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• 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.

• Charge & Current

Current is the rate of flow of charge. Charge is a physical property. The unit of charge is a Coulomb, C – the amount of charge that passes in 1 second when the current is 1 ampere: Q = It charge = current × time The charge on an electron is the smallest possible charge, and is known as the elementary charge, −e, where e = 1.60E−19 (E notation is used on EngineeringNotes - it is a web-friendly way of writing standard form, as exponents are not always compatible with different devices and browsers. 1.60E-19 is the same as 1.60 x10^-19. Don't confuse it with E for energy or the fundamental charge, e) A proton has the same charge, but positive, (+e). Because e is the smallest value Q can take, any charge value must be a multiple of e - therefore we say that charge is quantised. In most instances, charge is transferred via electrons - however there are a number of different so called 'Charge Carriers' depending on the material and state: In metals, the charge carriers are free, delocalised electrons (meaning they can move inside the material). Fluids, such as electrolytes, conduct with the use of ions. Gasses are insulators, but when enough voltage is applied electrons are torn out of atoms resulting in sparks – this is called ionisation. The magnitude of the current is dependent on number and speed of charge carriers. For example, a wire will have a larger current if there are more electrons flowing through it, or if there are the same number moving faster. Conventional Current Conventional current suggests that charge moves from positive to negative (high potential to low potential), however electrons move the opposite direction due to their positive attraction. Kirchhoff's Laws Being a fundamental physical property, charge must be conserved. It cannot be created nor destroyed - the total amount of electrical charge in the universe always remains the same. 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. Mean Drift Velocity Mean drift velocity is the average velocity of charge carriers in a conductor. While the effect of electricity travels at the speed of light, the carriers (such as the electrons) actually move very slowly in a domino effect. It is called ‘drift velocity’ because electrons do not all move in exactly the same direction, but randomly in all directions – it's just that if you were to take an average of all their directions, this would tend towards one direction. The current is dependent on the mean drift velocity, as shown in the current continuity equation: I = Anev current = cross sectional area × number density × elementary charge × velocity The number density of electrons per metre cubed gives rise to different levels of conductivity for different materials. It is a fixed property of a material. Conductors (such as metals) have a huge number of free electrons per unit volume, and so the drift velocity is small even for high currents. Semiconductors (such as silicon) have fewer charge carriers, so the drift needs to be greater to achieve the same current. Insulators have very few, if not 0 charge carriers. This means whatever you put in the formula; the current is always 0. This equation means that cross-sectional area has a vast effect on current. In order to maintain the same current in a narrower wire, a far higher drift velocity is required. The mean drift velocity is inversely proportional to the cross-sectional area

• Waves

• Thermal Physics

• Quantum Physics

• Circular Motion, SHM & Oscillations

• 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

• Astrophysics

• Cosmology

• 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

• Capacitors

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