Physics Notes

graveness gravitative subject argona strength at a diaphragm is specify as the gravitative tear per whole flock at that lay. Newtons law of gravitation The (mutual) gravitative twitch F mingled with dickens psyche peck M and m sepa outrankd by a dis bronzece r is given by F = GMm (where G Universal gravitational un vary) r2 or, the gravitational mash of among two put masses is proportional to the product of their masses inversely proportional to the squ ar of their sepa dimensionn. Gravitational written report strength at a point is the gravitational ram down per whole of measurement mass at that point. It is a vector and its S. I. whole is N kg-1.By definition, g = F / m By Newton Law of Gravitation, F = GMm / r2 Combining, order of g = GM / r2 Therefore g = GM / r2, M = Mass of intent creating the theme instance 1 presume that the public is a unvarying sphere of universal gas uninterrupted 6. 4 x 106 m and mass 6. 0 x 1024 kg, find the gravitati onal celestial sphere strength g at a point (a) on the surface, g = GM / r2 = (6. 67 ? 10-11)(6. 0 x 1024) / (6. 4 x 106)2 = 9. 77ms-2 (b) at elevation 0. 50 judgment of convictions the radius of above the nations surface. g = GM / r2 = (6. 67 ? 10-11)(6. 0 x 1024) / ( (1. 5 ? 6. 4 x 106)2 = 4. 34ms-2 Example 2 The securenessup due to gravitation at the Earths surface is 9. 0ms-2. Calculate the reanimateup due to gravity on a planet which has the equivalent density but twice the radius of Earth. g = GM / r2 gP / gE = MPrE2 / MErP2 = (4/3) ? rP3rE2? P / (4/3) ? rE3rP2? E = rP / rE = 2 consequently gP = 2 x 9. 81 = 19. 6ms-2 take for granted that Earth is a uniform sphere of mass M. The magnitude of the gravitational deplumate from Earth on a particle of mass m, located out facial expression Earth a surmount r from the centre of the Earth is F = GMm / r2. When a particle is released, it will downslope towards the centre of the Earth, as a result of the gravitational ra mp with an acceleration ag. FG = mag ag = GM / r2Hence ag = g Thus gravitational bailiwick strength g is also numeric entirelyy pit to the acceleration of free root. Example 1 A ship is at await on the Earths equator. Assuming the earth to be a perfect sphere of radius R and the acceleration due to gravity at the poles is go, express its app arnt cargo, N, of a torso of mass m in terms of m, go, R and T (the period of the earths rotation nearly its bloc, which is genius day). At the noth Pole, the gravitational liking is F = GMEm / R2 = mgo At the equator, Normal Re swear out motor on ship by Earth = Gravitational attraction unifying(a) hale N = mgo mR? = mgo mR (2? / T)2 Gravitational voltaic say-so variety at a point is defined as the pasture through (by an external agent) in take a unit mass from infinity to that point (without changing its kinetic cogency). ? = W / m = -GM / r Why gravitational authority determine atomic number 18 always prohibit? As the gravitational rive on the mass is photogenic, the eye socket of study make by an ext agent in poseing unit mass from infinity to either point in the report will be negative work as the force exerted by the ext agent is turnaround in deputation to the translation to ensure that ? KE = 0 Hence by the definition of negative work, all judges of ? re negative. g = - d? = gradient of ? -r graph Analogy E = -dV/dx dr Gravitational say-so naught U of a mass m at a point in the gravitational depicted object of a nonher mass M, is the work do in bringing that mass m non unit mass, or a mass from infinity to that point. U = m ? = -GMm / r Change in GPE, ? U = mgh only if g is constant over the distance h h radius of planet otherwise, must(prenominal) use ? U = m? f-m? i Aspects electric automobile Field Gravitational Field 1. Quantity inter playing with or producing the battle subject plain stitch stir up Q Mass M 2. Definition of Field Strength absorb p er unit constructive charge E = F / q Force per unit mass g = F / M 3. Force between two question Charges or Masses Coulombs Law Fe = Q1Q2 / 4 or2 Newtons Law of Gravitation Fg = G (GMm / r2) 4. Field Strength of isolated Point Charge or Mass E = Q / 4 or2 g = G (GM / r2) 5. Definition of effectiveness Work d iodin in bringing a unit verifying charge from infinity to the point V = W /Q Work make in bringing a unit mass from infinity to the point ? = W / M 6. say-so of isolated Point Charge or Mass V = Q / 4 or ? -G (M / r) 7. Change in Potential vigor ? U = q ? V ? U = m Total elan vital of a Satellite = GPE + KE = (-GMm / r) + ? (GMm / r) Escape bucket on of a Satellite By Conservation of animation, Initial KE + Initial GPE = Final KE + Final GPE (? mvE2) + (-GMm / r) = (0) + (0) Thus cope speed, vE = v(2GM / R) Note Escape speed of an objective lensive is case-by-case of its mass For a satellite in aeronaut go more or less, the unifying(a) force is prov ided by the gravitational force Must always state what force is providing the centripetal force before fol kickoffing eqn is use Hence GMm / r2 = mv2 / r = mr? 2 = mr (2? / T)2 A satellite does not move in the focus of the gravitational force ie it stays in its government note orbit because the gravitational force exerted by the Earth on the satellite is just sufficient to cause the centripetal acceleration but not enough to also pull it down towards the Earth. This explains also why the lunar month does not fall towards the Earth Geostationary satellite is one which is always above a au sotic point on the Earth (as the Earth rotates about its axis. For a geostationary orbit T = 24 hrs, orbital radius ( height) are obstinate value from the centre of the Earth, ang speeding w is also a fixed value rotates fr west to east. However, the mass of the satellite is not a crabbed value because the ke, gpe, the centripetal force are also not fixed values ie their values dep prohibi t on the mass of the geostationary satellite. A geostationary orbit must lie in the equatorial plane of the earth because it must promote in a plane where the centre of Earth lies since the net orce exerted on the satellite is the Earths gravitational force, which is directed towards the centre of Earth. Alternatively, may explain by masking why its impossible for a satellite in a non-equatorial plane to be geostationary. caloric physics Internal postal code is the sum of the kinetic zilch of the molecules due to its random motion the say-so skill of the molecules due to the intermolecular forces. Internal might is located by the values of the ongoing state and is independent of how the state is arrived at.You sens read also Thin Film Solar cellular telephoneThus if a arrangement beneath(a)goes a series of dislodges from one state A to another state B, its change in inbred zipper is the same, regardless(prenominal) of which course of study the changes in the p V it has taken to get from A to B. Since Kinetic expertness proportional to temp, and indispensable life force of the system = sum of its Kinetic Energy and Potential Energy, a lift in temperature will cause a hoist in Kinetic Energy and thus an growth in internal cleverness. If two bodies are in thermal equilibrium, in that location is no net blend of raise up postcode between them and they learn the same temperature. NB this does not imply they must bring the same internal energy as internal energy depends also on the tot of molecules in the 2 bodies, which is unknown here Thermodynamic (Kelvin) scale of temperature theoretical scale that is independent of the properties of whatever item substance. An absolute scale of temp is a temp scale which does not depend on the holding of any particular substance (ie the thermodynamic scale) Absolute null Temperature at which all substances have a minimum internal energy not zero internal energy. T/K = T/C + 273. 15, b y definition of the Celsius scale.Specific heat capacity is defined as the amount of heat energy indispensable to produce unit temperature change NOT by 1 K for unit mass NOT 1 kg of a substance, without causing a change in state. c = Q / m? T Specific latent heat of vaporisation is defined as the amount of heat energy undeniable to change unit mass of a substance from liquid mannikin to gaseous level without a change of temperature. Specific latent heat of fusion is defined as the amount of heat energy look ated to change unit mass of a substance from solid phase to liquid phase without a change of temperature L = Q / m for both cases of vaporisation runThe specific latent heat of vaporisation is greater than the specific latent heat of fusion for a given substance because * During vaporisation, in that location is a greater increase in volume than in fusion, * Thus more work is done a attractst atmospheric pressure during vaporisation, * The increase in vol also means the INCREASE IN THE (MOLECULAR) POTENTIAL ENERGY, hence, internal energy, during vaporisation more than that during melting, * Hence by 1st Law of Thermodynamics, heat supplied during vaporisation more than that during melting hence lv lf since Q = ml = ?U W. Note 1. the use of comparative terms greater, more, and 2. the increase in internal energy is due to an increase in the PE, NOT KE of molecules 3. the system here is NOT to be considered as an ensample gas system Similarly, you need to explain why, when a liq is boiling, thermal energy is cosmos supplied, and yet, the temp of the liq does not change. Melting Boiling Evaporation Occurrence Throughout the substance, at fixed temperature and pressure On the surface, at all temperaturesSpacing(vol) PE of molecules Increase s laxly Increase foolalizeificantly Temperature hence KE of molecules stiff constant during dish out Decrease for remaining liquid First Law of Thermodynamics The increase in internal energy of a syste m is tally to the sum of the heat supplied to the system and the work done on the system. ?U = W + Q ? U Increase in internal energy of the system Q Heat supplied to the system W work done on the system Need to recall the sign convention for all 3 terms Work is done by a gas when it expands work is done on a gas when it is ompressed. W = playing area under pressure volume graph. For constant pressure isobaric process, Work done = pressure x ? mess Isothermal process a process where T = const ? U = 0 for ideal gas ? U for a cycle = 0 since U ? T, ? T = 0 for a cycle Equation of state for an ideal gas p V = n R T, where T is in Kelvin NOT C, n no. of moles. p V = N k T, where N no. of molecules, kBoltzmann const Ideal Gas a gas which obeys the ideal gas compare pV = nRT FOR ALL VALUES OF P, V T Avogadro constant defined as the number of atoms in 12g of carbon-12.It is thus the number of particles (atoms or molecules) in one mole of substance. For an ideal gas, internal energy U = Sum of the KE of the molecules only since PE = 0 for ideal gas U = N x? m c2 = N x (3/2)kT for monatomic gas * U depends on T and number of molecules N * U ? T for a given number of molecules Ave KE of a molecule, ? m c2 ? T T in K not C Dynamics Newtons laws of motion Newtons First Law E real body continues in a state of rest or uniform motion in a satisfying agate drag of descent unless a net (external) force acts on it. Newtons Second LawThe rate of change of urge of a body is straight off proportional to the net force playing on the body, and the momentum change takes place in the way of the net force. Newtons Third Law When object X exerts a force on object Y, object Y exerts a force of the same type that is match in magnitude and opposite in stress on object X. The two forces ALWAYS act on different objects and they form an action-reaction p line of reasoning. unidimensional momentum and its conservation Mass is a measure of the amount of matter in a body, is t he plaza of a body which resists change in motion.Weight is the force of gravitational attraction (exerted by the Earth) on a body. linear momentum of a body is defined as the product of its mass and f number ie p = m v Impulse of a force (I) is defined as the product of the force and the time ? t during which it acts ie I = F x ? t for force which is const over the continuation ? t For a variable force, the impulse I = body politic under the F-t graph ? Fdt may need to count squares Impulse is equal in magnitude to the change in momentum of the body acted on by the force.Hence the change in momentum of the body is equal in mag to the area under a (net) force-time graph. Incorrect to define impulse as change in momentum Force is defined as the rate of change of momentum, ie F = m (v u) / t = ma or F = v dm / dt The one Newton is defined as the force needed to vivify a mass of 1 kg by 1 m s-2. pattern of Conservation of Linear urge When objects of a system interact, their fit momentum before and after interaction are equal if no net (external) force acts on the system. * The nub momentum of an isolated system is constant m1 u1 + m2 u2 = m1 v1 + m2 v2 if net F = 0 for all bangs NB Total momentum DURING the interaction/collision is also conserved. (Perfectly) elastic collision Both momentum kinetic energy of the system are conserved. Inelastic collision Only momentum is conserved, total kinetic energy is not conserved. Perfectly inelastic collision Only momentum is conserved, and the particles stupefy together after collision. (i. e. move with the same velocity. ) For all elastic collisions, u1 u2 = v2 v1 ie. relational speed of approach = relative speed of separation or, ? m1u12 + ? m2u22 = ? m1v12 + ? 2v22 In inelastic collisions, total energy is conserved but Kinetic Energy may be reborn into other forms of energy such as backbreaking and heat energy. modern of galvanizingity Electric reliable is the rate of flow of charge. NOT charged particles Electric charge Q musical give outage a point is defined as the product of the (steady) flow at that point and the time for which the current flows, Q = I t superstar coulomb is defined as the charge flowing per second fracture a point at which the current is one adenylic acid. Example 1 An ion beam of singly-charged Na+ and K+ ions is strait through vacuum. If the beam current is 20 ?A, calculate the total number of ions passing any fixed point in the beam per second. (The charge on severally ion is 1. 6 x 10-19 C. ) Current, I = Q / t = Ne / t where N is the no. of ions and e is the charge on one ion. No. of ions per second = N / t = I / e = (20 x 10-6) / (1. 6 x 10-19) = 1. 25 x 10-14 Potential divergency is defined as the energy transferred from galvanic energy to other forms of energy when unit charge passes through an galvanizingal device, V = W / Q P. D. = Energy Transferred / Charge = Power / Current or, is the ratio of the index finger supplied to the device to the current flowing, V = P / IThe volt is defined as the potency passing between 2 pts in a circuit in which one joule of energy is converted from electrical to non-electrical energy when one coulomb passes from 1 pt to the other, ie 1 volt = One joule per coulomb Difference between Potential and Potential Difference (PD) The authorisation at a point of the circuit is due to the amount of charge present on with the energy of the charges. Thus, the effectiveness along circuit drops from the positive terminal to negative terminal, and say-so differs from points to points. Potential Difference refers to the inconsistency in potential between any given two points.For example, if the potential of point A is 1 V and the potential at point B is 5 V, the PD crossways AB, or VAB , is 4 V. In addition, when in that respect is no energy want between two points of the circuit, the potential of these points is same and thus the PD across is 0 V. Example 2 A current of 5 mA passes through a medulla for 1 minute. The potential end across the bulb is 4 V. Calculate (a) The amount of charge passing through the bulb in 1 minute. Charge Q = I t = 5 x 10-3 x 60 = 0. 3 C (b) The work done to operate the bulb for 1 minute. Potential difference across the bulb = W / Q 4 = W / 0. Work done to operate the bulb for 1 minute = 0. 3 x 4 = 1. 2 J Electrical Power, P = V I = I2 / R = V2 / R Brightness of a lamp is fixed by the power dissipated, NOT by V, or I or R alone Example 3 A highschool- emf transmission line with a immunity of 0. 4 ? km-1 carries a current of 500 A. The line is at a potential of 1200 kV at the power station and carries the current to a city located 160 km from the power station. Calculate (a) the power loss in the line. The power loss in the line P = I2 R = 5002 x 0. 4 x 160 = 16 MW (b) the fraction of the transmitted power that is lost.The total power transmitted = I V = 500 x 1200 x 103 = 600 MW The fraction of power loss = 16 / 600 = 0. 267 electrical underground is defined as the ratio of the potential difference across a particle to the current flowing through it , R = VI It is NOT defined as the gradient of a V-I graph however for an ohmic conductor, its ohmic resistance equals the gradient of its V-I graph as this graph is a straight line which passes through the origin The Ohm is the resistance of a resistor if in that location is a current of 1 A flowing through it when the pd across it is 1 V, ie, 1 ? = One volt per adenylic acid Example 4In the circuit below, the voltmeter reading is 8. 00 V and the ammeter reading is 2. 00 A. Calculate the resistance of R. Resistance of R = V / I = 8 / 2 = 4. 0 ? Temperature characteristics of thermistors The resistance (i. e. the ratio V / I) is constant because metallic conductors at constant temperature obey Ohms Law. As V increases, the temperature increases, resulting in an increase in the bounty of vibration of ions and the collision frequency of negatrons with the lattice ions. Hence the resistance of the filament increases with V. A thermistor is do from semi-conductors.As V increases, temperature increases. This releases more charge carriers (electrons and holes) from the lattice, thus reducing the resistance of the thermistor. Hence, resistance decreases as temperature increases. In forwards bias, a diode has low resistance. In reverse bias, the diode has high resistance until the breakdown voltage is r to each oneed. Ohms law The current in a component is proportional to the potential difference across it provided fleshly conditions (eg temp) stay constant. R = ? L / A for a conductor of space l, uniform x-sect area A and resistivity ? Resistivity is defined as the resistance of a material of unit cross-section(a) area and unit length. From R = ? l / A , ? = RA / L Example 5 Calculate the resistance of a nichrome fit of length 500 mm and diameter 1. 0 mm, given that the resistivity of nichrome is 1. 1 x 10-6 ? m. Resistan ce, R = ? l / A = (1. 1 x 10-6)(500 x 10-3) / ? (1 x 10-3 / 2)2 = 0. 70 ? Electromotive force (Emf) is defined as the energy transferred / converted from non-electrical forms of energy into electrical energy when unit charge is locomote round a complete circuit. ie potential difference = Energy Transferred per unit charge E = WQEMF refers to the electrical energy generated from non-electrical energy forms, whereas PD refers to electrical energy being changed into non-electrical energy. For example, EMF Sources Energy Change PD across Energy Change Chemical Cell Chem Elec Bulb Elec Light Generator Mech Elec Fan Elec Mech thermocouple Thermal Elec Door Bell Elec Sound Solar Cell Solar Elec Heating element Elec Thermal Effects of the internal resistance of a source of EMF Internal resistance is the resistance to current flow within the power source.It reduces the potential difference (not EMF) across the terminal of the power supply when it is delivering a current. Consider the circuit below The voltage across the resistor, V = IR, The voltage lost to internal resistance = Ir Thus, the EMF of the cell, E = IR + Ir = V + Ir Therefore If I = 0A or if r = 0? , V = E Motion in a Circle Kinematics of uniform circular motion Radian (rad) is the S. I. unit for angle, ? and it can be related to degrees in the following way. In one complete revolution, an object rotates through 360 , or 2? rad. As the object moves through an angle ? , with notice to the centre of rotation, this angle ? s known as the angular displacement. Angular velocity (? ) of the object is the rate of change of angular displacement with respect to time. ? = ? / t = 2? / T (for one complete revolution) Linear velocity, v, of an object is its instantaneous velocity at any point in its circular path. v = prow length / time taken = r? / t = r? * The stress of the linear velocity is at a tangent to the circle described at that point. Hence it is sometimes referred to as the tangential velocit y * ? is the same for every point in the rotating object, but the linear velocity v is greater for points kick upst strains from the axis.A body sorrowful in a circle at a constant speed changes velocity since its stress changes. Thus, it always experiences an acceleration, a force and a change in momentum. Centripetal acceleration a = r? 2 = v2 / r in magnitude Centripetal force Centripetal force is the termination of all the forces that act on a system in circular motion. It is not a particular force centripetal means centre-seeking. Also, when asked to draw a diagram showing all the forces that act on a system in circular motion, it is wrong to include a force that is labelled as centripetal force. Centripetal force, F = m r ? 2 = mv2 / r in magnitudeA person in a satellite orbiting the Earth experiences cantlessness although the gravi correction strength at that height is not zero because the person and the satellite would both have the same acceleration hence the contac t force between man satellite / normal reaction on the person is zero Not because the field strength is minimum. D. C. Circuits Circuit signs Open Switch Closed Switch Lamp Cell Battery Voltmeter Resistor Fuse Ammeter Variable resistor Thermistor Light dependent resistor (LDR) Resistors in Series R = R1 + R2 + Resistors in Parallel 1/R = 1/R1 + 1/R2 + Example 1Three resistors of resistance 2 ? , 3 ? and 4 ? respectively are used to make the combinations X, Y and Z shown in the diagrams. List the combinations in order of increasing resistance. Resistance for X = 1/2 + 1/(4+3)-1 = 1. 56 ? Resistance for Y = 2 + (1/4 + 1/3)-1 = 3. 71 ? Resistance for Z = (1/3 + 1/2 + 1/4)-1 = 0. 923 ? Therefore, the combination of resistors in order of increasing resistance is Z X Y. Example Referring to the circuit drawn, determine the value of I1, I and R, the combined resistance in the circuit. E = I1 (160) = I2 (4000) = I3 (32000) I1 = 2 / 160 = 0. 0125 A I2 = 2 / 4000 = 5 x 10-4 AI3 = 2 / 320 00 = 6. 25 x 10-5 ASince I = I1 + I2 + I3, I = 13. 1 mAApplying Ohms Law, R = 213. 1 x 10-3 = 153 ? Example A battery with an EMF of 20 V and an internal resistance of 2. 0 ? is connected to resistors R1 and R2 as shown in the diagram. A total current of 4. 0 A is supplied by the battery and R2 has a resistance of 12 ?. Calculate the resistance of R1 and the power supplied to each circuit component. E I r = I2 R2 20 4 (2) = I2 (12) I2 = 1A Therefore, I1 = 4 1 = 3 AE I r = I1 R1 12 = 3 R1 Therefore, R1 = 4Power supplied to R1 = (I1)2 R1 = 36 W Power supplied to R2 = (I2)2 R2 = 12 W For potential divider with 2 resistors in series, Potential drop across R1, V1 = R1 / (R1 + R2) x PD across R1 R2 Potential drop across R2, V1 = R2 / (R1 + R2) x PD across R1 R2 Example Two resistors, of resistance 300 k? and 500 k? respectively, form a potential divider with outer junctions maintained at potentials of +3 V and -15 V. Determine the potential at the junction X between the resistors. The potential difference across the 300 k? resistor = 300 / (300 + 500) 3 (-15) = 6. 75 V The potential at X = 3 6. 75 = -3. 75 V A thermistor is a resistor whose resistance varies greatly with temperature.Its resistance decreases with increasing temperature. It can be used in potential divider circuits to monitor and control temperatures. Example In the intent on the right, the thermistor has a resistance of 800 ? when hot, and a resistance of 5000 ? when cold. Determine the potential at W when the temperature is hot. When thermistor is hot, potential difference across it = 800 / (800 + 1700) x (7 2) = 1. 6 VThe potential at W = 2 + 1. 6 V = 3. 6 V A Light dependent resistor (LDR) is a resistor whose resistance varies with the intensity of short falling on it. Its resistance decreases with increasing light intensity.It can be used in a potential divider circuit to monitor light intensity. Example In the stick out below, the resistance of the LDR is 6. 0 M in the dark but th en drops to 2. 0 k in the light Determine the potential at point P when the LDR is in the light. In the light the potential difference across the LDR= 2k / (3k + 2k) x (18 3) = 6 VThe potential at P = 18 6= 12 V The potential difference along the telegram is proportional to the length of the conducting wire. The sliding contact will move along wire AB until it finds a point along the wire such that the galvanometer shows a zero reading.When the galvanometer shows a zero reading, the current through the galvanometer (and the device that is being tribulationed) is zero and the potentiometer is said to be balanced. If the cell has negligible internal resistance, and if the potentiometer is balanced, EMF / PD of the unknown source, V = L1 / (L1 + L2) x E Example In the circuit shown, the potentiometer wire has a resistance of 60 ?. Determine the EMF of the unknown cell if the balanced point is at B. Resistance of wire AB= 0. 65 / (0. 65 + 0. 35) x 60 = 39 ? EMF of the test cell= 39 / (60 + 20) x 12 Work, Energy and PowerWork Done by a force is defined as the product of the force and displacement (of its point of application) in the direction of the force W = F s cos ? veto work is said to be done by F if x or its compo. is anti- match to F If a variable force F produces a displacement in the direction of F, the work done is determined from the area under F-x graph. May need to find area by run the squares. By article of faith of Conservation of Energy, Work Done on a system = KE grasp + GPE gain + Work done against friction Consider a rigid object of mass m that is ab initio at rest.To accelerate it uniformly to a speed v, a constant net force F is exerted on it, parallel to its motion over a displacement s. Since F is constant, acceleration is constant, Therefore, using the equation v2 = u2 +2as, as = 12 (v2 u2) Since kinetic energy is equal to the work done on the mass to bring it from rest to a speed v, The kinetic energy, EK = Work done by the forc e F = Fs = mas = ? m (v2 u2) Gravitational potential energy this arises in a system of masses where there are attractive gravitational forces between them.The gravitational potential energy of an object is the energy it possesses by virtue of its vista in a gravitational field. ductile potential energy this arises in a system of atoms where there are either attractive or repulsive short-range inter-atomic forces between them. Electric potential energy this arises in a system of charges where there are either attractive or repulsive electric forces between them. The potential energy, U, of a body in a force field whether gravitational or electric field is related to the force F it experiences by F = dU / dx.Consider an object of mass m being lifted vertically by a force F, without acceleration, from a certain height h1 to a height h2. Since the object moves up at a constant speed, F is equal to mg. The change in potential energy of the mass = Work done by the force F = F s = F h = m g h Efficiency The ratio of (useful) output energy of a work to the input energy. ie = Useful Output Energy x100% = Useful Output Power x100% insert Energy Input Power Power instantaneous is defined as the work done per unit time. P = Total Work Done = W Total beat tSince work done W = F x s, P = F x s = Fv t * for object travel at const speed F = Total resistive force equilibrium condition * for object root word to accelerate F = Total resistive force + ma Forces Hookes Law Within the limit of proportionality, the concomitant produced in a material is directly proportional to the force/load applied F = kx Force constant k = force per unit extension (F/x) Elastic potential energy/strain energy = Area under the F-x graph May need to count the squares For a material that obeys Hooke? s law, Elastic Potential Energy, E = ? F x = ? x2 Forces on Masses in Gravitational Fields A region of space in which a mass experiences an (attractive) force due to the presence of anot her mass. Forces on Charge in Electric Fields A region of space where a charge experiences an (attractive or repulsive) force due to the presence of another charge. Hydrostatic Pressure p = ? gh or, pressure difference between 2 points separated by a vertical distance of h Upthrust An upward force exerted by a fluid on a submerged or floating object arises because of the difference in pressure between the upper and set down surfaces of the object.Archimedes Principle Upthrust = weight unit of the fluid displaced by submerged object. ie Upthrust = Volsubmerged x ? fluid x g abrasional Forces * The contact force between two surfaces = (friction2 + normal reaction2)? * The component along the surface of the contact force is called friction * skirmish between 2 surfaces always opposes relative motion or attempted motion, and * Its value varies up to a maximum value called the static friction Viscous Forces * A force that opposes the motion of an object in a fluid * Only exists when there is (relative) motion Magnitude of viscous force increases with the speed of the object Centre of Gravity of an object is defined as that pt through which the entire weight of the object may be considered to act. A couple is a dyad of forces which tends to produce rotation only. mo of a Force The product of the force and the perpendicular distance of its line of action to the pivot tortuousness of a Couple The produce of one of the forces of the couple and the perpendicular distance between the lines of action of the forces. (WARNING NOT an action-reaction pair as they act on the same body. ) Conditions for Equilibrium (of an extended object) 1.The resultant force acting on it in any direction equals zero 2. The resultant moment about any point is zero If a mass is acted upon by 3 forces only and remains in equilibrium, then 1. The lines of action of the 3 forces must pass through a common point 2. When a vector diagram of the ternion forces is drawn, the forces will form a closed triangle (vector triangle), with the 3 vectors pointing in the same taste around the triangle. Principle of Moments For a body to be in equilibrium, the sum of all the anticlockwise moments about any point must be equal to the sum of all the clockwise moments about that same point.Measurement Base quantities and their units mass (kg), length (m), time (s), current (A), temperature (K), amount of substance (mol) Base Quantities SI Units consult Symbol Length metre m Mass kilogram kg Time second s Amount of substance mole mol Temperature Kelvin K Current ampere A lambent intensity candela cd Derived units as products or quotients of the base units Derived Quantities Equation Derived Units Area (A) A = L2 m2 Volume (V) V = L3 m3 Density (? ) ? = m / V kg m-3 Velocity (v) v = L / t ms-1 acceleration (a) a = ? v / t ms-1 / s = ms-2Momentum (p) p = m x v (kg)(ms-1) = kg m s-1 Derived Quantities Equation Derived Unit Derived Units Special Name Symbol Force (F) F = ? p / t Newton N (kg m s-1) / s = kg m s-2 Pressure (p) p = F / A Pascal Pa (kg m s-2) / m2 = kg m-1 s-2 Energy (E) E = F x d joule J (kg m s-2)(m) = kg m2 s-2 Power (P) P = E / t watt W (kg m2 s-2) / s = kg m2 s-3 Frequency (f) f = 1 / t hertz Hz 1 / s = s-1 Charge (Q) Q = I x t coulomb C A s Potential Difference (V) V = E / Q volt V (kg m2 s-2) / A s = kg m2 s-3 A-1 Resistance (R) R = V / I ohm ? (kg m2 s-3 A-1) / A = kg m2 s-3 A-2 Prefixes and their symbols to indicate decimal sub-multiples or multiples of both base and derived units Multiplying Factor Prefix Symbol 10-12 pico p 10-9 nano n 10-6 micro ? 10-3 milli m 10-2 centi c 10-1 decid d 103 kilo k 106 mega M 109 giga G 1012 tera T depends of physical quantities When making an estimate, it is only reasonable to give the figure to 1 or at most 2 significant figures since an estimate is not very precise. Physical Quantity Reasonable Estimate Mass of 3 cans (330 ml) of Coke 1 kgMass of a medium- coatd car 1000 kg Length of a football field 100 m Reaction time of a teen man 0. 2 s * Occasionally, students are asked to estimate the area under a graph. The usual rule of counting squares within the enclosed area is used. (eg. Topic 3 (Dynamics), N94P2Q1c) * Often, when making an estimate, a formula and a simple calculation may be involved. EXAMPLE 1 Estimate the average running speed of a typical 17-year-old? s 2. 4-km run. velocity = distance / time = 2400 / (12. 5 x 60) = 3. 2 ? 3 ms-1 EXAMPLE 2 Which estimate is pragmatic? Option ExplanationA The kinetic energy of a bus travelling on an expressway is 30000J A bus of mass m travelling on an expressway will travel between 50 to 80 kmh-1, which is 13. 8 to 22. 2 ms-1. Thus, its KE will be approximately ? m(182) = 162m. Thus, for its KE to be 30000J 162m = 30000. Thus, m = 185kg, which is an absurd weight for a bus ie. This is not a realistic estimate. B The power of a domestic light is 300W. A single light bulb in the house usually runs at about 20W to 60W. Th us, a domestic light is unlikely to run at more than 200W this estimate is rather high. C The temperature of a hot oven is 300 K. 300K = 27 0C. Not very hot. D The volume of air in a car tyre is 0. 03 m3. Estimating the width of a tyre, t, is 15 cm or 0. 15 m, and estimating R to be 40 cm and r to be 30 cm,volume of air in a car tyre is = ? (R2 r2)t = ? (0. 42 0. 32)(0. 15) = 0. 033 m3 ? 0. 03 m3 (to one sig. fig. ) Distinction between positive errors (including zero errors) and random errors and between precision and accuracy Random error is the type of error which causes readings to facing pages about the true value. Systematic error is the type of error which causes readings to deviate in one direction from the true value.Precision refers to the degree of agreement (scatter, spread) of repeated measurements of the same quantity. NB regardless of whether or not they are correct. Accuracy refers to the degree of agreement between the result of a measurement and the true va lue of the quantity. R Error Higher Less Precise v v vS Error HigherLess Accuratev v v prise the uncertainty in a derived quantity by simple addition of actual, fractional or luck uncertainties (a rigorous statistical treatment is not required). For a quantity x = (2. 0 0. 1) mm,Actual/ Absolute uncertainty, ? x = 0. 1 mm Fractional uncertainty, ? xx = 0. 05 Percentage uncertainty, ? xx 100% = 5 % If p = (2x + y) / 3 or p = (2x y) / 3, ? p = (2? x + ? y) / 3 If r = 2xy3 or r = 2x / y3, ? r / r = ? x / x + 3? y / y Actual error must be enter to only 1 significant figure, The number of decimal places a metrical quantity should have is determined by its actual error. For eg, suppose g has been initially calculated to be 9. 80645 ms-2 ? g has been initially calculated to be 0. 04848 ms-2. The final value of ? g must be recorded as 0. 5 ms-2 1 sf , and the appropriate recording of g is (9. 81 0. 05) ms-2. Distinction between scalar and vector quantities Scalar vector Definition A scalar quantity has a magnitude only. It is completely described by a certain number and a unit. A vector quantity has both magnitude and direction. It can be described by an arrow whose length represents the magnitude of the vector and the arrow-head represents the direction of the vector. Examples aloofness, speed, mass, time, temperature, work done, kinetic energy, pressure, power, electric charge etc. Common ErrorStudents tend to associate kinetic energy and pressure with vectors because of the vector components involved. However, such considerations have no bearings on whether the quantity is a vector or scalar. Displacement, velocity, moments (or torque), momentum, force, electric field etc. Representation of vector as two perpendicular components In the diagram below, XY represents a crosswise kite of weight 4. 0 N. At a certain instant, XY is inclined at 30 to the horizontal and the wind exerts a steady force of 6. 0 N at right angles to XY so th at the kite flies freely.By accurate scale drawing By calculations using sine and cosine rules, or Pythagoras? theorem elapse a scale diagram to find the magnitude and direction of the resultant force acting on the kite. R = 3. 2 N (? 3. 2 cm) at ? = 112 to the 4 N vector. Using cosine rule, a2 = b2 + c2 2bc cos A R2 = 42 + 62 -2(4)(6)(cos 30) R = 3. 23 NUsing sine rule a / sin A = b / sin B 6 / sin ? = 3. 23 / sin 30 ? = 68 or 112 = 112 to the 4 N vector Summing Vector Components Fx = 6 sin 30 = 3 NFy = 6 cos 30 4 = 1. 2 NR = v(-32 + 1. 22) = 3. 23 Ntan ? = 1. 2 / 3 = 22R is at an angle 112 to the 4 N vector. (90 + 22)Kinematics Displacement, speed, velocity and acceleration Distance Total length cover irrespective of the direction of motion. Displacement Distance moved in a certain direction. Speed Distance travelled per unit time. Velocity is defined as the rate of change of displacement, or, displacement per unit time NOT displacement over time, nor, displacement per seco nd, nor, rate of change of displacement per unit time speedup is defined as the rate of change of velocity. Using graphs to find displacement, velocity and acceleration * The area under a velocity-time graph is the change in displacement. The gradient of a displacement-time graph is the instantaneous velocity. * The gradient of a velocity-time graph is the acceleration. The SUVAT Equations of Motion The most important word for this chapter is SUVAT, which stands for * S (displacement), * U (initial velocity), * V (final velocity), * A (acceleration) and * T (time) of a particle that is in motion. Below is a list of the equations you MUST memorise, even if they are in the formula book, memorise them anyway, to ensure you can implement them quickly. 1. v = u +at derived from definition of acceleration a = (v u) / t 2. s = ? (u + v) t derived from the area under the v-t graph 3. v2 = u2 + 2as derived from equations (1) and (2) 4. s = ut + ? at2 derived from equations (1) and (2) Th ese equations return only if the motion takes place along a straight line and the acceleration is constant hence, for eg. , air resistance must be negligible. Motion of bodies falling in a uniform gravitational field with air resistance Consider a body moving in a uniform gravitational field under 2 different conditions Without Air ResistanceAssuming negligible air resistance, whether the body is moving up, or at the highest point or moving down, the weight of the body, W, is the only force acting on it, causing it to experience a constant acceleration. Thus, the gradient of the v-t graph is constant throughout its rise and fall. The body is said to undergo free fall. With Air Resistance If air resistance is NOT negligible and if it is projected upwards with the same initial velocity, as the body moves upwards, both air resistance and weight act downwards. Thus its speed will decrease at a rate greater than . 81 ms-2 . This causes the time taken to reach its maximum height reached to be lower than in the case with no air resistance. The max height reached is also reduced. At the highest point, the body is momentarily at rest air resistance beats zero and hence the only force acting on it is the weight. The acceleration is thus 9. 81 ms-2 at this point. As a body falls, air resistance opposes its weight. The downward acceleration is thus less than 9. 81 ms-2. As air resistance increases with speed, it eventually equals its weight (but in opposite direction).From then there will be no resultant force acting on the body and it will fall with a constant speed, called the terminal velocity. Equations for the horizontal and vertical motion x direction (horizontal axis) y direction (vertical axis) s (displacement) sx = ux t sx = ux t + ? ax t2 sy = uy t + ? ay t2 (Note If projectile ends at same level as the start, then sy = 0) u (initial velocity) ux uy v (final velocity) vx = ux + axt (Note At max height, vx = 0) vy = uy + at vy2 = uy2 + 2asy a (acceleration) ax (Note Exists when a force in x direction present) ay (Note If object is falling, then ay = -g) (time) t t Parabolic Motion tan ? = vy / vx ? direction of tangential velocity NOT tan ? = sy / sx Forces Hookes Law Within the limit of proportionality, the extension produced in a material is directly proportional to the force/load applied F = kx Force constant k = force per unit extension (F/x) Elastic potential energy/strain energy = Area under the F-x graph May need to count the squares For a material that obeys Hooke? s law, Elastic Potential Energy, E = ? F x = ? k x2 Forces on Masses in Gravitational Fields A region of space in which a mass experiences an (attractive) force due to the presence of another mass.Forces on Charge in Electric Fields A region of space where a charge experiences an (attractive or repulsive) force due to the presence of another charge. Hydrostatic Pressure p = ? gh or, pressure difference between 2 points separated by a vertical distance of h Upthrus t An upward force exerted by a fluid on a submerged or floating object arises because of the difference in pressure between the upper and lower surfaces of the object. Archimedes Principle Upthrust = weight of the fluid displaced by submerged object. ie Upthrust = Volsubmerged x ? fluid x g Frictional Forces The contact force between two surfaces = (friction2 + normal reaction2)? * The component along the surface of the contact force is called friction * Friction between 2 surfaces always opposes relative motion or attempted motion, and * Its value varies up to a maximum value called the static friction Viscous Forces * A force that opposes the motion of an object in a fluid * Only exists when there is (relative) motion * Magnitude of viscous force increases with the speed of the object Centre of Gravity of an object is defined as that pt through which the entire weight of the object may be considered to act.A couple is a pair of forces which tends to produce rotation only. Moment o f a Force The product of the force and the perpendicular distance of its line of action to the pivot Torque of a Couple The produce of one of the forces of the couple and the perpendicular distance between the lines of action of the forces. (WARNING NOT an action-reaction pair as they act on the same body. ) Conditions for Equilibrium (of an extended object) 1. The resultant force acting on it in any direction equals zero 2. The resultant moment about any point is zero If a mass is acted upon by 3 forces only and remains in equilibrium, then 1.The lines of action of the 3 forces must pass through a common point 2. When a vector diagram of the three forces is drawn, the forces will form a closed triangle (vector triangle), with the 3 vectors pointing in the same orientation around the triangle. Principle of Moments For a body to be in equilibrium, the sum of all the anticlockwise moments about any point must be equal to the sum of all the clockwise moments about that same point. Work , Energy and Power Work Done by a force is defined as the product of the force and displacement (of its point of application) in the direction of the force W = F s cos ?Negative work is said to be done by F if x or its compo. is anti-parallel to F If a variable force F produces a displacement in the direction of F, the work done is determined from the area under F-x graph. May need to find area by counting the squares. By Principle of Conservation of Energy, Work Done on a system = KE gain + GPE gain + Work done against friction Consider a rigid object of mass m that is initially at rest. To accelerate it uniformly to a speed v, a constant net force F is exerted on it, parallel to its motion over a displacement s. Since F is constant, acceleration is constant, Therefore, using the equation 2 = u2 +2as, as = 12 (v2 u2) Since kinetic energy is equal to the work done on the mass to bring it from rest to a speed v, The kinetic energy, EK = Work done by the force F = Fs = mas = ? m (v2 u2) Gravitational potential energy this arises in a system of masses where there are attractive gravitational forces between them. The gravitational potential energy of an object is the energy it possesses by virtue of its position in a gravitational field. Elastic potential energy this arises in a system of atoms where there are either attractive or repulsive short-range inter-atomic forces between them.Electric potential energy this arises in a system of charges where there are either attractive or repulsive electric forces between them. The potential energy, U, of a body in a force field whether gravitational or electric field is related to the force F it experiences by F = dU / dx. Consider an object of mass m being lifted vertically by a force F, without acceleration, from a certain height h1 to a height h2. Since the object moves up at a constant speed, F is equal to mg. The change in potential energy of the mass = Work done by the force F = F s = F h = m g hEfficiency The ratio of (useful) output energy of a machine to the input energy. ie = Useful Output Energy x100% = Useful Output Power x100% Input Energy Input Power Power instantaneous is defined as the work done per unit time. P = Total Work Done = W Total Time t Since work done W = F x s, P = F x s = Fv t * for object moving at const speed F = Total resistive force equilibrium condition * for object beginning to accelerate F = Total resistive force + ma kink Motion Displacement (y) Position of an moving ridger particle from its equilibrium position.Amplitude (y0 or A) The maximum magnitude of the displacement of an oscillating particle from its equilibrium position. power point (T) Time taken for a particle to undergo one complete cycle of oscillation. Frequency (f) hail of oscillations performed by a particle per unit time. Wavelength (? ) For a progressive expand, it is the distance between any two successive particles that are in phase, e. g. it is the distance between 2 consec utive crests or 2 troughs. Wave speed (v) The speed at which the roll upform travels in the direction of the propagation of the wave.Wave front A line or surface joining points which are at the same state of oscillation, i. e. in phase, e. g. a line joining crest to crest in a wave. Ray The path taken by the wave. This is used to indicate the direction of wave propagation. Rays are always at right angles to the wave fronts (i. e. wave fronts are always perpendicular to the direction of propagation). From the definition of speed, Speed = Distance / Time A wave travels a distance of one wavelength, ? , in a time interval of one period, T. The frequency, f, of a wave is equal to 1 / T Therefore, speed, v = ? / T = (1 / T)? f? v = f? Example 1 A wave travelling in the positive x direction is showed in the figure. Find the amplitude, wavelength, period, and speed of the wave if it has a frequency of 8. 0 Hz. Amplitude (A) = 0. 15 mWavelength (? ) = 0. 40 mPeriod (T) = 1f = 18. 0 ? 0. 12 5 sSpeed (v) =f? = 8. 0 x 0. 40 = 3. 20 m s-1A wave which results in a net transfer of energy from one place to another is known as a progressive wave. Intensity of a wave is defined as the rate of energy flow per unit time power per unit cross-sectional area perpendicular to the direction of wave propagation.Intensity = Power / Area = Energy / (Time x Area) For a point source (which would emit spherical wavefronts), Intensity = (? m? 2xo2) / (t x 4? r2) where x0 amplitude r distance from the point source. Therefore, I ? xo2 / r2 (Pt Source) For all wave sources, I ? (Amplitude)2 Transverse wave A wave in which the oscillations of the wave particles NOT movement are perpendicular to the direction of the propagation of the wave. Longitudinal wave A wave in which the oscillations of the wave particles are parallel to the direction of the propagation of the wave.Polarisation is said to occur when oscillations are in one direction in a plane, NOT just in one direction normal to the d irection of propagation. Only transverse waves can be polarized longitudinal waves cant. Example 2 The following stationary wave pattern is get under ones skined using a C. R. O. whose screen is graduated in cm squares. Given that the time-base is adjusted such that 1 unit on the horizontal axis of the screen corresponds to a time of 1. 0 ms, find the period and frequency of the wave. Period, T = (4 units) x 1. 0 = 4. 0 ms = 4. 0 x 10-3 sf = 1 / T = 14 x 10-3 250 Hz superposition principle Principle of Superposition When two or more waves of the same type meet at a point, the resultant displacement of the waves is equal to the vector sum of their individual displacements at that point. Stretched String A horizontal rope with one end fixed and another attached to a vertical oscillator. Stationary waves will be produced by the direct and reflected waves in the string. Or we can have the string stopped at one end with a pulley as shown below. Microwaves A microwave emitter placed a distance away from a metal plate that reflects the emitted wave.By moving a detector along the path of the wave, the nodes and antinodes could be detected. Air editorial A tuning fork held at the mouth of a open tube projects a sound wave into the column of air in the tube. The length of the tube can be changed by varying the water level. At certain lengths of the tube, the air column resonates with the tuning fork. This is due to the formation of stationary waves by the incident and reflected sound waves at the water surface. Stationary (Standing) Wave) is one * whose waveform/wave profile does not advance move, where there is no net transport of energy, and * where the positions of antinodes and nodes do not change (with time). A stationary wave is formed when two progressive waves of the same frequency, amplitude and speed, travelling in opposite directions are superposed. Assume boundary conditions are met Stationary waves Stationary Waves Progressive Waves Amplitude Varies from maximum at the anti-nodes to zero at the nodes. Same for all particles in the wave (provided no energy is lost). Wavelength Twice the distance between a pair of adjacent nodes or anti-nodes. The distance between two consecutive points on a wave, that are in phase. Phase Particles in the same segment/ between 2 adjacent nodes, are in phase. Particles in adjacent segments are in anti-phase. All particles within one wavelength have different phases. Wave Profile The wave profile does not advance. The wave profile advances. Energy No energy is transported by the wave. Energy is transported in the direction of the wave. Node is a region of destructive superposition where the waves always meet out of phase by ? radians. Hence displacement here is permanently zero or minimum.Antinode is a region of constructive superposition where the waves always meet in phase. Hence a particle here vibrates with maximum amplitude but it is NOT a pt with a permanent large displacement Dist between 2 successive nodes / antinodes = ? / 2 Max pressure change occurs at the nodes NOT the antinodes because every node changes fr being a pt of compression to become a pt of rarefaction half a period later Diffraction refers to the bed covering or bending of waves when they pass through an opening happy chance, or round an obstacle (into the shadow region). Illustrate with diag For significant diffraction to occur, the size of the gap ? ? of the wave For a diffraction bumpy, d sin ? = n ? , d = dist between successive slits grating spacing = reciprocal of number of lines per metre When a white light passes through a diffraction grating, for each order of diffraction, a longer wavelength red diffracts more than a shorter wavelength violet as sin ? ? ? . Diffraction refers to the spreading of waves as they pass through a narrow slit or near an obstacle. For diffraction to occur, the size of the gap should approximately be equal to the wavelength of the wave.Coherent waves Wav es having a constant phase difference not zero phase difference / in phase perturbation may be described as the superposition of waves from 2 ordered sources. For an observable / well-defined interference pattern, the waves must be coherent, have about the same amplitude, be unpolarised or polarised in the same direction, be of the same type. Two-source interference using 1. Water Waves Interference patterns could be observed when two dippers are attached to the vibrator of the ripple tank.The ripples produce constructive and destructive interference. The dippers are coherent sources because they are fixed to the same vibrator. 2. Microwaves Microwave emitted from a transmitter through 2 slits on a metal plate would also produce interference patterns. By moving a detector on the opposite side of the metal plate, a series of rise and fall in amplitude of the wave would be registered. 3. Light Waves (Young? s stunt man slit experiment) Since light is emitted from a bulb randomly, the way to obtain two coherent light sources is by splitting light from a single slit.The 2 beams from the double slit would then interfere with each other, creating a pattern of alternate bright and dark fringes (or high and low intensities) at regular intervals, which is also known as our interference pattern. Condition for Constructive Interference at a pt P Phase difference of the 2 waves at P = 0 or 2? , 4? , etc Thus, with 2 in-phase sources, * implies path difference = n? with 2 antiphase sources path difference = (n + ? )? Condition for Destructive Interference at a pt P Phase difference of the 2 waves at P = ? or 3? , 5? , etc With 2 in-phase sources, + implies path difference = (n+ ? ), with 2 antiphase sources path difference = n ? Fringe separation x = ? D / a, if aD applies only to Youngs effigy Slit interference of light, ie, NOT for microwaves, sound waves, water waves Phase difference betw the 2 waves at any pt X betw the central & 1st maxima) is (approx) propor tional to the dist of X from the central maxima. Using 2 sources of equal amplitude x0, the resultant amplitude of a bright fringe would be doubled 20, & the resultant intensity increases by 4 times not 2 times. IResultant ? (2 x0)2 Electric FieldsElectric field strength / intensity at a point is defined as the force per unit positive charge acting at that point a vector Unit N C-1 or V m-1 E = F / q F = qE * The electric force on a positive charge in an electric field is in the direction of E, while * The electric force on a negative charge is opposite to the direction of E. * Hence a +ve charge placed in an electric field will accelerate in the direction of E and gain KE & simultaneously lose EPE, while a negative charge caused to move (projected) in the direction of E will decelerate, ie lose KE, & gain EPE. Representation of electric fields by field lines Coulombs law The (mutual) electric force F acting between 2 point charges Q1 and Q2 separated by a distance r is giv en by F = Q1Q2 / 4 or2 where ? 0 permittivity of free space or, the (mutual) electric force between two point charges is proportional to the product of their charges inversely proportional to the square of their separation. Example 1 Two positive charges, each 4. 18 ? C, and a negative charge, -6. 36 ? C, are fixed at the vertices of an equilateral triangle of side 13. 0 cm. Find the electrostatic force on the negative charge. F = Q1Q2 / 4 or2= (8. 99 x 109) (4. 18 x 10-6)(6. 6 x 10-6) / (13. 0 x 10-2)2= 14. 1 N (Note negative sign for -6. 36 ? C has been ignored in the calculation)FR = 2 x Fcos300= 24. 4 N, vertically upwards Electric field strength due to a Point Charge Q E = Q / 4 or2 NB Do NOT substitute a negative Q with its negative sign in calculations Example 2 In the figure below, determine the point (other than at infinity) at which the total electric field strength is zero. From the diagram, it can be observed that the point where E is zero lies on a straight line whe re the charges lie, to the odd of the -2. 5 ? C charge. Let this point be a distance r from the left charge.Since the total electric field strength is zero, E6? = E-2? 6? / (1 + r)2 / 4 or2 = 2. 5? / r2 / 4 or2 (Note negative sign for -2. 5 ? C has been ignored here) 6 / (1 + r)2 = 2. 5 / r2 v(6r) = 2. 5 (1 + r) r = 1. 82 m The point lies on a straight line where the charges lie, 1. 82 m to the left of the -2. 5 ? C charge. Uniform electric field between 2 Charged Parallel Plates E = Vd, d perpendicular dist between the plates, V potential difference between plates Path of charge moving at 90 to electric field parabolic. beyond the pt where it exits the field, the path is a straight line, at a tangent to the parabola at exit.Example 3 An electron (m = 9. 11 x 10-31 kg q = -1. 6 x 10-19 C) moving with a speed of 1. 5 x 107 ms-1, enters a region between 2 parallel plates, which are 20 mm apart and 60 mm long. The top plate is at a potential of 80 V relative to the lower plate. Determ ine the angle through which the electron has been deflected as a result of passing through the plates. Time taken for the electron to travel 60 mm horizontally = Distance / Speed = 60 x 10-3 / 1. 5 x 107 = 4 x 10-9 s E = V / d = 80 / 20 x 10-3 = 4000 V m-1 a = F / m = eE / m = (1. 6 x 10-19)(4000) / (9. 1 x 10-31) = 7. 0 x 1014 ms-2 vy = uy + at = 0 + (7. x 1014)( 4 x 10-9) = 2. 8 x 106 ms-1 tan ? = vy / vx = 2. 8 x 106 / 1. 5 x 107 = 0. 187 Therefore ? = 10. 6 Effect of a uniform electric field on the motion of charged particles * Equipotential surface a surface where the electric potential is constant * Potential gradient = 0, ie E along surface = 0 * Hence no work is done when a charge is moved along this surface. W=QV, V=0 * Electric field lines must meet this surface at right angles. * If the field lines are not at 90 to it, it would imply that there is a non-zero component of E along the surface. This would contradict the fact that E along an equipotential = 0. Electric pot ential at a point is defined as the work done in moving a unit positive charge from infinity to that point, a scalar unit V ie V = W / Q The electric potential at infinity is defined as zero. At any other point, it may be positive or negative depending on the sign of Q that sets up the field. Contrast gravitational potential. Relation between E and V E = dV / dr i. e. The electric field strength at a pt is numerically equal to the potential gradient at that pt. NB Electric field lines point in direction of decreasing potential ie from high to low pot.Electric potential energy U of a charge Q at a pt where the potential is V U = QV Work done W on a charge Q in moving it across a pd ? V W = Q ? V Electric Potential due to a point charge Q V = Q / 4 or NB Substitute Q with its sign Electromagnetism When a conductor carrying a current is placed in a magnetic field, it experiences a magnetic force. The figure above shows a wire of length L carrying a current I and lying in a magneti c field of flux density B. Suppose the angle between the current I and the field B is ? , the magnitude of the force F on the conductor is iven by F = BILsin? The direction of the force can be found using Fleming? s Left Hand Rule (see figure above). Note that the force is always perpendicular to the plane containing both the current I and the magnetic field B. * If the wire is parallel to the field lines, then ? = 0, and F = 0. (No magnetic force acts on the wire) * If the wire is at right angles to the field lines, then ? = 90, and the magnetic force acting on the wire would be maximum (F = BIL) Example The 3 diagrams below each show a magnetic field of flux density 2 T that lies in the plane of the page.In each case, a current I of 10 A is directed as shown. Use Flemings Left Hand Rule to predict the directions of the forces and work out the magnitude of the forces on a 0. 5 m length of wire that carries the current. (Assume the horizontal is the current) F = BIL sin? = 2 x 10 x 0. 5 x sin90 = 10 N F = BIL sin? = 2 x 10 x 0. 5 x sin60 = 8. 66 N F = BIL sin ? = 2 x 10 x 0. 5 x sin180 = 0 N magnetised flux density B is defined as the force acting per unit current in a wire of unit length at right-angles to the field B = F / ILsin ? F = B I L sin ? ? Angle between the B and L NB write down the above defining equation & define each symbol if youre not able to give the affirmation form. Direction of the magnetic force is always perpendicular to the plane containing the current I and B even if ? ? 0 The Tesla is defined as the magnetic flux density of a magnetic field that causes a force of one newton to act on a current of one ampere in a wire of length one metre which is perpendicular to the magnetic field. By the Principle of moments, Clockwise moments = Anticlockwise moments mg x = F y = BILsin90 yB = mgx / ILy Example A 100-turn immaterial spin around 6. 0 cm by 4. 0 cm is pivoted about a horizontal axis as shown below. A horizontal uniform magne tic field of direction perpendicular to the axis of the coil passes through the coil. Initially, no mass is placed on the pan and the arm is kept horizontal by adjusting the counter-weight. When a current of 0. 50 A flows through the coil, equilibrium is restored by placing a 50 mg mass on the pan, 8. 0 cm from the pivot. Determine the magnitude of the magnetic flux density and the direction of the current in the coil.Taking moments about the pivot, sum of Anti-clockwise moments = Clockwise moment (2 x n)(FB) x P = W x Q (2 x n)(B I L) x P = m g x Q, where n no. of wires on each side of the coil (2 x 100)(B x 0. 5 x 0. 06) x 0. 02 = 50 x 10