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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.
- 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
- Nuclear & Particle Physics
Initially, Thomson proposed the ‘plumb pudding’ model for atoms, saying the atom was made up of a blob of positive charge with negatively charged electrons sprinkled in and about this. Rutherford’s Scattering experiment disproved this, by firing alpha particles at a gold foil and seeing where they emerged on a fluorescent screen. Most alpha particles passed straight through with very little deflection; however, some scattered a lot and others were reflected right back. This led to the conclusion that an atom was made up of largely empty space, with a very small, very dense, positively charged nucleus in the centre. This Nuclear Model is now widely adopted: The nucleus is surrounded by orbiting electrons in shells The nucleus is about one 10,000th of the size of the atom, but makes up most the mass The protons and neutrons are around 2000 times more massive than the electron Nuclear density is always ~1017 kg m-³; atomic density is far less Isotopes are atoms with the same proton number but a different nucleon number – the different number of neutrons does not affect the chemical properties, however can make the atom unstable. The radius of the nucleus is directly proportional to the cube root of the nucleon number: R = r(0) x ³√A nuclear radius = r(0) x ³√nucleon number r(0) can be taken as 1.4 E-15 The Four Forces There are four forces that govern the laws of physics - the strong and weak nuclear, the electromagnetic, and gravitational. The nucleons are held together by the strong nuclear force, but as the nucleus gets bigger and the electrostatic repulsion outweighs it on the outside, the nucleus becomes unstable. Strong Nuclear Force Very short range: attractive to ~3 fm; repulsive to ~0.5 fm Works equally between all nucleons At very small separations, it must be repulsive, else the nucleus would collapse Weak Nuclear Force The only thing that can change one quark into another Also very small range Acts between all particles Electromagnetic Force Infinite range Acts between all charged particles Can be attractive or repulsive Gravitational Force Infinite range Acts on all objects with mass Always attractive Fundamental Particles All particles come in particle/anti-particle pairs, where each part has the same mass, but opposite charge. Classification of Matter All matter is categorised into hadrons and leptons: Quarks The hadrons are largely made up of up, down and strange quarks, and their respective anti-quarks. All of these have a charge, and the combinations of these quarks is what gives protons their +1 and neutrons their 0 charge. Quark notation consists of just writing the letters next to each other: Protons are made up of uud Neutrons are made up if udd Decay We can express beta decay in terms of quarks: Radioactivity Radioactive decay is both spontaneous (cannot be induced) and random (cannot be predicted), and typically takes one of four forms: Alpha emission happens in heavy nuclei (large nucleon numbers) Beta-minus emission happens in neutron-rich nuclei (many more neutrons than protons) Gamma radiation is emitted from nuclei with too much energy Half-life, T(1/2) is the average time taken for activity, mass of radioactive nuclei, or number of radioactive nuclei to halve. The decay constant, λ, is a measure of the rate of radioactive decay. T(1/2) = ln(2) / λ half life = ln(2) / decay constant Activity, A, is the number of particles decaying per second. A = λN Activity = decay constant x number of undecayed nuclei remaining The Number of undecayed nuclei decreases exponentially (it halves for every fixed period of time, the half life). Investigating decay You can investigate half-life by plotting a graph of activity against time, for samples such as protactinium-234: Shake bottle containing uranium salt, its decay products (including protactinium-234), and two solvents Wait for two solvents to separate – the protactinium-234 will be in the top layer Record activity of top layer Record again every minute Subtract background radiation from all results Plot graph of activity against time Determine half-life from exponential graph Similarly, you can simulate radioactive decay with dice with one side coloured differently. Shake the pot, tip the dice on a table and remove all the ones with the coloured side facing up. Repeat this until no dice left, plotting a graph of the numbers left each round. Carbon Dating Radioactive isotopes can be used to date object that are more than tens of thousands of years old, using the proportion of carbon-14 in them. Plants take this in as carbon dioxide from the atmosphere, and animals then eat the plants. All living things contain the same percentage of carbon-14, and using its half-life (5730 years), we can estimate something’s age by comparing current carbon-14 levels. Mass Defect, Binding Energy & E = mc² Einstein observed that all mass (and matter) is, in fact, energy. He relates the two with his energy-mass equation: E = mc² Energy = mass x speed of light² Pair Production Pair production is when energy is converted into matter – if two photons collide with enough energy, they may create a third proton, paired with an antiproton. Typically, you get electron-positron pairs formed, because of their low mass (and therefore less energy is required). The minimum energy the photon must have is that of the rest masses of the particle and antiparticle produced. Annihilation Annihilation is the opposite of pair production – all the mass of a particle and its antiparticle is converted into energy. This happens all the time, hence we do not see many antiparticles. Mass Defect & Binding Energy The mass of a nucleus is less than the mass of its constituent parts – the difference in mass is known as the mass defect. This is converted into energy. The binding energy is the energy required to separate all of the nucleons in a nucleus and is equivalent to the mass defect. The binding energy per unit of atomic mass, u, is about 930 MeV u-1 To compare binding energy on different nuclei, we look at binding energy per nucleon. This is the curve of binding energy per nucleon for all atoms: The higher the binding energy per nucleon, the more stable the nucleus, because more energy is required to remove nucleons from the nucleus Iron-56 is the most stable nucleus, as it has the highest binding energy per nucleon Nuclei below Iron-56 will undergo fusion (which increases binding energy per nucleon dramatically, hence releases a lot of energy) Nuclei above Iron-56 will undergo fission to split into two smaller nuclei, so binding energy per nucleon increases too, but not as dramatically as with fusion. The change in binding energy = energy released. This is why all nuclear reactions release or require energy. Nuclear Fission & Fusion The Induced Fission process A thermal (low energy) neutron is fired at a U-235 nucleus Uranium must be enriched, as natural occurring Uranium only has 0.7% U-235; but 2.5-3% required for fission reactions The larger the nucleus, the more unstable Nucleus becomes very unstable, and splits into two smaller nuclei releasing two or three fast (high energy) neutrons The new, smaller nuclei have a higher binding energy per nucleon, and as such are more stable The change in binding energy per nucleon means energy is released in the form of kinetic energy of neutrons The Moderator (water) slows down the fast neutrons to be thermal neutrons, to maximise probability of starting another fission reaction Control rods (made of boron) absorb excess neutrons, to ensure the average reaction creates one neutron for every neutron that is lost To maintain a safe chain reaction Critical mass is the exact mass of nuclear fuel required to keep the reaction going at a steady rate Coolant (heavy water or CO2) around the reactor core removes the thermal energy and transfers it to the electrical generation system (turbines powered by steam) Nuclear waste High Level waste (used fuel rods) are stored in cooling ponds for a year before being sealed in glass blocks and buried deep in steel cases Intermediate Level waste (empty fuel rods, old components) are encased in cement in metal drums, and stored underground. Low Level waste (Packaging, clothes) are cleaned, compacted, placed in steel containers and sunk or buried Nuclear Fusion When two light nuclei have enough energy to overcome the electrostatic repulsion between them, they can get close enough for the strong nuclear force to bind them This releases a lot of energy, because the binding energy per nucleon increases so dramatically This energy helps maintain conditions to allow more fusion reactions to occur A lot of energy is required to overcome the electrostatic repulsion, however The energy released per reaction is less in fusion than in fission, but because the fusionable nuclei have such a small mass in comparison, one mole of reactants gets many, many more reactions.
- Forces & Newton's Laws
First off, it is important to distinguish between mass and weight. Mass is a physical property of an object, dependent on the amount of matter the object is made up of. The weight of an object is the gravitational force acting on its mass. On earth, where g = 9.81, the weight is 9.81 x the mass: weight = mass x gravity W = mg = 9.81m Newton's Laws of Motion Newton's 1st Law: An object will stay at a constant velocity, or motionless, unless it experiences a resultant force. Newton's 2nd Law: The resultant force on an object is directly proportional to the rate of change of momentum of the object, and acts in the same direction. Therefore, F = Δp / Δt, which is commonly written as F=ma Newton's 3rd Law: For every action, there is an equal but opposite reaction (equal in magnitude and type of force). Representing Forces Weight acts down from an object's centre of gravity. Other forces can act in any direction, and so to avoid confusion it is best to draw a force diagram: The length of arrow should represent the magnitude of the force. Single headed arrows represent forces. If a system is in equilibrium - meaning there is no net resultant force and it is motionless or at constant velocity - then all the force arrows will make a closed shape. The example above would give a triangle. So yeh, trigonometry is used a lot... Drag & Terminal Velocity Though not modelled mathematically at A-Level, when objects fall through fluids (liquids & gasses) there is significant drag that slows them down. This means that unless in a vacuum, there is no such thing as free fall as acceleration is not constant. The magnitude of the drag force depends largely on the shape and texture of the object, the density of the fluid, and the speed of the object - the greater the speed, the greater the drag force, until it equals the weight of the object. Terminal Velocity When this equilibrium is reached, there is no net resultant force on the object anymore, and so (according to Newton's Laws) it stops accelerating. This constant speed is known as terminal velocity. Terminal velocity can be investigated in fluids by connecting the object to a light polystyrene ball over a pulley and measuring the speed of this once the object is dropped into the fluid, such as water or glycerol. Moments Moments are the turning forces acting on an object, around a pivot. M = Fx moment = force × perpendicular distance from pivot The units of moments are Nm When an object is in equilibrium, there is neither a net force nor a net moment in any direction. The Principle of moments can be used for this: the sum of all clockwise moments must equal the sum of all anticlockwise moments for an object to be in equilibrium. When we model a rod or object as uniform, it means we assume all the weight is evenly distributed throughout, and only acts about the centre. Torque Couples Torque Couples are when there are equal moments on either side of a pivot, acting in the same rotational directions (either both clockwise or anti-clockwise). This causes the object to spin. The torque of a couple (a pair or equal but opposite forces) is the net moment of the pair, and is simply found as follows: Torque = Fd torque = force × distance between forces Density & Pressure Density is a fixed property of an object/material, and is defined as the mass per unit volume. ρ = m / v density = mass / volume The units of density are kg/m³ To determine the volume of irregular objects, a eureka can is used. The volume of the object is equal to the volume of water displaced. Pressure Pressure is the normal contact force per unit area. p = F / A pressure = force / cross-sectional area The units of pressure are Pascals (Pa) - this is equivalent to N/m² There is constant pressure in fluids, because of the weight of the fluid above a given point. This depends on the depth - at sea level, atmospheric pressure is ~101 kPa, but at the top of Everest, it is ~30kPa. This can be calculated using the density and depth of a fluid such as water: p = hρg pressure = height x density x gravity The pressure does not depend on cross-sectional area, as this cancels out in the derivation: W = mg weight = mass of column x gravity W = ρVg mass = density x volume W = ρAhg volume = area x height p = ρAhg/A pressure = force (weight) / area p = pgh A / A cancels Archimedes' Principle Since pressure in a fluid only depends on height, when an object is submerged in the fluid, there will be a greater pressure at the base of it than the top of it (as the base of it is deeper). This causes a resultant force acting up on the object, and is known as upthrust. According to Archimedes' Principle: The upthrust on an object submerged (either partially or fully) in a fluid is equal to the weight of the fluid displaced. Upthrust = xρgA upthrust = thickness x density x g x area If the upthrust is greater than or equal to the weight of the object, the object will float.
- Motion, Momentum & Impulse
Everything in the universe is in motion, and there are basic laws of mechanics that govern this. Velocity is the rate of change of an object's position, and therefore has direction. This makes it a vector quantity, unlike speed which is scalar (magnitude but no direction). velocity = distance / time The units of velocity are m/s Acceleration is the rate of change of velocity, and so is the mathematical derivative of this. acceleration = change in velocity / change in time The units of acceleration are m/s² Motion Graphs We can plot motion on two main types of graph - it is important to know the properties of each. Displacement-Time Graphs The Gradient is the velocity - draw a tangent to find the instantaneous velocity Horizontal line represents zero velocity Velocity-Time Graphs The Gradient is the acceleration The Area beneath the graph is the displacement Linear Motion When acceleration is constant (e.g. free fall when we ignore air resistance), we can use SUVAT equations to work out the variables: Linear motion can be investigated with trolley cars, air gliders, ticker tape, light gates, data loggers and motion capture. Acceleration of free fall, g , should always be taken -9.81 unless otherwise stated. This can be looked at experimentally with electromagnetic trap doors triggering a timer, and light gates. g = 9.81 m/s^2 Stopping Distance Stopping distances are made up of thinking distance (the distance travelled in the time it takes the driver to react) and the braking distance (the distance travelled during the deceleration to standstill). Projectile Motion At A-Level, projectile motion is modelled as free fall - meaning air resistance is ignored. Of course, in reality, this is not the case (see Forces & Newton's Laws). Ignoring air resistance allows us to model the motion with constant acceleration, so we can use SUVAT, giving a neat, symmetrical curve: In projectiles, it is important to split up horizontal and vertical components and to treat them as completely independent. Ignoring air resistance for both components, we can assume that horizontal velocity remains constant, whereas vertical velocity decelerates with gravity. Momentum For objects of changing acceleration or mass, using F = ma does not suffice. Therefore, momentum is used. Momentum is a vector quantity (it has both magnitude and direction) dependent on the mass and velocity of an object. p = mv momentum = mass x velocity The units of momentum are kg m/s Conservation of Momentum When dealing with collisions in closed systems, you can use the principle of conservation of momentum to calculate the velocity of the objects before or after the collision. It is vital to establish a positive and negative direction, and not to get these confused, when doing collision calculations. There are two types of collisions: Perfectly Elastic Both momentum and kinetic energy are conserved Inelastic Only momentum is conserved - kinetic energy is lost to other forms It is important to note that in both collisions, the total energy is still conserved. Impulse In a collision, the resultant force on an object that causes the acceleration or deceleration changes dramatically over a very short period of time. This is very hard to analyse, and so we look at impulse instead. Newton's Second Law tells us that the net force us equal to the rate of change of momentum. F = Δp / Δt Rearranging this gives F x Δt = Δp Change of momentum is equivalent to impulse I = Ft Impulse = Force x time The Units of Impulse are Newton Seconds, Ns, or kg m/s Collisions can be plotted on force-time graphs, where the area under the line is the impulse. Car Safety If a car of mass 1000kg, travelling at 100km/h crashes into a wall and is brough to a stop, there is a very large change of momentum. If this happens quickly, a massive force is exerted on the car and driver. If this happens slowly, the same impulse will exert a far smaller force Think of it as flattening the curve - the area is the same, but shallower. Therefore, cars are designed with 'crumple zones' to spread the impulse out as much as possible and minimise the force on the driver. The whole bonnet is deigned to be crushed, the seat belt is slightly elastic, and the airbags blow up in front of you, all to maximise the time taken for you to hurtle forward.