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- Energy, Power & Resistance
Potential Difference Potential Difference (p.d.) is the work done per unit charge – the amount of energy converted from electrical to another form per coulomb of charge passing through a component. W = VQ work done = p.d. × charge Electromotive Force Electromotive Force is the amount of energy converted into electrical energy per coulomb of charge through the source: W= εQ work done = e.m.f x charge Essentially, p.d. is voltage used up in a circuit, whereas e.m.f. is the voltage provided to the circuit by a cell or power supply. Both p.d. and e.m.f. are often referred to as voltage, because they both shave the Volt as their unit. The Volt is 1J of energy per 1C of charge. Accelerating Electrons A cathode-ray tube (also called an electron gun) is used to accelerate electrons. A hot metal filament releases electrons by thermionic emission, and then these are accelerated in a vacuum between the filament and a metal plate with a small hole in it. The two are connected and a very high p.d. is applied, making the filament a cathode and the plate an anode. The electrons released from the filament are attracted to the positively charged anode plate, and pass through the hole. This creates a concentrated beam of high velocity electrons. When an electron is accelerated in this way, the energy transferred equals the wok done on the electron, which equals the kinetic energy: eV = 1/2 m v^2 work done on electron = gain in kinetic energy of electron Resistance All conductors have resistance that obstructs the flow of charge through them. The higher the resistance of a conductor, the more energy is required to push the charge through, as more energy is lost in the process. Ohm's Law Ohm's Law states that the current flowing through a conductor is directly proportional to the p.d. across it, provided that the physical conditions (such as temperature) remain constant. V = IR p.d. = current x resistance The units of resistance are Ohms, Ω, defined as the resistance of a conductor when 1 volt produces a current of 1 ampere through that conductor. I-V Characteristics To investigate the I-V relationships of components, a test circuit as shown can be set up and the current recorded for a variety of potential differences. Taking averages increases reliability, measuring to 3 sig-figs increases accuracy. A graph can be plotted to see its ohmic or non-ohmic properties. The ammeter must be connected in series, and the voltmeter in parallel. A Resistor/Metallic Conductor obeys Ohm's Law - the I-V characteristic is a straight line through the origin, where the inverse of the gradient is the resistance (R = V/I). A Filament Lamp is an example of temperature affecting resistance. As the p.d. increases, so does the brightness and temperature of the bulb and with it, its resistance. This is because at higher temperatures, the positive ions in the filament have greater internal energy, meaning they vibrate more. Therefore, the probability of the charge carriers colliding with them is greater. In this instance, Ohm's law does not apply. A Diode only works with current flowing thorough it in one direction - this is known as the forward bias. Typically, there is a threshold of around 0.6V before they conduct at all. Light Dependent Resistors and NTC Thermistors (Negative Temperature Coefficient) lose resistance as the energy on them increases. This is because they are semiconductors, that release more electrons for conducting when they have more energy. This means the electron density increases, so the current increases and resistance decreases. Resistivity Resistivity is a property of a material, as different structures make for different conducting properties. It is defined as the resistance from one side to another of a 1m cube. R = ρL/A resistance = resistivity × length / cross-sectional area This relationship means that resistance is directly proportional to length, while it is inversely proportional to the cross-sectional area: R ∝ L R ∝ 1/A Resistivity is measured in ohm meters, Ωm Investigating Resistivity To determine resistivity, you need to work out the cross-sectional area using a digital caliper. Then, clamp the wire in place with a voltmeter in parallel and an ammeter in series with a fixed power supply. Vary the length of wire (use a ruler) and measure voltage and current to establish the resistance. Repeat at several different lengths. The resistivity of a metal increases with temperature. This is because charge is carried by moving electrons, but heat makes this motion harder as the ions in the structure vibrate more (see I-V characteristics of a filament lamp above). The resistivity of a semiconductor decreases with temperature, as more electrons are released for conducting (See LDR/Thermistors above). Energy & Power Power is the rate of transfer of energy, where one Watt is one joule of work done per second. P = IV power = current x voltage Using the equation V = IR, we can rewrite this power equation to other, often more useful forms: If power is energy per second, then we can use it to work out the electrical energy (work done): W = Pt work done = power x time Again, this can be rewritten in multiple ways: A joule is a very small unit of measuring energy. In fact, it is so small that it is rarely used commercially, because we typically use thousands or millions of joules at a time. Therefore, the Kilowatt-hour is used domestically: 1 kWh = 3.6 million joules This can quickly be calculated: 1 kW = 1000W 1 h = 60x60s = 3600s 1 kWh = 1000W x 3600s = 3,600,000 Ws = 3,600,00 J Energy Transferred (kWh) = Power (kW) x Time (h)
There are two forms of waves - progressive and stationary. Progressive waves transfer energy and can either be transverse, where the vibrations are perpendicular to the direction of travel such as EM waves, or longitudinal, where the vibrations are parallel to the direction of travel such as sound waves. Wave Motion Displacement, x, is how far the wave has moved from its undisturbed position Amplitude, A, is the maximum magnitude of displacement Wavelength, λ, is the length of one whole wave cycle, e.g. from peak to peak Wave speed, v, is the speed at which the wave moves – the distance it travels per second Period, T, is the time taken for the whole cycle to complete Frequency, f, is the number of cycles passing a given point per second Phase is a measurement of the position of a certain point along the wave Phase difference is the amount one wave lags behind another. It is measured in degrees or radians, where one wavelength represents 360° or 2π Determining Frequency Frequency can be determined using a cathode ray oscilloscope to measure voltage. It displays waves from a signal generator as a function of voltage (y-axis) and time (x-axis), from which the time period can be calculated, and this used to work out frequency. The units of frequency are Hertz, Hz, or /s The wave equation: λ = 1/T Frequency = 1 / Time Period If you know the frequency and the wavelength, you can use the wave equation to work out speed: v = fλ wave speed = frequency × wavelength Intensity is the measure of how much energy a wave is carrying. For example, the ‘brightness’ or ‘loudness’ of light and sound waves are just their intensity. I = P/A Intesnisty = Power / Area It is proportional to amplitude squared: Intensity ∝ Amplitude Squared Reflection, Refraction, Diffraction & Polarisation Reflection Waves are reflected when they change direction at a boundary between two different media but stay in the same medium. This means the wavelength and frequency stay the same, so the angle of incidence equals the angle of reflection. Refraction Refraction occurs when a wave moves from one medium into another, and it is the way in which the wave changes direction when this happens. This change in direction is a result of the wave speeding up when entering a less dense medium and bending away from the normal, or slowing down when entering a more dense medium and bending towards the normal. This happens because the frequency remains constant but the wavelength changes. Diffraction Diffraction occurs when waves pass through narrow openings – the effect is most noticeable when the gap is a wavelength or less across. This can be demonstrated by a ripple tank or shining monochromatic light through a slit, onto a screen. Polarisation Polarisation leaves waves vibrating in only one direction, e.g. 2D not 3D. Only transverse waves can be polarised, and two perpendicular polarising filters allow no waves through. This can be demonstrated using polarising filters for visible light, or microwave transmitters and receivers along with a metal grille for microwaves. Microwave transmitters transmit vertically polarised waves, so you only need one grille. Electromagnetic Waves Electromagnetic waves are a unique subset of transverse waves, because they can travel and transfer energy in no medium - they can travel through a vacuum. This is known as self-propagating, meaning they can sustain themselves due to their electric and magnetic fields, acting at right angles to one another. All electromagnetic waves travel at the same speed in a vacuum, but at different frequencies and wavelengths. The speed of light in a vacuum is 3.0E8 m/s E.M. waves, like all transverse waves, can be reflected, refracted, diffracted and polarised. The Electromagnetic Spectrum Refraction of Light Different materials slow down light by different amounts. We express this property of the material as the refractive index: n = c/v refractive index = speed of light in a vacuum / speed of light in the material Similarly, Snell’s law works this out using angles: Total Internal Reflection As well as a refractive index, materials also have a critical angle. sin C = 1/n sin of the critical angle = 1 / refractive index θ < C When θ is less than the critical angle, some of the light is internally reflected, but most emerges and speeds up, bending away from the normal. θ = C When θ is the same as the critical angle, the light nor emerges nor reflects - it refracts along the edge between the two media. θ > C When θ is greater than the critical angle, all the light is internally reflected at the same angle, θ, from the normal. This is called total internal reflection (TIR) Superposition Superposition occurs when two or more waves meet and pass through each other. When the meeting happens, the two waves instantaneously combine to form a resultant wave of the sum of the individual amplitudes. Then, the two waves separate and continue unchanged. This moment of interference can be either constructive or destructive: constructive interference is when a peak and a peak give a bigger peak destructive interference is when a peak and a trough cancel out to give a smaller peak or no peak at all (if the two waves have the same amplitude) Waves must be coherent to interfere – have the same wavelength and frequency, and be a fixed phase difference apart. Destructive interference is use in noise-cancelling headphones. A microphone detects the sound waves from the surroundings, and the speaker generates the exact same waves but in antiphase. Path & Phase Difference Path difference is how much further one wave has travelled than the other to get to the point of interference. If the path difference is a whole number of wavelengths, constructive interference occurs. This is known as being in phase and is commonly described as a multiple of 360˚or 2π radians. If the path difference is a half number of wavelengths, destructive interference happens. This is known as being in antiphase, when the waves are 180˚or π radians apart. Investigating Interference Interference can be demonstrated with two parallel speakers connected to the same signal generator. When passing some distance in front of and parallel to them, the volumes will vary from loud to quiet depending on whether the path difference is a whole or half number of wavelengths. Exactly the same experiment can be carried out with a microwave generator, a double slit, and a microwave receiver. Young's Double Slit A good way of demonstrating interference using visible light is with laser and a double slit. Laser light is, by definition, coherent, and if the slits are about the same length as a wavelength, the light is diffracted. The diffraction patterns from each slit cross over one another, interfering when the waves meet. This gives a pattern like this: Young’s experiment was the first evidence for the wave nature of light. Newton believed light to be a series of particles, like air, but Huygens believed it to be a wave. The experiment can be used to calculate the wavelength of the light source by measuring the slit separation, a, the fringe spacing, x, and the distance between the slit and the screen, D. λ = ax / D wavelength = slit seperation × fringe spacing / slit to screen separation Slit separation is sometimes labelled a, sometimes d A diffraction grating can be used to give a sharper image on the screen than a double slit, as when there are hundreds of slits per mm more beams reinforce the pattern. This means the bright spots are brighter and narrower and the dark spots are darker and wider. Measuring the fringe width from the zero order to the nth order allows you to calculate the angle of diffraction. This can be used to work out the wavelength more accurately: d sinθ = nλ slit separation × sin(angle of diffraction) = order × wavelength Stationary Waves Stationary Waves occur when two identical waves meet from opposite directions (identical in speed, frequency and length). The two waves superpose to form a series of nodes (no amplitude) and antinodes (maximum amplitude). This happens all the time when waves reflect on themselves. Phase difference on stationary waves is different: in between adjacent nodes, particles in a stationary wave are in phase, yet their amplitudes are different on different sides of a node, particles are in antiphase, as they reach the negative of the amplitude at the same time. Unlike a progressive wave, there is no net energy transfer by a stationary wave as the two waves go in opposite directions. The wavelength of a progressive wave is the distance between two adjacent nodes, whereas on a stationary wave the wavelength is the distance between two nodes/antinodes. Forming Stationary Waves Stationary waves can be formed with a microwave generator. When the waves are reflected off a metal sheet a stationary wave forms (if the distance is correct). A receiver will read a series of minimum intensities at the nodes, and maximum intensities at the antinodes. Harmonics When a string is taught, there is a node at each end. If this is plucked, or caused to vibrate, it will form a note – the first harmonic – at its fundamental frequency. However, at the same time stationary waves of half the wavelength and double the frequency will form, creating the second, third, fourth harmonic etc. The same occurs inside a hollow tube. If the tube is open at only one end, a node forms on the closed end and an antinode at the opening. This means only odd multiples of the fundamental frequency are possible, as each harmonic is ¼ of a wavelength, rather than a half. A tube that is open at both ends, however, can form any integer multiple of the fundamental frequency, as the first harmonic is also half a wavelength.
- Electrical Circuits
First off, a recap of Kirchhoff's laws: Kirchhoff's 1st Law Since charge is conserved in a circuit, none can be lost at path junctions. The rate of flow of this charge is also unchanged, meaning the total current going into a branch equals the total current coming out of the branch, regardless of how many points are on the branch. Current/charge is conserved around a circuit – when the current reaches a branch it splits so that the total current in the branches is equal to the current before splitting. Kirchhoff's 2nd Law This law applies the principle of conservation of energy to electrical circuits. It says that the amount of energy being put into the circuit (the E.M.F.) must equal the amount of energy coming out of the circuit (the P.D.): Electrical energy is conserved in a circuit - the sum of the e.m.f.s around any closed loop equals the sum of the p.d.s around the closed loop. Practically, these laws mean that in series, the voltage across all components adds up to the supply voltage and the current is the same across each component, while in parallel, the total voltage in each branch is the same and the current in the branches add up to the pre-branching current. Resistors in Circuits The layout of resistors in a circuit makes a big difference. In Series, the values of the resistors are just added together to find the total resistance. In parallel, however, the sum of one over each resistance value gives one over the total resistance. Internal Resistance All sources of e.m.f. have internal resistance, because the electrons collide with the atoms inside the power supply – this is what causes them to warm up. The terminal p.d. is the voltage used in the circuit, or the p.d. across the load resistance, R. If there was no internal resistance, the terminal p.d. would equal the e.m.f. of the supply. The energy wasted per coulomb of charge in overcoming the internal resistance is known as the lost volts, v. There are three equations associated with internal resistance: ε=V+v e.m.f. = terminal p.d. + lost volts ε=I(R+r) e.m.f. = current (load resistance+internal resistance) ε=V+Ir e.m.f. = terminal p.d. + (current × internal resistance) It is easiest to think of internal resistance as a normal resistor connected in series with the power supply and then use normal electricity laws and equations to work things out. Investigating Internal Resistance ε = V + Ir rearranges to V = −Ir + ε, the equation of a straight line (y = mx + c). This can be used experimentally to work out internal resistance: measure the terminal p.d. and current in the circuit as you vary the load resistance plot these on a graph of V against I The gradient will be the internal resistance and the y-intercept the e.m.f. Additionally, putting a voltmeter at either end of an e.m.f. source will tell you a value very slightly smaller than its e.m.f., as voltmeters have a very high resistance but a small current still passes through. Good to know: When the circuit is open, and no current is drawn, the e.m.f. equals the terminal p.d. When the circuit is shorted, the terminal p.d. is 0 and the e.m.f. is at a maximum. Potential Dividers Potential dividers are amongst the most common circuit types, because they have the simple but crucial ability to vary the p.d. across an output when connected to a fixed input. Potential dividers consist of two or more resistors are connected in series so that the voltage is split across them to add up to the e.m.f. The p.d. is divided in the ratio of resistances: Potential dividers can be used to produce a varying output voltage, either manually with a variable resistor or automatically with an LDR or thermistor. The diagram above shows a variable resistor. As the resistance across R2 changes, so does the ratio of resistances (because R1 has a fixed resistance). This means the voltage across each resistor also changes. The higher the resistance on R2, the higher the output voltage. Potentiometers A potentiometer is a variable resistor with three terminals – it s essentially a normal potential divider, but the two resistors are combined into one variable resistor. As the contact moves towards the top, the ratio of resistances changes the output p.d. increases. As the contact moves down, the output p.d. decreases. At the top, Vout = Vin, whereas at the bottom, Vout = 0. Potentiometers are used for volume control, dimmer switches etc.
- Quantum Physics
In 1900, Max Planck discovered that EM energy travelled in little packets rather than as a continuous wave (like charge, EM radiation is quantised - hence the name), suggesting EM radiation was, in fact, particulate. These little packets of energy are called photons. The photon model is used to explain how EM radiation interacts with matter, while the wave model explains its propagation through space. The energy of a photon is directly proportional to its frequency: E = hf energy = Planck′s constant × frequency Planck’s Constant is 6.63 E−34 This can be combined with the wave equation for the speed of light in a vacuum: E = hc/λ energy = Planck′s constant × 3.00 E8 / wavelength Rearranging this shows that photon energy is inversely proportional to the wavelength. Electron Volts Because photon energy is so small (a red photon has an energy of 3.00 E-19 J), the electron volt, eV, is used. It is defined as the energy transferred to/from an electron when it moves through a p.d. of 1 Volt: 1 eV = 1.60 E−19 J 1 J = 6.25 E18 eV Investigating Planck's Constant Planck’s constant can be investigated by seeing at exactly what p.d. an LED goes on – the threshold voltage. This, multiplied by the elementary charge, gives the wave energy, and since we know the frequency of the light, we can deduce Planck’s Constant. Repeating this for a number of different colours will give a more accurate result. The Photoelectric Effect When UV radiation is shone onto certain metals, like zinc, electrons are released from the surface of the metal. These electrons are called photoelectrons, and a gold leaf electroscope can be used to demonstrate this occurring: The electroscope is given a negative charge, causing the gold leaf to stand erect (as like charges repel) UV light is shone onto the zinc plate If enough electrons are photoelectrons, the electroscope loses its charge and the gold leaf falls down There are three key observations from the photoelectric effect: Photoelectrons are only emitted if the UV radiation shining onto the zinc plate is above a certain frequency - this is called the threshold frequency. If the higher the frequency is above the threshold, the greater the kinetic energy of the emitted photoelectron. If the radiation is above the threshold frequency, photoelectron emission is immediate and instantaneous - suggesting that it is brought about by one particle of radiation hitting the surface (a photon). Increasing the intensity of the UV radiation does not increase the kinetic energy of the photoelectron emitted. Instead, increasing the intensity causes more photoelectrons to be emitted. The Work Function The threshold frequency is explained because an electron will only be emitted if the photon has enough energy to do so. The minimum energy for emission is the work function, ϕ: Einstein's Photoelectric Equation The energy transfer is explained in Einstein’s Photoelectric equation: It is the maximum KE, because some electrons require more energy to be emitted than others, and as such will have less KE after they are emitted. This is because they are closer to the positive ions in the metal, and as such more energy is required to overcome the attraction between the electron and the positive metal ion. The photoelectric equation can be re-written as the equation of a line, where Planck’s constant is the gradient and the work function the y-intercept. Wave-Particle Duality We have already established above that EM radiation, a wave, can have particle-like properties. This means that particles, like electrons, can also have wave-like properties... Electron Diffraction Electrons are accelerated through a vacuum tube They collide with a polycrystalline graphite film The electrons pass through the spaces between the carbon atoms They fall onto a fluorescent screen, which shows a series of concentric circles where they land This shows the polycrystalline carbon acts as a diffraction grating - the gaps in the carbon structure are roughly the size of the wavelength of the accelerated electrons. The de Broglie wavelength The smaller the voltage, the slower the electrons, the wider the gaps between rings – this shows the wavelength of a particle is inversely proportional to its momentum: λ= h/p wavelength = Planck′s Constant / momentum The larger a particle, the harder it is to observe its wave like properties, as its momentum is so large, giving it a minuscule wavelength. This is why it is generally only seen with electrons.
- Thermal Physics
The temperature of a substance is defined by the amount of internal energy it has. Generally, warmer substances will transfer thermal energy to cooler substances until they are the same temperature - this is known as thermal equilibrium. In every day life, we use either °C or °F to measure temperature. The degree, °, however, shows that the temperature is relative to a substance-specific standard. Degrees centigrade are related to water, with 0 °C being its freezing point and 100 °C its boiling point. Degrees Fahrenheit is jut a mess with water being liquid between 32 °F and 212 °F. In physics, using a relative temperature scale is not practical. Instead, absolute temperature is used, the units of which are Kelvin, K. Zero Kelvin is not relative to any particular substance - rather it is the absolute minimum possible internal energy (where particles have no kinetic energy at all, so do no vibrate) T = θ + 273 Absolute temperature (in K) = Relative temperature (in °C) + 273 A change of 1 K is the same as a change of 1 °C T is used for absolute temperature, θ for relative Absolute zero is -273 °C Solids, Liquids & Gasses The kinetic model explains the three phases of matter: Solids Solids have their particles arranged in a regular pattern, all touching one another. They vibrate around fixed positions, and are held together by strong electrostatic attractive forces. The greater the thermal energy, the greater kinetic energy the particles have, so they vibrate faster. Liquids The particles in a liquid are further spaced apart, but generally still touching. They are not fixed into a regular arrangement, but are free to move around - this allows liquids to flow. Due to this, liquids adapt to the shape of a container. The greater the thermal energy, the faster the particles move (as they have more KE). Gasses In a gas, the particles are completely free to move, and do so at a high speed. They fill the space they are confined to, and collide with the walls of the container. This is why gasses exert pressure on a container. Again, the more thermal energy, the more KE the particles have. Brownian Motion Liquids and gasses are grouped together as fluids, because their particles are free to move. Their movement is random and in a zigzag (caused by collisions with other particles and boundaries). This is known as Brownian motion, and supports the kinetic model. This can be demonstrated with pollen particles suspended in water, or smoke particles in air, viewed through a microscope. Internal Energy All matter has energy contained within it, known as internal energy. Internal energy is the sum of the randomly distributed kinetic and potential energies of the particles in a system The kinetic energy of a particle is dependent on its mass and speed. As seen above, kinetic energy is proportional to temperature. At 0 K, the particles have 0 kinetic energy. The potential energy is down to the electrostatic interactions between different particles. This is never 0, even at absolute zero. Therefore at absolute zero, internal energy is at its minimum but not zero. As temperature increases, so does internal energy. Changes of Phase During a change of phase, the internal energy of the substance changes. This is due to a change in potential energy not a change in kinetic energy. When the substance changes phase, its temperature stays the same but its molecular structure changes. For example, ice melts at 273 K, or 0 °C. In the process, the potential energy of the substance increases, but both the ice pre-melt and water post-melt are at 273 K, 0 °C. Similarly when water boils, both the water immediately before evaporating and the gas immediately after are at 100 °C because the kinetic energy is constant throughout changes of phase. This often causes confusion because changes of state are seen to be brought on by a change in temperature, but this is not strictly the case. Thermal Properties All materials have certain thermal properties that are fixed to that material. Specific Heat Capacity Specific heat capacity, c, is defined as the amount of energy required to raise the temperature of 1 kg of a substance by 1 K (or 1 °C). It is a fixed property of a substance/material. E = mcΔθ Energy change = mass x specific heat capacity x change in temperature (in K) Specific Latent Heat The specific latent heat of a substance, L, is defined as the energy required to change the phase of the substance per unit mass while at a constant temperature. L = E/m Specific latent heat = energy supplied / mass There are two forms of specific latent heat: Specific Latent Heat of Fusion is the energy required to change a substance from a solid to a liquid Specific Latent Heat of Vaporisation is the energy required to change a substance from a liquid to a gas The specific latent heat is the change of potential energies that occurs in a substance when it changes phase, increasing its internal energy. Remember that the kinetic energies remain constant. Ideal Gasses When working with gasses, using mass as a measurement is not practical (a kg of air is over 150,000 teaspoons!). Instead, we use the number of particles, often measured in moles. One mole contains 6.03 E23 particles This is known as Avogadro's constant. To find the number of moles, n, multiply the number of particles by Avogadro's constant. To find the number of particles, N, divide the number of moles by Avogadro's constant. Small n for the smaller number (number of moles), big N for the bigger number (number of particles) Kinetic Theory for Ideal Gasses We use the kinetic theory (see above) for gasses. This means we make certain assumptions, which is why we call them ideal gasses: The gas contains a large number of molecules The gasses follow Brownian Motion - move at random directions with random speeds The molecules occupy a negligible volume compared to the volume of the gas All collisions with other molecules or the container walls are perfectly elastic, meaning that both momentum and kinetic energy is conserved The collisions are of negligible duration compared to the time between collisions Only forces during collisions are worth noting (electrostatic interactions are negligible) When the gas molecules collide with the walls of the container, they change direction. This means there has been a change in momentum of -2mu, where m is the mass of the molecule and u its speed. Before the collision, its velocity is +u, after the collision it is -u (-u - +u = -2u). The change in momentum is the same as the impulse, Ft exerted on the wall. This impulse (due to Newton's third law) exerts a very small force on the wall, but many many collisions are taking place. The force of all of these collisions combined is what exerts a pressure on the walls, pressurising the container. Gas Laws Boyle's Law Boyle's law states that when temperature and mass of a gas remain constant, the pressure of the gas, p, is inversely proportional to its volume, V, and so pV is constant. p ∝ 1/V pV = constant The Pressure Law The Pressure law states that when the volume and mass of an ideal gas remain constant, the pressure, p, is directly proportional to its absolute temperature in Kelvin, K. p ∝ T p/T = constant This can be used to workout an estimate for absolute zero by submerging a flask of air in a water bath, and connecting it with an air tight tube to a pressure gauge. As the temperature of the water increases, so does the temperature of the air. This, in turn, increases the pressure inside it. Plotting a graph of pressure against temperature shows direct proportion as it will have a straight line of best fit. The line does not go through the origin, but extrapolating it backwards to meet the x-axis will give an estimate for absolute zero. The two equations above can be combined: pV/T = constant For one mole of an ideal gas, the constant in this equation is the molar gas constant, R, 8.31 J/K/mol Adding this to the equation gives pV/T = R. For when you have a different number of moles, n: pV/T = nR This is known as the ideal gas equation. Charles's Law Charles's Law states that when the pressure of an ideal gas is fixed, the volume, V, of the gas is directly proportional to its absolute temperature, T. Root Mean Square (r.m.s.) Speed When looking at the energy of a gas, we need to know the average velocity of all the particles in the gas. However, we cannot calculate this, because velocity is a vector quantity, and so has direction: all the particles are travelling in different directions so cancel out. This means the actual average velocity is 0 m/s. Instead, we use root mean square speed, the square root of the mean of the square of the velocities of the gas particles. √mean of square of speed √c̅² The units of c.m.s are m²s-² The mean square speed (not rooted) is used in the pressure & volume equation: pV = 1/3 Nm c̅ ² Where p is the pressure, V the volume, N the number of particles in the gas, m the mass, and c̅ ² the mean square speed. If you have the number of moles, multiply by Avogadro's constant to find the number of particles. The Boltzmann Constant The Boltzmann constant, k, is equal to the molar gas constant divided by Avogadro's constant: k = 1.38 E-23 This can be substituted into the ideal gas equation: pV = NkT Kinetic Energy of Gasses Combing the above equations relates average kinetic energy to absolute temperature: 1/3 Nm c̅ ² = pV = NkT 1/3 Nm c̅ ² = NkT Equating the equations 1/3 m c̅ ² = kT Cancelling the number of particles, N 2/3 x 1/2 m c̅ ² = kT Rewriting 1/3 as 3/2 x 1/2 1/2 m c̅ ² = 3/2 kT Dividing by 2/3 This is now in the form of the kinetic energy equation, KE = 1/2 mv². Therefore this is the equation for mean kinetic energy of a gas: KE = 1/2 m c̅ ² = 3/2 kT Internal Energy of Ideal Gasses Internal energy is the sum of potential and kinetic energies in a system. One of the ideal gas assumptions, however, is that the electrostatic potential energy is negligible - this means that the internal energy of an ideal gas is the sum of kinetic energies in the particles.
- Circular Motion, SHM & Oscillations
The S.I. unit for angles is not degrees, but radians. This is the angle subtended by a circular arc with a length equal to the radius of the circle: 1 rad is equal to about 57.3° There are 2π in a circle, and half that in a semicircle. Therefore, to convert from degrees to radians, divide the angle in degrees by 180/π to convert from radians to degrees, multiply the angle in radians by 180/π Circular Motion While the velocity of linear motion is defined as displacement / time, the velocity of circular motion is defined as angle / time. Therefore, it is called angular velocity, and is given the Greek letter omega, ω. ω = θ/t angular velocity = angle / time This equation is rarely used, however, as it is far easier to work with the time period and frequency of circular motion. Time period, T, is the time taken for it to complete one circle. Frequency, f, refers to the number of complete rotations (of 2π rads) per second. This gives rise to the far more helpful equations: ω = 2π/T angular velocity = 2π / time period ω = 2πf angular velocity = 2π x frequency The units of angular velocity are radians per second. We can calculate the linear velocity using the angular velocity: v = ωr linear velocity = angular velocity x radius Centripetal Force Even if the object undergoing circular motion is travelling at a constant speed, its velocity is always changing (due to its direction changing). This means the object is always accelerating – this is known as centripetal acceleration, and always acts towards the centre of the circle. a = v²/r centripetal acceleration = linear velocity² / radius a = ω²r centripetal acceleration = angular velocity² x radius According to newton’s first law, if an object is accelerating, there must be a net force acting on it. This is the centripetal force, which is constantly acting perpendicular to the velocity, into the centre of the circle. This force is what causes the object to travel in a circular path. From Newton’s second law (F = ma) we can derive: F = mv²/r centripetal force = mass x velocity² / radius F = mω²r centripetal force = mass x angular velocity² x radius Despite the force constantly changing direction, the object’s velocity remains perpendicular. This means there is no motion in the direction of the force, and so no work is done on the object, and its kinetic energy remains the same. Centripetal force can be investigated with a bung on the end of a string, a marker on it, a set of masses, and a glass tube. Simple Harmonic Motion All objects move slowly with simple harmonic motion (SHM) - either on an atomic scale (where the atoms in a solid vibrate) or on a larger scale like a spring or pendulum system. SHM is defined in terms of acceleration and displacement: An object moving with SHM oscillates from side to side about a midpoint Displacement is the distance from the midpoint Amplitude is the maximum displacement (the furthest distance from the midpoint) A restoring force constantly pushes or pulls the object back to the midpoint The magnitude of the restoring force depends on the displacement and makes the object accelerate towards the centre The period is the time taken to make one complete oscillation (from the midpoint out towards one side, back in through the midpoint and out to the other side, and back to the midpoint) The frequency is how many complete oscillations are made each second. It is important to note that frequency and period are independent of amplitude, because they are constant for any given oscillation. Angular Frequency Angular frequency, ω, is the magnitude of the vector quantity angular velocity in circular motion. Just like in circular motion, angular frequency is given by the following two equations: ω = 2π/T angular frequency = 2π / time period ω = 2πf angular frequency = 2π x frequency Acceleration Simple harmonic Motion occurs when an oscillation’s acceleration is directly proportional to its displacement from the midpoint. a = −ω²x acceleration = - (angular frequency)² x displacement Acceleration always acts towards the midpoint, and is at a maximum at each end (where displacement = amplitude). As the object passes through the midpoint, its acceleration is zero. a(max) = − ω²A maximum acceleration = - (angular frequency)² x amplitude Velocity The velocity changes depending on where in the oscillation the object is. v = ± ω√(A² - x²) velocity = ± angular frequency x √(amplitude² - displacement²) At each end, velocity is 0m/s (because it changes direction here). In the midpoint, velocity is at its maximum: v(max) = ωA maximum velocity = angular frequency x amplitude Displacement Displacement varies with time, and is measured from the midpoint out. If at t=0 seconds, the object is at its maximum displacement, use cosine: x = A cos(ωt) If at t=0 seconds, the object is in the midpoint, use sine: x = A sin(ωt) Displacement, Velocity & Acceleration Graphs Phase Difference Like waves, we use phase difference, ϕ, to denote the difference to compare two identical oscillations and the positions in their cycles. If two identical oscillations both reach their maximum positive displacement at the same time, they are perfectly in phase with a phase difference of 0 rad. If one reaches its maximum positive displacement when the other reaches its maximum negative displacement, they are exactly in antiphase with a difference of π rad. Energy of Oscillations When an object undergoes SHM, energy is constantly transferring from KE to GPE and back. As the object moves towards its maximum amplitude, it gains height and as such GPE. This comes from a reduction in KE. As the object returns to its midpoint, it loses GPE as this is transferred back into KE. The maximum GPE is when KE is zero when the displacement is at its maximum. The Maximum KE is when GPE is zero when the displacement is zero. Resonance Simple harmonic oscillations can either be free or forced. Free oscillations have no transfer of energy to or from their surroundings. For example, if a mass on a spring that has been pulled down and released has no external factors, it will oscillate at its natural frequency at the same amplitude forever. In reality, this never happens because factors like air resistance reduce the amplitude dramatically. Forced oscillations happen when there is an external driving force, at a set driving frequency. This inputs energy to the system. All substances, systems and structures have a natural frequency. This is the frequency at which they oscillate freely. Resonance If the driving frequency is equal to the natural frequency, resonance occurs. This is when the system gains more and more energy from the driving force, and so the amplitude quickly increases. Sometimes, this can be strong enough to destroy the system entirely. Damping The forces that oppose SHM are known as dampening forces. The most common example is air resistance. Systems are often deliberately damped to stop them oscillating or minimise effects of resonance. The heavier the damping, the faster the amplitude decreases – however it always happens at an inverse square rate. Light Damping reduces the amplitude by a very small amount each time. Heavy Damping reduces the amplitude quickly, and within a few cycles, the oscillations become negligible. Critical damping is when it never oscillates: the amplitude is reduced in the shortest possible time. Damping affects resonance and the natural frequency as well:
The universe is everything that exists. It contains many galaxies (ours is the Milky Way) that are clusters of stars and planets. Within galaxies, there are many solar systems. These contain one star, the planets that orbit this star, the planetary satellites (such as moons or artificially launched satellites), asteroids, and comets. According to Kepler's first law, all objects have elliptical orbits. However, planets tend to be more on the circular side, whereas comets have intensely elliptical orbits. Life Cycle of Stars Formation of Stars Nebulae - Stars form when nebulae (large clouds of gas and interstellar dust) slowly contract under gravitational forces. Protostars - When the cloud of dust gets dense enough, it splits up into protostars. The loss in gravitational potential energy in bringing all the dust/gas together causes the protostar to heat up. Fusion - Eventually the protostar becomes hot enough for fusion to occur (a few million degrees), where hydrogen nuclei combine to form helium. Main Sequence - The process of fusion releases a vast amount energy, keeping the core of the star hot until it has burnt through all the hydrogen fuel. This phase is relatively stable, because while the gravitational force acts to compress the star further, the radiation pressure from the emitted photons and the pressure of the gas push outwards. Main Sequence The lifetime of a star depends on its mass: a larger star burns through its fuel much faster than smaller stars. The threshold for small and large stars is the Chandrasekhar limit (around 1.4 solar masses) - it is the maximum mass for which electron degeneracy pressure (that keeps the star going) can counteract the gravitational force (that tries to collapse the star). Low Mass Stars have a core mass of up to 1.4 solar masses (the mass of the sun), and a lifetime of five to ten billion years. Massive Stars have cores heavier than 1.4 solar masses and a lifespan of just a few hundred-thousand years. After the main sequence, the star enters the final phase. Again, this is dependent on the size of a star. Death of a Low Mass Star Becomes a Red Giant: as the fuel runs out, there is less energy opposing gravity, so the core pulls into itself, heating up. Before the core can contract completely, there is enough energy for the helium atoms start to combine to form carbon and oxygen atoms. This process also emits a huge amount of radiation, and prevents the star from collapsing completely, and actually forces the outermost layers of the star out, making it bigger than it was in its normal phase. Because it is larger for the same material, its heat is spread out over a larger surface area: this is why it appears redder. Eventually, the helium fuel runs out, however there will never be enough energy for carbon and oxygen to fuse. Initially, the gravitational forces pull the star in on itself once again, however the helium still in the shell will continue to undergo thermonuclear reactions. This provides more energy than the force of gravity, and so the shell separates further and further from the core, causing clouds of glowing gas, ionised by the UV radiation from the core. This is known as a planetary nebula. The exposed core is known as a white dwarf and is immensely dense – over a tonne per teaspoon. Once it reaches about earth size, the electron degeneracy pressure (the pressure exerted by the electrons) stop it collapsing any more. These last for billions of years and slowly cool down and become dimmer, before eventually becoming a black dwarf – the corpse of a past small star. Death of a Massive Star Similar to the break down of a small star, as the hydrogen fuel runs out, helium becomes the fuel. Initially the star contracts and then expands far beyond its initial size. However, they are far bigger: Red Supergiants can have a radius up to 1500 times larger than the sun, whereas a normal reg giant would only be 200-800 times larger. At the end of the red supergiant phase, once all the helium fuel has been used up and fused into carbon and oxygen, the star pulls in on itself. No nebula forms as the outer shell of the star is pulled all the way in instead of separating. As this happens, the star becomes denser and denser until it cannot withstand its own gravitational force. The whole core collapses, resulting in an explosion into a supernova. This leaves behind one of two things: A Neutron Star: this is similar to a white dwarf – the immensely dense core left behind – held only together by neutron degeneracy as thermal energy is not enough to overcome gravity. They are only about 20km in diameter, and spin up to 600 times a second (called Pulsars). They emit radio waves in two beams as the rotate that sometimes sweep past the earth as radio pulses. This forms when the core mass is 1.4 – 3 solar masses. A Black Hole: If neutron degeneracy is not sufficient to overcome gravity, the star will continue to collapse into a point of singularity: an infinitely small, infinitely dense point. This becomes the centre of a black hole. Around the black hole is a region where the gravitational pull is so strong that even light is sucked in – the boundary of this region is called the event horizon. This forms when the core mass is >3 solar masses. Luminosity Luminosity is the measure of how bright an object is, and for stars, this can be plotted against temperature on a Hertzprung-Russell diagram: The main sequence, red giants, super giants and white dwarfs can be plotted in different areas on these graphs because they are so stable and last such a long period of time. You do not plot transitional stars as the transitions are so rapid. Electromagnetic Radiation from Stars In isolated gas atoms, electrons can only exist within certain well-defined energy levels, where n=1 is the ground state. Electrons can move down energy levels by emitting a photon, hence the energy levels have negative values. The difference in energy between the levels is equal to the energy of the emitted photon: E = hf difference in energy = Planck constant x frequency Using the wave equation, v = fλ, we can rewrite this as: E = hc/λ difference in energy = Planck constant x speed of light / wavelength As gasses are heated, many electrons will move into higher energy levels. When they fall back down to the ground state, photons are emitted. Using a diffraction grating, this emitted light can be separated into an emission line spectrum, where each line corresponds to a certain wavelength of emitted light brought about by the corresponding energy change. Different atoms have different discrete energy levels, and as such produce different spectral lines. This can be used to identify elements within stars. Continuous Spectra contain all possible wavelengths – e.g. the spectrum of white light. When white light is shone through a diffraction grating, each order has a spectrum as each wavelength of light within the white light is diffracted by a different amount. Red is always on the outside, and violet on the inside. The zero order remains white, as the waves just pass straight through without being diffracted. Hot objects emit continuous spectra in the visible and infrared regions. Extremely hot objects may also reach shorter wavelengths, like the ultraviolet region. Emission and Absorption Line Spectra Emission Line Spectra show the wavelengths of the emitted photons as lines, when passed through a diffraction grating. They form when a cloud of gas is excited (heated up). When this happens, electrons move up into higher energy levels. As the gas cools, the electrons fall back down into lower levels, releasing the energy as photons. The lines on the emission spectra correspond to the wavelengths of these emitted photons. Absorption Line Spectra are the exact opposite to their corresponding emission line spectrum. When light with a continuous spectrum of energy pass through a cold gas, electrons in their ground states are excited by photons of the correct wavelengths. These wavelengths are then missing from the continuous spectrum, leaving black lines. Light emitted from stars has to travel through a large amount of gas at its surface before travelling to earth, which absorbs particular wavelengths, leaving behind an absorption spectrum. The wavelengths can be calculated with the angle from the zero-order line (see section on waves and diffraction gratings) using the equation: d sinθ = nλ slit separation x sin(angle of diffraction) = order number x wavelength To calculate θ, we can apply the small angle approximation, tanθ ≈ θ: θ = x/D angle = fringe separation / grating-to-screen distance Wien's Law As the surface temperature of a star increases, the most common wavelength emitted becomes shorter. This is known as the peak wavelength, λ(max). Wien’s Displacement Law can be used to estimate a star’s peak surface temperature, in kelvin: λ(max) ∝ 1/T Stephan's Law The luminosity of a star is defined as the total energy the star emits per second – its power output. It depends on temperature, as well as surface area and can be calculated using Stefan’s Law: L = 4πr²σT⁴ Luminosity = 4π x radius² x the Stephan Constant x Temperature⁴ The Stefan constant is taken as 5.67 E-8 Wm-² K-⁴
- Gravitational Fields
A gravitational field is a region where an object experiences a non-contact force because of its and another object’s mass. Gravitational fields are always attractive, and as such the values are always negative. Spherical objects, or things modelled as spheres such as planets and satellites, can be modelled as a point with mass at its centre. Field Lines map gravitational fields, including their strength. The closer together, the greater the field strength. Gravitational field strength is given by: g = F/m gravitational field strength = gravitational force / mass Newton's Law of Gravitation Newton’s Law of Gravitation states that the gravitational force experienced by two objects interacting is directly proportional to the product of their masses and inversely proportional to the square of their separation: F= −GMm/r² Where M and m are the masses of the two objects, G is the gravitational constant, and r is the separation between the centre of the objects. This means the distance between them, plus their radii. The gravitational field strength at a certain point from a single point mass is given as the same, divided by one of the masses: g = −GM/r² Close to the surface of a planet, gravitational field strength can be modelled as a uniform field, and numerically equals the acceleration of free fall (9.81ms-²). G is the gravitational constant: 6.67 E-11 Planetary Motion The motion of planets around their star can be described using Kepler's laws: Each planet moves in an ellipse around the sun, with the sun at one focus. A line joining the sun to a planet will sweep out equal areas in equal times. The square of the period of orbit is directly proportional to the cube of the radius: T² ∝ r³ The time the planet takes to travel between A and B is the same as the time taken between D and C. The shaded areas are the same. Kepler's 3rd Law The circular motion on an orbiting object occurs because there is a constant centripetal force acting on it due to the gravitational force of the focus. Using the equations for circular motion and Newton's law of gravitation we can derive Kepler's third law: F = mv²/r Centripetal force F = -GMm/r² Newton's law of gravitation T = 2πr/v Time period of circular motion mv²/r = -GMm/r² Equating centripetal force and Newton's law of gravitation v² = -GM/r m and r cancel v = √(GM/r) Square-rooting both sides T = 2πr / √(GM/r) Sub into time period of circular motion T² = 4π²r² / (GM/r) Rearrange the fractions T² = 4π²/GM r³ Kepler's 3rd law Here we can see that the coefficient in Kepler's 3rd Law is 4π²/GM, and that time period is independent of mass of orbital (the M represents the mass of the planet/star being orbited). Geostationary Satellites Geostationary satellites are satellites that orbit above the earth's equator, and have a time period of exactly 24 hours. This means they are always above the same area of land/sea. This makes them incredibly useful for communication, as you do not have to constantly realign transmitters and receivers, both on earth and the satellite. From Kepler's third law, we can work out the radius of a geostationary satellite: 24 hours = 86,400 seconds 86,400² = 7,464,960,000 7,464,960,000 x GM = 2.99 E24 2.99 E24 / 4π² = 7.57 E22 ³√7.57 E22 = 42,000,00 m = 42,000 km Gravitational Potential and Energy Gravitational potential at a point is the work done to move a unit mass from infinity to the point. It is always negative, with it reaching a maximum of 0 at infinity. In a radial field, Gravitational Potential is given as: V(g) = −GM/r Gravitational potential = - G x mass of object / distance from centre of object Instead of measuring the potential from infinity, we tend to measure difference in potential as an object gains or loses height, or between two objects. Therefore, you always need to set a zero point, from which to calculate the change in potential. Gravitational Potential Energy When an object is moved, work is done against gravity. The amount of energy required to do this work depends on the mass of the object and the potential difference: E = m x V(g) Energy = mass x gravitational potential Combining this with the equation for gravitational potential gives: E = GMm/r Energy = G x the two masses / the separation Force-distance graphs show how the magnitude of the force sue to the gravitational field changes with distance. Since F = GMm/r², multiplying by r gives the energy equation above. Therefore, the area under the curve is equal to the work done in moving the object. Escape Velocity For something to escape a gravitational field, its kinetic energy must be at least equal and opposite to its gravitational potential energy – this is known as escape velocity: v = √(2GM/r) escape velocity = Square root of (2 x G x mass of planet / orbital radius) Note that the mass is of the planet/star causing the gravitational field. This is because the mass of the orbiting object trying to escape cancels: 1/2 mv² + -GMm/r = 0 Total KE + GPE = 0 for escape velocity 1/2 v² + -GM/r = 0 Cancel the m's 1/2 v² = GMm/r Add -GM/r to both sides v² = 2GM/r Multiply by two v = √(2GM/r) Square root both sides
Capacitors store electrical charge. They are made up of two conducting plates separated by a dielectric (insulator) or gap. A positive and a negative charge builds up on the opposite plates, but the insulator stops the charge from moving – this creates a potential difference via a uniform electric field. The charges on each plate are equal and opposite. Capacitance is the of a capacitor is defined as the charge stored per unit p.d. across the capacitor. C =Q/V capacitance = charge / potential difference The units of capacitance are Farads, F, or Coulomb per Volt, C/V. Capacitor Networks As a general rule of thumb, everything about capacitors is the opposite to how it would be with resistors: 1/C = 1/C(1) + 1/C(2) … Combined Capacitance in series C = C(1) + C(2) … Combined Capacitance in parallel Energy Stored Work is done removing negative charge from one plate and depositing it on the other plate – this is the same as the energy stored by the capacitor, and can be found as the area under a p.d. – charge graph: W = 1/2 QV Energy stored = 1/2 x charge x p.d. This equation can be rewritten using the capacitance equation, C = Q/V, rearranged as Q = CV W = 1/2 V²C Energy Stored = 1/2 voltage² x capacitance Or, using the capacitance equation written in the form V = Q/C: W = 1/2 Q²/C Energy stored = 1/2 charge² / capacitance Uses for Capacitors as Energy Stores Flash Photography Before LEDs, you would need a short burst of high current to give a bright flash. A capacitor discharging quickly achieved this. Back-up Power Supplies If there is a power outage, many large capacitors can be used to keep key systems running for a short period of time. Smoothing out p.d. When AC is converted to DC, capacitors charge up at the peaks and discharge at the troughs to supply a constant, smooth power output. Charging & Discharging The p.d. and charge across a capacitor increase over time as the capacitor is charged, but the current reduces – when this reaches 0 A, the capacitor is fully charged and the p.d. across it is equal to that of the power supply, V(0). To measure capacitance, p.d. and current across a charging capacitor, you need to set up a circuit where the capacitor is in series to a resistor. To measure discharge, however, the resistor and capacitor must be in parallel. As soon as the capacitor is connected to the power supply, electrons flow onto the plate connected to the negative terminal. This creates a negative charge on one of the plates, so the electrons on the other plate are repelled towards the positive terminal, leaving that plate positive. An equal but opposite charge builds on each plate, creating a p.d. between them. To begin with, the current is high, but it drops as electrostatic repulsion makes it harder for electrons to be deposited. For discharging, the Charge and p.d. graphs are the opposite (or negative), but the current graph is the same. The capacitor is fully discharged when both the current and p.d. are 0. Time Constant & Exponential Decay The time taken for a capacitor to charge or discharge depends on two things: the capacitance, as this affects the amount of charge that can be transferred at a given voltage the resistance, as this affects the current in the circuit However, the pattern of charging/discharging is always the same: exponential decay. This means that for a set period of time, the charge, p.d. or current decreases by the same ratio. This is modelled with the number e: C is the capacitance of the capacitor, and R the resistance in the circuit. Because these are constant, they are known as the time constant, τ: τ = CR time constant = capacitance x resistance The time constant is the time taken for the charge, p.d. or current to fall to 37% (1/e) of its initial value, or for the p.d. or charge to rise to 63% of its maximum value. The larger the resistor in series with the capacitor, the longer it takes to charge/discharge. The time for the capacitor to charge/discharge fully is around 5τ.
Instead of using metres and kilometres, distances are measured in far larger units: Astronomical units, AU: the mean distance between the Earth and the Sun. ~150 million km Light-Years, ly: the distance and electromagnetic wave travels in one year. ~ 9.5x1015 m Parsecs, pc: the length of the adjacent side of a right-angled triangle with angle 1 arcsecond and opposite side length 1 parsec. ~ 3.1x1016 m The angle is measured in arcseconds because of stellar parallax – objects further away appear to be moving more slowly than object closer to earth, and nearby stars appear to move differently depending on where in the earth’s orbit we are. Parallax in arcseconds, p, can be calculated for a known distance, d, measured in parsecs: p = 1/d parallax (in arcseconds) = 1 / distance (in parsecs) d = 1/p distance (in parsecs) = 1 / parallax (in arcseconds) 1 second of arc = (1/3600)° The Cosmological Principal The Cosmological Principal states that, on a large scale, the universe is: homogeneous (every part is the same as another) isotropic (looks the same in every direction) and the laws of physics apply everywhere. We apply this principle when investigating the universe, allowing us to use standard mechanics to solve problems on a cosmological scale. Doppler Shift & Hubble's Law The Doppler Shift is the way in which frequency and wavelength change with the motion of the wave's source, relative to you. This happens because the waves bunch together in front of the source and stretch out behind it. The extent of bunching and stretching depends on the sources velocity. Red shift occurs when a light source moves away from us, as the wavelengths become longer and the frequencies become lower, so the light shifts towards the red end of the spectrum. Blue shift occurs when a light source moves towards us, as the opposite happens and the light shifts towards the blue end of the spectrum. The same principal applies to sound waves: as the source moves towards you, the frequency increases and wavelength decreases; as it moves away the frequency decreases and wavelength increases. For example, a high speed train approaching, passing, and disappearing from a level crossing. The amount of shift can be calculated with the Doppler Equation: Δλ/λ ≈ Δf/f ≈ v/c Δλ and Δf represent the difference between observed and emitted wavelength and frequency. v is the velocity of the source, and c the speed of light, 3 E8 m/s. Hubble's Law Until the early 20th century, it was believed the universe was infinite in both space and time – Steady State Theory. However, Edwin Hubble noticed that the spectra from far away galaxies all show red shift, implying they are moving away from us, and that the universe is expanding. The amount of galactic red shift gives the recessional velocity, and Hubble’s Law shows that recessional velocity is directly proportional to the distance of receding galaxies: v ≈ H(0) x d recessional velocity ≈ the Hubble constant x distance The Hubble constant, H(0), can be taken as roughly 65-80 km s-1 Mpc-1 (velocity is in kms-1 and distance in Mpc). There is not yet a universally accepted accurate reading. To find the Hubble constant in SI units (s-1), velocity must be in ms-1 and distance in m. The Big Bang Theory If the universe is expanding and cooling down (conservation of energy principal), then back in time, it must have been smaller and hotter. At time t=0, it must have been at a point of singularity (infinitely dense and infinitely hot). This is the Big Bang Theory. Cosmic Microwave Background Radiation is the evidence for the Big Bang: if there were a point of singularity from which very rapid expansion happened, there would be a great deal of gamma radiation produced. This would still be observable today, and it is, but as microwave radiation, because as the universe has expanded, the wavelength has been stretched and lost energy. This was picked up accidentally by Penzias and Wilson in the 60’s. The age of the universe can be estimated as 1/the Hubble constant. However, this assumes that the universe has been expanding at the same rate for all of time (which is unlikely), and since we do not have an accurate value for the Hubble constant, we can only roughly estimate the universe’s age to be about 13.8 bn years. It is difficult to establish how large the universe is, as we can only see the ‘observable universe’ – a sphere around the earth with a radius of 13.8 billion light years (as that is how far light can have travelled in time). Taking into account the expansion of the universe, it is probably around 46-47 billion ly in radius. The Evolution of the Universe This is the current theory of the evolution of the universe. The times given is the age of the universe at each point, so the amount of time after the big bang. 0 s (Big Bang) Time and space are formed, the universe is at a point of singularity (infinitely dense and hot). 10^-35 s Universe is expanding rapidly, known as inflation. There is no matter yet, and temepratures are around 10^28 K. 10^-6 s First fundamental particles (quarks, leptons etc.) form. We do not know exactly how this happens, but we know it involves the Higgs Boson. 10^-3 s Quarks combing to become hadrons (protons and neutrons) through pair production. 1 s Matter production stops, temperature dropped to about 10^9 K. 100 s Protons and neutrons fuse to form light elements like deuterium (hydrogen-2) and helium 380,000 years Universe is cold enough for heavier atoms to form as the nuclei take on electrons. The microwave background radiation originates from this time. 30 million years Stars begin to form, and the fusion that occurs in them form the first heavy elements (anything after lithium). 200 million years The Milky Way forms, as clouds of hydrogen and stars are pulled together by gravitational forces. 9 billion years Our solar system forms out of the supernova of a dead larger star 10 billion years The earth forms 11 billion years Primitive life on earth begins 13.8 billion years (today) Temperature has dropped to about 2.7K, diameter of universe is probably around 46 - 47 billion light years. Dark Energy & Matter In the 30’s, Fritz Zwicky calculated the mass of cluster galaxies based on their velocity and luminosity. The two measurements gave vastly different results, with velocity giving a far more massive result. This implies the existence of mass that cannot be seen – dark matter. In the 70’s, Vera Rubin observed that stars at the edges of galaxies were moving faster than they should be given their mass, again, implying the existence of dark matter. Since all things with mass are attracted to one another through gravity, we would expect the expansion of the universe to be slowing down. However, in the 90’s it was discovered that the expansion is, in fact, speeding up. This acceleration is only possible if there is some undetectable energy that spans all the universe – dark energy. Today, it is estimated that there is approximately five times as much dark matter as there is ordinary matter, and that this makes up around a quarter of the entire universe. It is also believed that around 70% of the universe is dark energy. This leaves only 5% of the universe to be ordinary, detectable matter.
- Electric Fields
Electric fields are regions where non-contact electrical forces can be felt by charged objects. They are generated by electrical charges. Field lines show the path a positive test charge would take from positive to negative. The closer they are, the stronger the field. Lines never stop in empty space and must never cross. Field lines go from positive to negative, so the point charge below is negative: Uniformly charged spheres can be modelled as a point charge at its centre, with field lines leaving/entering at right angles all around, to infinity. Closer to the point, you can see that the field lines are closer together. Therefore, the field strength must be greater. In all radial fields, the field strength is proportional to the distance from the point charge via an inverse square law. Electric field strength is force per unit charge: E = F/Q Electric field strength = Force / Charge The units of electric field strength and newtons per coulomb, N/C Coulomb's Law The attractive/repulsive force between two point charges can be calculated using Coulomb's law: F = Qq / 4πε₀r² Force = Product of two charges / 4π x ε₀ x separation² ε₀ is the permittivity of free space, and can be taken as 8.85 E-12. Since electric field strength is calculated as E = F/Q, coulomb's law can be used to calculate field strength of a point charge by dividing by one charge: F = Q / 4πε₀r² Force = Charge / 4π x ε₀ x separation² Electric vs Gravitational Fields Newton's law of gravitation is very similar to coulomb's law, and the fields have many other similarities, too. However, there are some differences. Similarities Point masses/charges both produce radial fields Force and field strength are inversely proportional to distance² Force is proportional to product of masses/charges Newton's and Coulomb's laws are in same format, with different coefficients and mass and charge respectively Differences Electric fields can be attractive or repulsive; gravitational is always attractive Mass produces the field for gravitational fields, charges for electrical Uniform Electric Fields In a uniform electric field, the field strength is the same everywhere: E = V/d Field strength = p.d. / distance The field lines are parallel to one another and evenly spaced. Capacitors & Electric Fields Parallel plate capacitors work by using a uniform electric field across the insulator between the two plates. Remember that the charge on each plate is equal but opposite. There are two basic types of parallel plate capacitors, those with a vacuum between the two plates as the insulator, and those with an insulating material (dielectric). For capacitors with a vacuum, the permittivity of free space, ε₀, and the area, A, of the plates (just one plate - the areas are the same) is required to calculate capacitance: C = ε₀A/d capacitance = permittivity of free space x area / plate separation For capacitors with a dielectric, the permittivity, ε, of that material must be calculated: ε = ε₀ x ε(r). ε(r) is the relative permittivity. Therefore, capacitance is: C = εA/d capacitance = permittivity x area / plate separation Motion of Charged Particles in Uniform Electric Fields Charged particles in a uniform electric field accelerate towards the oppositely charged plate at a uniform rate. Charged particles moving perpendicular to the field move in a curved shape, just like projectile motion. This is because a particle of charge Q experiences a constant force (F = EQ) acting parallel to the field lines. The work done increases, and so does its kinetic energy. This means it accelerates at a uniform rate. If the velocity of the particle has components perpendicular to the field lines, this will remain unchanged, resulting in a curved path. Electrical Potential and Energy Electric potential is the work done in bringing the unit charge from infinity to a point. Electric potential at infinity is 0. In a radial field, electric potential is given as V = Q/4πε₀r Electrical potential = charge / 4π x ε₀ x distance The units of electric potential are volts. When Q is positive, so is V: the force is repulsive. When Q is negative, so is V: the force is attractive. The absolute magnitude of V is greatest at the surface of the charge and decreases as the distance increases. Graphs of force against distance look the same, but the area under the graph is equal to the work done in moving the unit charge. Therefore, electrical potential energy (the work done) is given as: E = Vq Electrical potential energy = electrical potential x charge E = Qq/4πε₀r Electrical potential energy = product of charges / 4π x ε₀ x separation Electrical Potential and Capacitance Since capacitance is given as C = Q/V, we can substitute in the equation for electrical potential, V, to get the capacitance of an isolated sphere: C = 4πε₀R Capacitance = 4π x ε₀ x radius of the sphere
- Magnetism & Electromagnetism
A magnetic field is a region in which a magnetic material experiences a force, and is caused by a moving charge or a permanent magnet. Just like gravitational and electric fields, magnetic fields are represented by field lines - except instead of talking about positive and negative end, we talk about north and south poles. The closer the field lines, the stronger the magnetic field. Identical to charges, like poles repel and opposite poles attract. The diagram above shows the field patterns from permanent bar magnets, but magnetic fields arise all the time from current-carrying wires: For a straight current-carrying wire, the field lines are arranged in concentric circles in a plane perpendicular to the wire and direction of current. This is found using the right-hand grip rule. A flat current-carrying coil produces the same pattern, but all the way around the coil, so the field is three-dimensional. A solenoid produces this pattern on an elongated scale. Magnetic Flux Density A current that is perpendicular to a uniform magnetic field will induce a force. The direction of this force can be established with Fleming's left-hand rule: The magnitude of the force causing the motion is proportional to the magnetic flux density, B, of the magnetic field. This is defined as the force on one meter of wire carrying a current of 1 amp, perpendicular to the magnetic field. F = BIL sinθ Force = flux density x current x length of wire x sin(angle with magnetic field) If the wire is perpendicular, the equation is just F = BIL, as sin(90°) = 1 If the wire is parallel, there is no force, as sin(0°) = 0 This relationship can be investigated with a known length of wire on a scale, running perpendicular through a uniform magnetic field. When a known current passes through the wire, the scale will read a change in mass. This, multiplied by g, is equivalent to the force exerted on the wire. The units of magnetic flux density are Tesla, T, defined as 1Wb/m (see electromagnetism, below) Motion of Charged Particles F = BIL is the equation used for calculating the force on a current-carrying wire perpendicular to a uniform magnetic field. This is not to be confused with the following equation, used for charged particles perpendicular to a uniform magnetic field: F = BQv Force = flux density x charge x velocity Circular Motion According to Fleming's left hand rule, the motion of a charged particle must always be perpendicular to its motion. This means that the articles will travel on a curved or circular path, as the force induced acts as a centripetal force: Velocity Selectors A velocity selector is a device that makes use of electric and magnetic fields to produce a fine stream of particles at a given velocity. The two fields are perpendicular to one another, and the particles are charged before entering the device. The particles will experience an upwards force from the magnetic field, according to Fleming's left hand rule, of magnitude BQv. The particles experience a downward force from the electric field, because the positive charge of the particle is attracted to the negative end of the field. The magnitude of this force is given as EQ. Unless the upwards and downwards forces are the same, the particle will experience a resultant force and be deflected. Therefore, only particles where BQv = EQ will continue in a straight line. The charges cancel, so the selected velocity is given as: v = E/B selected velocity = electric field strength / flux density This is used in mass spectrometers to identify ions and particles, based on their mass. Electromagnetism Magnetic flux, ϕ, is defined as the product of the perpendicular component of the flux density through a given area. More or less, this means the number of field lines passing through a particular area. It is calculated as: ϕ = BA cosθ magnetic flux = flux density x area x cosθ Magnetic flux linkage, Nϕ, is the product of the flux, ϕ, and the number of coils, N. ϕ = BAN cosθ magnetic flux linkage = flux density x area x number of coils x cosθ Both flux and flux linkage have Weber, Wb, as their units. However, to avoid confusion, Weber Turns are sometimes used as the units for flux linkage. Electromagnetic Induction When a conductor moves through a magnetic field and experiences a change in magnetic flux, an e.m.f. is induced. This happens because the electrons in the conductor move to one side of it, according to Fleming's left-hand rule. This leads to a resultant positive charge on the opposite side. Faraday's Law Faraday's Law states that the magnitude of e.m.f. induced is directly proportional to the rate of change of magnetic flux linkage: ε = -Δ(Nϕ) / Δt e.m.f. = - change in magnetic flux linkage / change in time. ε = -Δ(BAN cosθ) / Δt (using the equation for flux linkage above) Lenz's Law Lenz's law states that the direction of the induced e.m.f opposes that of the change in flux that has taken place. This is why Faraday's law above has a negative sign in it. A.C. Motors & Generators A.C. Motor In the setup above, when a current passes through the red wire rectangle, it moves clockwise. This is because, according the Fleming's left hand rule, the motion, magnetic field, and current must all by perpendicular. There is a split ring commutator to change the direction of current every half turn - else the motor would always change direction. To increase the speed, the current, magnetic field strength, or number of coils should be increased. A.C. Generator For a generator, we sue the right hand rule instead. Each finger still represents the same thing. Note that the current in the rectangular wire coil is going the opposite way to in the motor, and that there are two slip rings instead of a split ring commutator - this is why the current provided is alternating. The e.m.f is induced because of the change in magnetic flux experienced by the wire coil as it rotates. Therefore, faster rotations cause a higher current. This can also be achieved by more coils or stronger magnets. Transformers Transformers are made up of a laminated soft iron core with at least two wire coils on it, the primary and secondary. When an AC current is applied to the primary coil, a constantly changing magnetic flux is induced in the iron core. This, in turn, induces an e.m.f. in the secondary coil. An iron core is used because it is so easily and efficiently magnetised and demagnetised. This ensures that, even though the transformers produce a large amount of heat, all the magnetic flux produced by the first coil is transferred to the second. The core is laminated (meaning it is made up of many very thin insulated layers) to prevent losing to much heat from induced currents in the core itself - these are known as eddy currents. For an ideal transformer, we can use the turn-ratio equation to calculate the output voltage: n(s) / n(p) = V(s) / V(p) primary coils / secondary coils = primary voltage / secondary voltage Assuming the transformer is 100% efficient, we can assume that power is conserved. Therefore, since P=IV, we can assume that V(p) x I(p) = V(s) x I(s) Primary voltage current = secondary voltage x current Therefore, we can see, that if voltage is stepped up, then current must step down and vice versa. Step-up transformers increase voltage and decrease current Step-down transformers decrease voltage and increase current