoms and HL Chemistry 1#2.2 - Electrons in atoms for more information.
An electron in an atom will only become excited if a photon of exactly the right amount of energy strikes it. That energy can be found using the formula: \[\Delta E=hf\]
where \(E\) is the energy of the photon at frequency \(f\), and \(h\) is Planck’s constant (\(\pu{6.63\times 10^{-34} Js}\) or \(\pu{4.14\times10^{-15} eVs}\)).
An electron that de-excites will emit a photon of that exact energy and thus frequency to return to its previous state.
Binding energy
According to Einstein’s theory of special relativity: \[\Delta E=\Delta mc^2\]
Neutrons in the nucleus hold the protons together via strong nuclear forces that somewhat act like glue. An increase in neutrons increases the strong nuclear force. In smaller nuclei, \(N=Z\), but in larger nuclei, \(N>Z\) as more neutrons are required to keep the nucleus stable as the number of protons increases.
The mass of a stable nucleus is always less than the sum of the masses of the individual nucleons (the mass defect) as some of the mass is converted to energy during the formation of a nucleus. The energy used is known as the binding energy of a nucleus. \[E_\pu{binding} = \pu{mass defect}\times c^2\]
As such, the binding energy of a nucleus is also the energy required to separate it completely into individual nucleons.
Atomic mass is measured relative to the mass of a carbon-12 atom, which is exactly 12 u (unified atomic mass units). \[\pu{1 u}=\pu{1.661\times10^{-27} kg}=\pu{931.5 MeVc^-2}\]
A higher binding energy per nucleon results in more energy required to break it apart and thus it being more stable.
(Source:
Kognity)
!!! note It is required to know the general shape of the curve, that \(~8.8 \pu{ MeV}\) is the maximum, and that the end boundaries are \(0\) and \(~7.5 \pu{ MeV}\). It is also required to know the elements at each of those points (hydrogen-1, iron-56/nickel-62, and uranium-238).
Since a greater binding energy per nucleon is more energetically favourable, nuclei to the right of iron-56 fission (split) while those to the left fuse (combine) to release energy — changes that would increase binding energy per nucleon are likely to occur because of this.
Radioactive decay
!!! definition - An alpha particle is a helium-4 nucleus (2 protons, 2 neutrons). - A beta particle is an electron. - A gamma ray is a photon.
Radioactivity is the emission of ionising (will make ions) radiation due to changes of a nucleus. It is random and spontaneous — it is unaffected by external factors such as other nuclei decaying.
Nuclear equations are similar to chemical equations but show how nuclei change in a nuclear process by tracking the atomic and mass numbers. A nuclear equation is balanced so that the sum of the atomic and the sum of the mass numbers on both sides are equal.
\[A\to B+C\]
Alpha decay occurs when the strong nuclear force is unable to hold a large nucleus together and emits an alpha particle. The alpha particle is positive and can barely penetrate paper. The two particles move in opposite directions with momentums equal in magnitude.
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Kognity)
\[\ce{^A_Z N → ^{A-4}_{Z-2} N' + ^4_2 He}\]
\(\ce{^4_2 He}\) may be replaced by \(\ce{^4_2\alpha}\).
!!! example Radium-226 alpha decays to radon-222.
Beta-minus decay (\(\beta^-\)) occurs when a neutron decays into a proton and releases a beta-minus particle (an electron and an electron antineutrino). It can penetrate up to 3 mm of aluminum. Where \(\overline{v}_e\) is the antineutrino:
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Kognity)
\[\ce{^1_0n → ^1_1p + ^0_{-1}e + ^0_0\overline{v}_e}\]
!!! note The bar over the electron antineutrino identifies it as an antiparticle.
The beta-minus particle can be written explicitly over the electron as \(^0_{-1}\beta\).
In terms of the mother nucleus, the reaction results in the mass number staying the same while the atomic number increases by one.
Beta-plus decay (\(\beta^+\)) occurs when a proton decays into a neutron and releases a positron (an antielectron with a positive charge) and an electron neutrino (\(v_e\)) \[\ce{^1_1p → ^1_0n + ^0_1e + ^0_0v_e}\]
The positron can be written as a beta-plus particle as \(^0_1\beta\).
In terms of the mother nucleus, the reaction results in the mass number staying the same while the atomic number decreases by one.
Gamma decay occurs when an excited nucleus transfers its energy to a high-energy photon with frequencies in the gamma region of the electromagnetic spectrum. There is no change in mass nor atomic number. A nuclide with an asterisk \(*\) indicates it as excited. This emits ionising radiation which is not good for living beings.
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Kognity)
\[\ce{^A_ZX^* → ^A_ZX + ^0_0\gamma}\]
Detecting radiation
As radiation cannot be seen, it must be detected experimentally.
A Geiger counter utilises a gas-filled tube with a wire in the centre at high voltage. When a charged particle passes through, gas is ionised which cascade onto the wire to produce a pulse.
A cloud chamber contains vapour that turns into liquid droplets when ionising particles pass though, resulting in visible lines showing the path of the particles. A magnetic field can spiral the particle in a certain direction which allows for its charge to be identified.
Half-life
The half-life of an element is the time required for half of the nuclides in a sample to decay — it is always the same no matter the number of initial nuclides.
As such, this means that the number of parent nuclei decreases by 50% of its current value each half-life.
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Kognity)
!!! example Radioactive dating analyses the ratio between carbon-14 and carbon-12 to determine the age of plant nmatter. As the ratio of C-14 and C-12 in the atmosphere has been relatively comstant, living plants maintain that balance by constantly exchanging carbon. Dead plants’ carbon-14 slowly decay at a known rate, so the ratio can be used to determine the time since the plant died.
Nuclear reactions
A nuclear reaction occurs when a nucleus is hit by another nucleus or subatomic particle and a different nuclide is formed (nuclear transmutation). In such a collision, energy and momentum must be conserved. Generally, neutrons are most commonly used in these reactions as they are not affected by Coulomb force exerted by the protons or electrons.
The reaction energy \(Q\) is the difference in mass between the initial and final states multiplied by \(c\) squared. In the sample reaction \(a+X → Y+b\): \[Q=[(M_a + M_X) - (M_Y + M_b)]c^2\]
- If \(Q\) is positive, the reaction is exothermic and will occur at any amount of initial kinetic energy.
- If \(Q\) is negative, an initial kinetic energy equal to \(Q\) is required (the activation energy).
Nuclear fusion occurs when two lighter nuclei combine into a heavier one, releasing energy in the process.
Nuclear fission occurs when a heavy nucleus splits into two lighter nuclei. Along with some excess neutrons, energy is released. The two split pieces are usually somewhat unequal.
Fission in reactors
The energy release of nuclei is very large — the energy density per unit mass is much higher than any other conventional source.
As nuclei get smaller, their stablility increases as the number of neutrons also decreases, so excess neutrons can set off a chain reaction by reacting with more nuclei.
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Kognity)
However, neutrons that are ejected often have too much energy and must be moderated to slow down to prevent a critical mass where the number of reactions is self-sustaining, leading to overheating and reactor meltdown. A moderator is a material surrounding fuel rods to slow down incoming neutrons — usually heavy water.
Control rods are also inserted into the reaction core to control the rate of reaction. These absorb the neutrons from the moderator and the amount absorbed can be adjusted by raising the rods partially up to all the way from the reactor.
Nuclear power is superior to other types of energy generation in that:
- it has a high power output due to high energy density
- there are large reserves of nuclear fuel on Earth
- there are no greenhouse gases emitted to generate power
Nuclear power has the following issues in that:
- waste is highly radioactive with long half-lives, rendering removal and storage of nuclear waste a major issue
- initial startup costs are expensive
- strict maintenance is required due to the risk of nuclear meltdown
- fissionable fuel can be recovered and used for destructive weapons
- mining uranium is unhealthy — miners are exposed to harmful radiation and waste material from mines deemed not pure enough is not easy to dispose
Nuclear fusion
Nuclear fusion generates energy per unit mass an order of magnitude greater than can be achieved with fission. The sun takes hydrogen and fuses it into helium. Heavier stars can fuse elements up to iron-56. \[\ce{4 ^1_1 H → ^4_2 He + 2 e^+ + 2 v_e + 2\gamma}\]
Nuclear fusion power has the following issues in that it is currently unsustainable for more than a few seconds because:
- the temperature required for the reaction is greater than 100 million degrees Celsius
- it requires more energy input to heat the sample than is obtained from the fusion reaction
- materials currently known cannot withstand the temperature making containment difficult — currently magnetic fields are used to hold the particles in place
The Standard Model
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Kognity)
Elementary particles
An elementary particle is a subatomic particle that is not composed of other particles.
Particles currently thought to be elementary as of January 2021 include bosons, quarks, and leptons:
Bosons/Force exchange particles
!!! definition - Virtual particles are bosons that do not have infinite range.
Bosons are particles that carry/exchange forces between particles. The theory of exchange forces posits that all forces are due to particles exchanged between elementary particles. There are four types of bosons that can be roughly categorised by their effect.
Gluons (\(\pu{g}\)) are exchanged for matter to feel strong nuclear force: the strongest interaction between particles. These particles are heavy (120 MeV/c2) and short-lived, thus giving the force a very short range.
Photons (\(\pu{\gamma}\)) are exchanged for matter to feel electromagnetic force: the second strongest force responsible for magnetism and electric force that only act on charged particles. These particles have a rest mass of zero and travel for an infinite distance until they are absorbed.
The W+, W-, and Z0 bosons are together responsible for the weak nuclear force and are the third strongest force. These particles have a heavy rest mass (80 GeV/c2 for Ws, 91 GeV/c2 for Z) and so are even more limited in range than gluons.
Gravitons are responsible for gravitational force: the weakest force. These particles are massless and so they have infinite range.
The Higgs field and Higgs boson are responsible for elementary particles obtaining their mass because of magical fields and rainbows.
Quarks
!!! definition - A hadron is any particle made of quarks. - A baryon is any hadron made of three quarks. An antibaryon is any particle made of three antiquarks. - A meson is a hadron made of exactly one quark and one antiquark involved in the strong interaction. - A fermion is any particle with mass (hadrons or leptons)
Gluons (strong force) only interact with quarks, which are heavier, more tightly bound elementary particles. There are six quarks with different properties:
| Charge | |||
|---|---|---|---|
| \(\frac{2}{3}\)e | up (u) | charm (c) | top (t) |
| \(-\frac{1}{3}\)e | down (d) | strange (s) | bottom (b) |
!!! reminder e is the elementary charge (\(\pu{1.6\times10^{-19} C}\)).
- All quarks have a baryon number of \(\frac{1}{3}\).
- All quarks have a strangeness number of 0 except for the strange quark, whose number is equal to -1.
- All quarks have their own respective antiquark: an antiparticle with opposite charge and baryon number but otherwise identical mass.
- The quark confinement theory states that a singular quark cannot be isolated from its hadron.
Every particle has its own antiparticle with the same properties but with opposite quantum numbers. In practice, this indicates that mass stays the same while baryon number, lepton number, and charge are opposite. When a corresponding quark and antiquark meet, annihilate each other to become energy. They are denoted by a bar over their letter.
!!! example An up antiquark (also known as “u-bar”) is written as ū.
!!! note - Protons are composed of two up quarks and one down quark (uud). - Neutrons are composed of one up quark and two down quarks (udd).
Leptons
Leptons are lighter and more loosely bound elementary particles compared to quarks. They do not participate in the strong interaction. All leptons have a lepton generation/family which is based on their relative mass. A higher mass indicates a higher generation.
| Charge | Generation 1 (LI) | Generation 2 (LII) | Generation 3 (LIII) |
|---|---|---|---|
| -1e | electron (e) | muon (µ) | tau (\(\tau\)) |
| 0 | electron neutrino (\(\pu{v_e}\)) | muon neutrino (\(\pu{v_\mu}\)) | tau neutrino (\(\pu{v_\tau}\)) |
- All leptons have a lepton number of 1.
Elementary particle interactions
In any interaction, the following are true:
- charge is conserved
- the baryon number is always conserved
- the lepton number of each family is always conserved
- the strangeness number is always conserved in strong and electromagnetic interactions
!!! example A lepton number of \(\pu{L_{III} = 1}\) on one side becoming \(\pu{L_{II} = 1}\) on the other is impossible as lepton family must be kept consistent during interactions.
Feynman diagrams
A Feynman diagram provides a graphic representation of particle interactions to predict the outcome of a particle collision.
Generally, the time axis is left-to-right but can be specified to be otherwise. The following assumes time moves from left to right.
Fermions are represented by straight lines with arrows. Particles have their arrows pointing forward in time while antiparticles point backward (even though they still move in the direction of time).
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Kognity)
Bosons/force exchange particles are represented by wiggly lines with no arrow.
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Kognity)
Particles only interact at a vertex where left refers to the state before the interaction while the right refers to the state afterward. A vertex must have one arrow pointing toward and one away from the vertex. Conservation laws apply at each vertex.
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Kognity)
Contents of hadrons must be shown. (See the last example for an example.)
Feynman diagram examples
!!! example An electron being repelled by another electron due to
Coulomb repulsion:
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Kognity)
!!! example Beta decay:
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Kognity)
!!! example Some weak interaction that violates strangeness:
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Kognity)
Energy sources
!!! definition - A primary energy source is one that is not transformed and used directly by the consumer, such as burning wood for heat. - A secondary energy source is one that is converted from a primary source, such as electricity. - Proved reserves are the resources that are known to be obtainable. - Production are the actual reserves and placed in the market in a certain time interval. - The expectancy of a product is the estimated time its proved reserves will last given its production (proved reserves ÷ production). - The specific energy (\(E_{SP}\)) of a source is the energy obtained per unit mass of fuel (J/kg). - The energy density (\(E_D\)) of a source is the energy obtained per unit volume (J/m3).
If a fuel source can be replenished with a “reasonable” amount of time — one human generation, or 50–100 years — it is considered to be renewable. The world still largely uses non-renewable energy sources.
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Kognity)
Electricity is the most common secondary energy source due to its convenience and ease of transport.
Sankey diagrams
Sankey diagrams show the transfer of energy in a system via directed lines proportional to quantity of energy. Arrows pointing away indicate energy degradation — losses in energy transformation.
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Kognity)
To minimise electrical losses during transportation, high voltage, low resistance wires with high cross-sectional areas are used to reduce resistance to reduce power loss.
Power generation
A moving magnetic field produces an electromotive force as alternating current via induction and is how the large majority of power generation is handled.
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Kognity)
In a nutshell, a source of thermal energy such as burning fossil fuels is used to boil water whose steam is then used to turn a turbine to generate power before condensing and repeating the cycle.
Coal and oil-powered power plants have efficiencies of around 40% while natural gas is slightly higher at 50% as the gas itself can be used to turn a turbine.
In nuclear power plants, the coolant fluid is instead used to turn a turbine.
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Kognity)
As only uranium-235 is fissile, but uranium-238 is significantly more common (99.3% U-238 to 0.7% U-235), uranium must first be enriched until the concentration of U-235 is ~3%. Gaseous diffusion is a form of enrichment by forming uranium hexafluoride gas and then spinning it in a centrifuge to force separation of U-238 and U-235 based on mass. The energy used in enriching uranium is substantial and should be included in Sankey diagrams.
Moderators are used to encourage fission as they slow down neutrons that are going too fast to fission (most) to a speed more suitable for fission.
Wind
The kinetic energy of wind can be harnessed to generate power. As convection currents provide the greatest airflow near large bodies of water, wind farms are often constructed there. The wind turns rotor blades which turn a turbine to generate power. It is a source of clean and renewable energy.
Assuming all wind kinetic energy is converted to mechanical energy, where \(P\) is the power generated, \(A\) is the area of the circle that the blades spin around, \(\rho\) is the density of the air, and \(v\) is the speed of the wind in the direction of the blades: \[P=\frac{1}{2}A\rho v^3\]
(Source:
Kognity)
| Advantages | Disadvantages |
|---|---|
| Renewable | Wind strength is inconsistent |
| Wind is widely available | Turbine blades may kill birds |
| Does not emit greenhouse gases | Many of them are needed to replace one fossil fuel plant, requiring lots of space so they don’t interfere with each other |
Hydro
In hydroelectric plants, a dam is often used to increase the height of a reservoir so that it falls and spins a turbine to generate power. As such, the energy generated is roughly equal to the gravitational potential energy of the water. Where \(\Delta h\) is the average height of the water from the turbine: \[E=mg\Delta h\]
(Source:
Kognity)
During times of lower demand, dams often have a pumped storage system that pumps water back into the reservoir for use during higher demand.
Tidal
A tidal barrage generates energy via the kinetic energy of water moving during changes in tide using a multi-directional turbine.
Photovoltaic
!!! definition Intensity is the power delivered per unit area (watts per square metre).
Photovoltaic (PV) cells are made of silicon doped with phosphorus and boron impurities to convert sunlight directly into electricity. Light from the sun frees electrons in the silicon to produce a current.
(Source:
Kognity)
The solar constant \(S\) of \(\pu{1.36\times10^3 W/m^2}\) determines the intensity of the sun’s light that reaches the Earth. At different latitudes and between seasons, the intensity changes because the Earth is round and is tilted, respectively.
Solar heating
Instead of converting between multiple forms of energy, solar heating directly converts the sun’s energy to heat, increasing efficiency drastically. By using insulation, a black substance, and a glass top, the heat from the sun is trapped and absorbed into water where it is used to heat things.
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Kognity)
Thermal energy transfer
!!! definition - Radiation is the transfer of energy via electromagnetic waves emitted away from an object. No medium is needed. - Convection is the transfer of thermal energy via another medium away from an object. - Conduction is the transfer of thermal energy via physical contact.
Black bodies
!!! definition Emissivity (\(e\)) is a dimensionless value from 0 to 1 indicating the ability of an object to emit radiation relative to a black body (which has an emissivity of 1). Darker and duller surfaces have an emissivity closer to 1 while shinier and whiter ones are closer to 0.
All bodies with an absolute temperature will emit radiant energy in the form of electromagnetic waves. The temperature of the body determines the wavelengths and power of the radiation emitted. A perfect emitter has an emissivity of 1 and is known as a black body, absorbing all electromagnetic radiation.
Generally, as the temperature of a body goes down, its peak power density is reduced and its peak wavelength increases.
Wien’s displacement law relates the temperature of a black body to the waves it emits. Where \(\lambda_\text{max}\) is the peak wavelength in metres and \(T\) is the temperature of the body in kelvin: \[\lambda_\text{max}\times T=\pu{2.9\times10^{-3} m\cdot K}\]
The Stefan-Boltzmann law relates the specifications of a body to the power it emits. Where \(P\) is the power emitted by the body, \(A\) is its surface area, \(T\) is its temperature, \(e\) is its emissivity, and \(\sigma\) is the Stefan-Boltzmann constant (equal to \(\pu{5.67\times10^{-8} Wm^{-2}K^{-4}}\)): \[P=e\sigma AT^4\]
In problems where the environment temperature is different from the temperature of an object, there will be power loss. The net power emitted by a body will be: \[P_\text{net}=e\sigma A(T_1^4-T_2^4)\]
The solar radiation reaching earth is equal to \(\pu{S= 3.9\times10^{26} W}\) with the assumption that it is a black body.
Albedo
Derived from \(I=\frac{P}{A}\), the intensity at a point in space can be related to the power of the radiation emitted by the source (\(P\)) and the distance between the two (\(d\)): \[I=\frac{P}{4\pi d^2}\]
!!! example The solar constant is derived in this way by substituting \(d\) as the distance from the Earth to the sun.
As Earth and most other planetary bodies are not flat disks pointed at the sun, in reality the sun’s intensity is reduced to a quarter due to the formula for a sphere. Therefore, the power absorbed/incident to the Earth is equal to, where \(S\) is the solar constant: \[P_\text{in}=(1-\alpha)\frac{S}{4}A\]
Albedo (\(\alpha\)) is the ratio of power from incident rays reflected or scattered to the power absorbed by a body, ranging from 0 to 1. A black body has albedo 0. On average, Earth’s albedo is equal to \(0.3\) due primarily to the atmosphere but also clouds and ice. \[\alpha=\frac{\text{energy scattered/reflected}}{\text{energy absorbed}}\]
Greenhouse gases are responsible for remaining increases in temperature. By absorbing and then re-emittng their natural frequencies of electromagnetic radiation (infrared for greenhouse gases), they delay the release of radiation back into space and heat up the atmosphere.
Radiation absorption by greenhouse gases
All molecules have a natural frequency at which they absorb radiation from the electromagnetic spectrum and resonate at. The natural frequency of greenhouse gases is in the infrared region, which is what the Earth re-emits solar radiation as. Therefore, greenhouse gases absorb this radiation and them re-emit it in all directions, “trapping” some of the radiation. Resonance is also the phenomenon responsible for the protection from ultraviolet radiation by the ozone layer.
Photoelectric effect
Wave-particle duality posits that everything can be described as either a particle or a wave, and that all particles will show some wave characteristics and all waves will show some particle characteristics, so really what the hell is happening?
The photoelectric effect is the phenomenon in which electrons are emitted when electromagnetic radiation hits a material. It was theorised that EM radiation travelled in discrete energy packets known as quanta, which held a defined amount of energy that could not be divided smaller. Where \(h\) is Planck’s constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength of the light: \[E=\frac{hc}{\lambda}\]
!!! example The application of the photoelectric effect in the real world has led to large advances in technology, such as in photodiodes in light sensors — semiconductors where electrons are freed by indicent light to raise the conductivity of the material, and charged coupled devices in cameras — the “film” of the modern digital camera.
It was later observed that for a given material, electrons were ejected when light shone on a surface only if the light’s energy/frequency was greater than a certain threshold. Below that threshold, electrons were not ejected regardless of the intensity of the light.
Electrons have a “binding energy” that hold them to the nucleus of an atom. To release an electron from the nucleus, energy greater than that binding energy must be provided. This binding energy, known as the work function \(W_0\), is therefore the minimum energy required for a surface to eject electrons.
As such, where \(E_k\) is the kinetic energy of the ejected electron, \(E_{ph}\) is the energy of the incident photon/radiation, and \(W_0\) is the work function of the surface: \[E_k=E_{ph}-W_0\]
Electrons are ejected and thus the photoelectric effect observed only if \(E_k>0\). The equation above shows that the kinetic energy of an ejected electron is determined only by the wavelength/frequency of the incident radiation and not by the intensity of the light.
The intensity of incoming radiation effectively represents the number of photons striking per unit area of a surface, so while it does not affect whether electrons are ejected, it affects the number of electrons that are ejected only if they are determined to be ejected.
In a light frequency-kinetic energy graph,
- the x-intercept represents the threshold frequency \(f_0\): the minimum frequency required to liberate electrons at all.
- the y-intercept represents the work function \(-W_0\): the “binding energy” of the electrons.
Momentum of photons
It was also observed that during photon-electron collisions that momentum and kinetic energy were conserved, further reinforcing the idea of wave-particle duality of light. Where \(h\) is Planck’s constant, \(f_i\) and \(f_f\) are the initial and final frequencies of the light, \(m\) is the mass of the electron, and \(v_{ei}\) and \(v_{ef} are the initial and final velocities of the electron:\)\(hf_i+\frac{1}{2}mv^2_{ei}=hf_f + \frac{1}{2}mv^2_{ef}\)$
In general, the momentum of a photon is equal to, where \(h\) is Planck’s constant and \(\lambda\) is the wavelength of the light: \[p=\frac{h}{\lambda}\]
Matter waves
Particles/waves cannot act like particles and waves at the same time. For a given observation, it adopts the property of one or the other. In reality, all particles exhibit wave properties sometimes and all waves exhibit particle properties sometimes. Each particle has a wave function that determines how likely it is to be somewhere at any point in time.
By equating the equations for momentum of photons and particles, the (de Broglie) wavelength of a particle can be determined. Where \(\lambda\) is the wavelength of the particle, \(m\) is its mass, \(v\) is its velocity, and \(h\) is Planck’s constant: \[\lambda=\frac{h}{mv}\]
Therefore, wavelengths of “particles” are only really significant for small masses at high speeds rather than large masses at lower speeds.
!!! example A 50 kg mass moving at 16 m/s is has a wavelength many orders of magnitude smaller than a quark and will not display any observable wave behaviours.
The discovery of wave-particle duality has led to advancements in technology such as the scanning electron microscope.
Special relativity
Einstein’s theory of special relativity states that time and space are relative depending on the frame of reference of the observer, and light travels at the same speed of \(\pu{3.0\times10^8 m/s}\) in a vacuum no matter how it is observed in all inertial frame of reference.
- An inertial reference frame is one in which the law of inertia in the frame holds true. Only frames of reference moving at a constant velocity or at rest are inertial, and the same laws of physics apply in all inertial frames of reference.
Time dilation and length contraction
The faster an observer moves, to ensure that it appears to them that light travels at \(c\), time slows down for the observer. Observers in inertial frames of reference will experience time at a slower rate — this phenomenon is known as time dilation.
For two inertial reference frames, where \(t_s\) is the time observed between two events (stationary/proper time) at the same location and at rest relative to a stationary observer, \(t_m\) is the time observed between two events in a different frame of reference (moving time), \(v\) is the speed difference between the frames of reference, and \(\gamma\) is the Lorentz factor: \[ \gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \\ t_m=\gamma t_s \]
Space is also relative. An observer moving at a higher constant velocity will have space contract — this phenomenon is known as length contraction.
Where \(L_s\) is the length/distance measured of an object at the same location and at rest relative to a stationary observer, and \(L_m\) is the length/distance from a different reference frame: \[L_m=\frac{L_s}{\gamma}\]
From different inertial reference points, there can never be agreement on whether events are simultaneous. Because time slows down for an observer moving faster, one event will also appear to happen first.
Matter-energy equivalence
The total energy of an object in an inertial frame of reference relative to another is related to its velocity and its mass. \[ E_t=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}} \\ E_k=E_t-E_\text{rest} \]
At rest, the energy of an object is related to its mass. \[E_\text{rest}=mc^2\]