Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
PHYS113 Electricity and Electromagnetism Semester 2; 2002 Set B Notes (WK-10 final) Professor B. J. Fraser 1 Uses in Technology Accelerators (1929) (Giancoli Section 44.2, p1115) Van der Graaf HV Accelerator Works because E-field inside Gaussian sphere is zero 1m sphere 3 x 106 V Up to 20 MV produced Precipitators (See Figure shown) Remove dust and particles from coal combustion -ve wire @ 40 - 100 kV E-field particles to wall > 99% effective. Photocopiers (1940) (Giancoli Example 21.5, p555) Image on +ve photoconductive drum Charge pattern -ve toner pattern Heat fixing +ve paper. 2 5. What is Capacitance? What is a Capacitor? Charge-carrying conductors are surrounded by an electric field. Field can do work on other charges. Capacitance describes the energy stored in the electric field between 2 equal but oppositely charged conductors. charge on either Capacitance = p.d. between them Q C V Unit = Farad, F, = C/V Large unit, - usually use mF, nF, pF. A capacitor is a device comprising a pair of conducting surface, plates, carrying charge with a p.d. and a fixed separation between them. 3 Parallel Plate Capacitor +Q Area, A - + + - - -Q + + -- +++ - + + d From Gauss’ law, Q Q 2 EA E for each plate: 0 2 0 A E For 2 || plates: Q 0 A But electric field, E = V/d, Qd 0 A Q V C Q 0 A V Qd C 0 A d 4 Cylindrical Capacitor +Q 2a 2b -Q l Inner conductor radius = a, Inner conductor radius = b, length = l. b Vb Va E d s a 2k From Gauss’ law, E r b b dr b V Er dr 2k 2k ln r a a a Thus: C Q C V Q Q V 2kQ b ln l a l b 2k ln a Important for practical capacitors & shielded cables 5 Capacitance of an Isolated Sphere Potential of a sphere is: Q V 4 0 r Thus, the capacitance is: Q C 4 0 r V Example: Capacitance of the Earth is: 7.1 x 10-4 F 6 Energy in Capacitors The electric field contains energy q Work to move dW Vdq dq C a charge dq is: For total charge, Q: Q Q q 1 Q2 W dW dq q dq C C0 2C 0 The work is stored as potential energy in the electric field: Q 2 CV 2 QV U 2C 2 2 Since electric field is: E = V/d 0E 2 CV 2 0 A 2 2 0 E 2 U E d Ad volume 2 2d 2 2 energy Energy density = volume 0E 2 U u i.e. energy E2 volume 2 7 Alternative Energy Storage Worked Example An alternative energy proposal is for the storage of energy in electric fields of capacitors. Find the E field required to store 1J in a volume of 1 m3 in a vacuum. 0E 2 u J m-3 2 E 2u 0 2 1 -12 8.85 x 10 4.75 x 105 N C-1 I.e. Large ! 8 Capacitors in Electric Circuits In circuits, capacitors appear in parallel or series combinations. Parallel Capacitors: Charge is stored on the plates of both capacitors. Qtotal = Q1 + Q2 Since Q = CV & each capacitor has the same p.d. across it: Ctotal Qtotal V Q1 Q2 V Ceq C1 C2 Q1 + + - C1 - Q2 C2 + - 9 Capacitors in Electric Circuits Series Capacitors: The magnitude of the charge on each plate is the same, Q. The potential difference is summed across the capacitors: Vbattery = V1 + V2 where V1 = Q/C1 , V2 = Q/C2 , etc. Q Q Q Ceq C1 C2 1 1 1 Ceq C1 C2 V1 + V2 - + C1 C2 + - 10 Dielectrics Dielectric - nonconducting material between the plates of a capacitor. Examples: air, paper, plastic, glass. It has 2 important properties: Dielectric Strength: Size of E-field (V/m) that causes dielectric to fail (stop insulating). Arc or short circuit (Typ. ~ 106 V/m) Correct dielectric can increase max operating voltage of capacitor. Dielectric Constant: Molecular dipoles in the dielectric material align with the electric field. Reduces effective field to E/k, where k is a constant. 11 Dielectric Constant The capacitance therefore increases as well: C = k C0 where k = dielectric constant This allows capacitors to be made smaller by using high k dielectrics. The energy density in the electric field is also reduced to: u = u0 / k Arises because it takes work to insert the dielectric Piezo-electricity. 12 6. Electric Currents What is an electric current? An electric current is an organised movement of charges. Usually but not always electrons. Charges move due to applied E-field. By definition, Q I av average current: t dQ and at any time, I dt instantaneous current is: Unit: 1 Ampere, A = C/s (‘amp’) Typ. household currents ~ few amps In electronic circuits, ~ mA, mA, nA. By convention, direction of current flow chosen for +ve charges. I.e from +ve to -ve. Electrons actually moving other way. 13 Currents in Materials (or, Why Do Lights Turn On?) Current comprises charges flowing across an area dA at velocity v: dA I J d A A v where J = current density = I/A for small A. The number of charges passing through A is n x A. Q I nq vd A t And: J = n q vd where vd = drift velocity of charges and: n = number density. 14 Current in a Wire Worked Example A light draws a current of 0.5 A through a copper wire of diameter 1.0 mm. Find the drift velocity of electrons in the wire. The density of copper is 8.92 g cm-3. I I nq vd A vd nqA What is n? number of electrons n unit density N A Cu M Cu 6.02 x 10 8.92 23 63.5 8.5 x 10 22 electrons cm -3 8.5 x 10 28 electrons m -3 15 Current in a Wire A = cross-sectional area of wire = r2 = (5 x 10-4) 2 I vd nqA I nqr 2 0. 5 8.5 x 1028 1.6 x 10-19 2.5 x 10-7 4.7 x 10 -7 ms -1 Thus, it takes 6 hours for an electron to move 1 m. Why do the lights turn on so quickly? 16 Resistance and Conductivity: Ohms ain’t Ohms! The rate at which charges move in a conductor due to an electric field depends on magnitude of the field. Thus: J = s E where s = conductivity & depends on geometry & properties of conductor. This is known as Ohm’s Law. Not all materials obey Ohm’s law. I.e. not all materials are linear. Metals at increasing temperature & semiconductors don’t obey Ohms law. These are non-ohmic conductors. For a wire of length l & area A. E = V / l A E = J / s J I sV A l vd l 17 Conductivity of Materials 1 l l V I I s A A where = resistivity = 1 / s Thus: V=IR where R = l/A = resistance Unit = Volt / Amp: = V / A I I Ohmic conductor (e.g. resistor) V V Non-ohmic conductor (e.g. diode) A resistor is a device built with a specified resistance ( ‘s M’s) Units of resistivity are .m, & conductivity ( .m)-1. (mho). Good conductor has low . 18 Electric Power or, How Bright is your Light? Charges lose energy in flowing in a material (supplied by the battery). For a small charge dq moving through a p.d. V, dU = V dq Thus, power is given by: P dW dq V Dt dt But I = dq/dt: P = VI Unit = Watt, 1W = 1 J/s For an ohmic material the power dissipated (mostly in heat) is: 2 V P IV I 2 R R 19 Car Starter Motor Worked Example A car starter motor draws 500 A through a wire of resistance 0.01 . Find the voltage drop and the power loss in the cable. V = IR = 500 x 0.01 = 5 V P = I2 R = (500)2 x 0.01 = 2 500 W 20 7. Direct Current Circuits Sources of EMF The electric energy that drives charges around a circuit is called the electromotive force (emf) Not a force but an energy. A source of emf increases the potential energy of charges in a circuit (“pumps them up”) By definition, the emf () is given by: dW dq Unit is the Volt (= J/C) 21 Equivalent Circuits and Thevenin’s Theorem All circuits, no matter how complex can be reduced to a simple equivalent circuit This circuit has a source of emf, , and a resistance, R. This is Thevenin’s theorem I + - R The net potential energy around the circuit is: - I R = 0 22 Internal Resistance All real sources of emf have some internal resistance that: Reduces the output terminal voltage Limits the power that can be delivered by the emf source. I Rint + - R V = - I Rint 23 Sources of EMF Name Converts Battery Electrical Chemical Generator Electrical Mechanical Solar Panel Radiation Electrical Thermocouple Electrical Heat MHD Magnetic Electrical d r b R around Potential a circuit: c I + a' - a Ir IR V a a' b c d 24 Resistors in Circuits Combinations of resistances in circuits may be in series or parallel. Resistors in series Same current in each resistor. Voltage across each is: Vr = I R V1 I V2 I R1 + R2 I V Around the circuit loop. V = I (R1 + R2) Therefore: Req = R1 + R2 25 Resistors in Circuits Resistors in parallel Same p.d. across each resistor. Current is shared between resistors. I1 1 1 I I1 I 2 V R1 R2 R1 I2 I R2 V + 1 1 1 Req R1 R2 - In household circuits appliances are connected in parallel. Xmas tree lights are often in series. 26 Circuit Analysis: Kirchoff’s Laws Complex circuits involving multiple loops are analysed using Kirchoff’s Laws. First Law: At a Junction The sum of the currents entering and leaving the junction is zero. Statement of conservation of charge I in I out Second Law: Around a Circuit Loop The sum of potential changes is zero The potential is conserved. Statement of conservation of energy V 0 loop 27 Circuit Analysis Example Worked Example Find the current in each branch of the circuit shown below. 2 + 10 V - 3 1 + 5V - 4 10 28 Circuit Analysis Solution Pick a junction and assign arbitrary current directions and sum to zero. It doesn’t matter if the initial guess of current direction is wrong since the answer will just be a -ve value! 2 I1 1 I3 I2 10 I1 + I2 = I3 (1) 29 Circuit Analysis Solution Sum potential drops around first loop. Start here I1 - 2 + - 1 + + 10 V a - 3 + + 5V - - 4 + I2 Mark all voltage rises & drops depending on the current. Current flow from +ve to -ve ! 10 - 2 I1 + I2 - 5 + 4 I2 - 3 I1 = 0 - 5 I1 + 5 I2 - 5 = 0 I1 - I2 = 1 (2) 30 Circuit Analysis Solution Sum around the other loop. + 10 V - - 3 + - 2 + I1 I3 + 10 - 10 - 2 I1 -10 I3 - 3 I1 = 0 - 5 I1 - 10 I3 + 10 = 0 I1 + 2 I3 = 2 (3) Solve simultaneously & check ! I1 = 0.8 A, I2 = - 0.2 A, I3 = 0.6 A 31 Maximum Power Transfer Practically we are interested in the amount of power that can be transferred from source to load. The max. amount of power will be transferred from any source (with internal resistance, r) to a load (of resistance, R) when R equals r. Recall that: Rr the power delivered to the load is: P I 2R R - Rint + I I 2 R r 2 R When is P a maximum? 32 Maximum Power Transfer Easiest to plot P as a function of R/r. Can also calculate dP/dx = 0 where x = R/r but this is tricky! P Pmax 1 x = R/r Max value when x = 1 or R = r Maximum power transfer theorem. That’s why there are several output sockets on the back of a stereo amplifier - so its resistance can be matched to that of the speakers. 33 Impedance The maximum power transfer theorem is an example of impedance matching. Any medium through which energy is transferred has a certain resistance to the flow - an impedance. It turns out that for any system involving a transfer of energy from a supplier to a receiver we need the impedance of the supplier and receiver to be equal in order to transfer the max. energy. Thus, we can consider the impedance of a wire or a string or an ear, etc. Impedance matching is a common problem in transport of electrical signals 34 Measuring Instruments Analogue meters comprise a coil of wire mounted on a pivot between magnets. Current passing through the coil causes a deflection of the needle. The basic moving-coil meter is the D’Arsonval galvanometer. A current of ~ 1mA gives a full scale deflection (fsd). Internal resistance of meter is the meter resistance (RM). 35 Ammeters and Voltmeters An ammeter uses a resistive shunt to bypass a known fraction of the current (e.g. 999 mA). Is Rs + - IM RM A voltmeter uses a series resistance to extend the measurement range. A known fraction of the voltage is dropped across the resistance. Rs + I A RM V For an ideal ammeter: RM 0 For an ideal voltmeter: RM 36 Designing an Ammeter Worked Example A galvanometer of resistance 75 has a full scale deflection of 1.5 mA. Design a meter to measure 1A at fsd. VM Is 1A Rs + IM RM A IM = 1.5 mA Is = 1.0 - 0.0015 = 0.9985 A VM = IM RM = Is Rs Rs = IM RM / Is = (0.0015 x 75) / 0.9985 Rs = 0.113 37 Designing a Voltmeter Worked Example Design a meter to measure 25V at fsd using the same galvanometer. Vs Vm Rs + I RM V V V = Vs + VM = I Rs + I RM Rs = (25 / 0.0015) - 75 Rs = 16 591 38 RC Circuits & Time Constants At d.c. capacitors are an open circuit I.e there is no electrical path. The plates of a capacitor will charge or discharge if the current varies with time. The rate at which this happens depends upon the series resistance of the circuit and the size of the capacitor. The series resistance limits the current flowing into the capacitor. The characteristic time is called the time constant of the circuit. t=R C Units: .F 39 RC Time Constants R I + - + - C q C C (1-1/e) Charging RC q t I0 Discharging I0 / e RC t 40 RC Circuits Around the loop: q IR 0 C dq q R 0 dt C dq q 1 q C dt R RC RC t q C 1 e RC and t dq I I 0 e RC dt The time constant property of RC circuits is essential in timedependent circuits, e.g. oscillators & filters 41 Electricity in the Home What is a fatal current? Why does house wiring have 3 wires? How do fuses work? What is a circuit breaker? What is an ELCB? What current can be drawn from power points? What is an “off-peak” system? 42 What is a Fatal Current? 1.0 0.2 Amperes 0.1 0.01 V = 240 V DEATH R = 1.5 k Extreme breathing difficulty Muscular paralysis Can’t let go Painful Mild sensation R = 0.5 M Threshold of sensation 0.001 43 Why does House Wiring have 3 Wires? The three wires are live (hot), neutral and earth. Actually, only two wires come into your house - live & neutral. Live is the high potential side of the transformer while the neutral is connected to ground at the transformer. But - neutral may be at a different potential to earth by the time it gets to your house! The earth wire is the local earth (water pipe, earth stake). All electrical devices in a metal case have the case connected to earth. Ensures that if the live wire touches the case then the least resistant path to earth is through the earth wire & not through you! 44 How do Fuses Work? Fuse is a small metallic strip designed to melt when the current exceeds a certain value. Fuse wire in Woolies rated at 8 A & 16 A for example. But, bear in mind that plain fuse wire does take a finite time to melt. In some cases, this means that the wire has time to pass a much higher value of current than its rating! Special fuses are available - quick blow fuses have a spring that applies tension to the fuse wire - if it starts to melt it is pulled thinner and blows quickly. 45 What is a Circuit Breaker? What is an ELCB ? More modern homes have the fuses replaced by a circuit breaker (CB). When the current exceeds a certain value the CB acts as a switch & opens the circuit. A common design involves the use of a bimetallic strip. When the current exceeds a certain value the strip heats up and bends. The bending strip breaks the circuit. Can be slow - many CB’s now incorporate electromagnets. An earth leakage circuit breaker (ELCB) is a device that detects very small currents (mA) to ground. If a current is detected then the power is switched off in a few ms. Could save your life! 46 What Current can be Drawn from Power Points? Its important to be able to calculate the max current that can be drawn. Typically, a power circuit is fused at 16 A in Australia. Light circuits are fused at 8 A. Thus, for a single circuit the total current load must not exceed 16 A. But - most appliances quote the power drawn and not the current. Just need to remember that P = IV and that mains voltage is 240 V. Max power load on a single circuit is: P = 16 x 240 = 3.8 kW In many old houses in Newcastle all of the sockets in the house are on a single circuit!!! Be careful when turning stuff on! especially in winter! 47 What is an “off - peak” system? Demand for electricity is not spread out evenly during the day or year. This presents problems for the power supply companies and the management of the power distribution network. To encourage more even use of power the cost of electricity supplied during low demand periods “off peak” is reduced. This usually occurs after 11pm and is measured by a separate meter box. The meter box is activated by a high frequency signal transmitted down the power cable to your house. Usually operates water heaters and household storage heaters. Off-peak power is also used to store energy - e.g. hydro-systems. 48