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PHYS113 Electricity and Electromagnetism Semester 2; 2002. Set B Notes (WK-10 final) Professor B. J. Fraser. 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 10 6 V
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PHYS113 Electricityand ElectromagnetismSemester 2; 2002 Set B Notes (WK-10 final) Professor B. J. Fraser
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.
charge on either Capacitance = p.d. between them 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. 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.
Parallel Plate Capacitor +Q - + - -Q + - + + - - - + + - + - - + + - Area, A d • From Gauss’ law, for each plate: • For 2 || plates: • But electric field, E = V/d,
2a 2b Cylindrical Capacitor +Q -Q l • Inner conductor radius = a, Inner conductor radius = b, length = l. • From Gauss’ law, • Thus: Important for practical capacitors & shielded cables
Capacitance of an Isolated Sphere • Potential of a sphere is: • Thus, the capacitance is: Example: • Capacitance of the Earth is: 7.1 x 10-4 F
energy Energy density = volume Energy in Capacitors • The electric field contains energy • Work to move a charge dq is: • For total charge, Q: • The work is stored as potential energy in the electric field: • Since electric field is: E = V/d i.e. energy E2
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. I.e. Large !
C1 Q1 + - + - Q2 C2 + - 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:
+ - V1 V2 + - C1 C2 + - 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.
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.
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.
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, average current: • and at any time, 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.
dA v Currents in Materials(or, Why Do Lights Turn On?) • Current comprises charges flowing across an area dA at velocity v: • where J = current density = I/A for small A. • The number of charges passing through A is n x A. • And: J = n q vd • where vd = drift velocity of charges • and: n = number density.
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. • What is n?
Current in a Wire • A = cross-sectional area of wire = p r2 = p (5 x 10-4) 2 • Thus, it takes 6 hours for an electron to move 1 m. • Why do the lights turn on so quickly?
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 = sE • 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 E = J / s A vd l
I I V V Non-ohmic conductor (e.g. diode) Ohmic conductor (e.g. resistor) Conductivity of Materials • where r = resistivity = 1 / s • Thus: V = I R • where R = r l/A = resistance • Unit = Volt / Amp: = V / A • 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 r.
Electric Poweror, 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: • But I = dq/dt: P = VI • Unit = Watt, 1W = 1 J/s • For an ohmic material the power dissipated (mostly in heat) is:
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
7. Direct Current CircuitsSources 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 (e) is given by: • Unit is the Volt (= J/C)
I e R + - 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, e, and a resistance, R. • This is Thevenin’s theorem • The net potential energy around the circuit is: e - I R = 0
I Rint R e + - 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. V = e - I Rint
e b d r R + c Ir - IR a a' V a a' b c d Sources of EMF Name Converts Battery Chemical Electrical Generator Mechanical Electrical Solar Panel Radiation Electrical Thermocouple Heat Electrical MHD Magnetic Electrical • Potential around a circuit: I
+ - 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 • Around the circuit loop. V = I (R1 + R2) • Therefore: V1 V2 I R1 R2 I I V Req = R1 + R2
I1 R1 I2 R2 V I + - Resistors in Circuits Resistors in parallel • Same p.d. across each resistor. • Current is shared between resistors. • In household circuits appliances are connected in parallel. • Xmas tree lights are often in series.
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 Second Law:Around a Circuit Loop • The sum of potential changes is zero • The potential is conserved. • Statement of conservation of energy
+ + - - Circuit Analysis Example Worked Example • Find the current in each branch of the circuit shown below. 10 V 3 2 5 V 1 4 10
2 I1 1 I3 I2 10 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! I1 + I2 = I3 (1)
+ - Circuit Analysis Solution • Sum potential drops around first loop. • Mark all voltage rises & drops depending on the current. • Current flow from +ve to -ve ! Start here I1 10 V + - 3 2 - - + + a 5 V - + - + 1 4 I2 10 - 2 I1 + I2 - 5 + 4 I2 - 3 I1 = 0 - 5 I1 + 5 I2 - 5 = 0 (2) I1 - I2 = 1
10 V - + - + 3 2 + 10 - - + Circuit Analysis Solution • Sum around the other loop. • Solve simultaneously & check ! I1 I3 10 - 2 I1 -10 I3 - 3 I1 = 0 - 5 I1 - 10 I3 + 10 = 0 (3) I1 + 2 I3 = 2 I1 = 0.8 A, I2 = - 0.2 A, I3 = 0.6 A
I Rint R e + - 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: • the power delivered to the load is: • When is P a maximum?
P Pmax x = R/r 1 Maximum Power Transfer • Easiest to plot P as a function of R/r. • Can also calculatedP/dx = 0 • where x = R/r but this is tricky! • 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.
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
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).
Rs Is + - A IM RM Rs + - V I RM Ammeters and Voltmeters • An ammeter uses a resistive shunt to bypass a known fraction of the current (e.g. 999 mA). • A voltmeter uses a series resistance to extend the measurement range. • A known fraction of the voltage is dropped across the resistance. • For an ideal ammeter: RM 0 • For an ideal voltmeter: RM
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. 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 VM Rs Is + - A IM 1 A RM
Rs + - V I RM Designing a Voltmeter Worked Example • Design a meter to measure 25V at fsd using the same galvanometer. V = Vs + VM = I Rs + I RM Rs = (25 / 0.0015) - 75 Rs = 16 591 Vs Vm V
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 varieswith 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. • Units: . F t = R C
R + e C I q - C e C e (1-1/e) Charging R C t q + I0 - Discharging I0 / e R C t RC Time Constants
RC Circuits • Around the loop: • and • The time constant property of RC circuits is essential in time-dependent circuits, e.g. oscillators & filters
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?
What is a Fatal Current? V = 240 V 1.0 0.2 DEATH R = 1.5 k 0.1 Extreme breathing difficulty Amperes Muscular paralysis Can’t let go Painful 0.01 Mild sensation R = 0.5 M Threshold of sensation 0.001
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!
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.
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!
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!
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.