Lecture 10: PN Junction & MOS Capacitors
|
|
- Valentine Austin
- 6 years ago
- Views:
Transcription
1 Lecture 10: P Junction & MOS Cpcitors Prof. iknej eprtment of EECS
2 Lecture Outline Review: P Junctions Therml Equilibrium P Junctions with Reverse Bis ( MOS Cpcitors ( : Accumultion, epletion, Inversion Threshol oltge C Curve eprtment of EECS
3 Results of MT #1 Goo Job! AG MI MAX ST E This is only 17% of your gre Homework 15% Lbortory 20% Miterm #1 17% Miterm #2 18% Finl 30% eprtment of EECS
4 P Junction in Therml Equilibrium Contct potentil evelops between P n region iffusion current blnce by rift current epletion region is spce-chrge region where the concentrtion of free crriers is low The epletion region is chrge ue to the immole bckgroun ions (onors n cceptors Use the epletion Approximtion to estimte the chrge ensity clculte the electric fiels n potentil vrition using electrosttics in 1 eprtment of EECS
5 Hve we invente bttery? Cn we hrness the P junction n turn it into bttery? n p A th ln ln ni n i th ln 2 ni A? umericl exmple: 15 A m ln 60m log 20 n i 600m eprtment of EECS
6 Contct Potentil The contct between P junction cretes potentil ifference Likewise, the contct between two issimilr metls cretes potentil ifference (proportionl to the ifference between the work functions When metl semiconuctor junction is forme, contct potentil forms s well If we short P junction, the sum of the voltges roun the loop must be zero: mn pm n p 0 pm mn ( mn pm eprtment of EECS
7 P Junction Cpcitor Uner therml equilibrium, the P junction oes not rw ny current But notice tht P junction stores chrge in the spce chrge region (trnsition region Since the evice is storing chrge, it s cting like cpcitor Positive chrge is store in the n-region, n negtive chrge is in the p-region: q x q po x no eprtment of EECS
8 Reverse Bise P Junction Wht hppens if we reverse-s the P junction? < 0 Since no current is flowing, the entire reverse se potentil is roppe cross the trnsition region To ccommote the extr potentil, the chrge in these regions must increse If no current is flowing, the only wy for the chrge to increse is to grow (shrink the epletion regions eprtment of EECS
9 Current Uner Reverse Bis n p n E0 E0 n p p X X Uner therml equilibrium current is zero If we pply reverse s, we re incresing the brrier ginst iffusion current rift current is low since the fiel only moves minority crriers cross junction In fct, current is not zero but very smll since the minority crrier concentrtion is low. Minority crriers within one iffusion length of junction cn contribute to reverse s current. This is more or less inepenent of the pplie s 0 ( eprtment of EECS
10 eprtment of EECS EECS 105 Fll 2003, Lecture 10 oltge epenence of epletion With Cn reo the mth but in the en we relize tht the equtions re the sme except we replce the builtin potentil with the effective reverse s: s n p q x x X 1 1 ( 2 ( ( ( ε n s n x q x ε 1 ( 2 ( 0 p s p x q x ε 1 ( 2 ( 0 X X 1 ( 0
11 Chrge ersus Bis As we increse the reverse s, the epletion region grows to ccommote more chrge Q J ( qx p ( q 1 Chrge is not liner function of voltge This is non-liner cpcitor We cn efine smll signl cpcitnce for smll signls by breking up the chrge into two terms Q J ( v QJ ( q( v eprtment of EECS
12 eprtment of EECS EECS 105 Fll 2003, Lecture 10 erivtion of Smll Signl Cpcitnce From lst lecture we foun otice tht J J v Q Q v Q ( ( R p j j j x q Q C C 1 ( 0 j p j C x q C s s p j q q q x q C ε ε
13 eprtment of EECS EECS 105 Fll 2003, Lecture 10 Physicl Interprettion of epletion Cp otice tht the expression on the right-hn-sie is just the epletion with in therml equilibrium This looks like prllel plte cpcitor! s j q C ε s s s j X q C ε ε ε ( ( s j X C ε
14 A rible Cpcitor (rctor Cpcitnce vries versus s: C C j j0 Appliction: Rio Tuner eprtment of EECS
15 iffusion Resistor -type iffusion Region Oxie P-type Si Substrte Resistor is cpcitively isoltion from substrte Must Reverse Bis P Junction! P Junction cretes istribute cpcitnce with substrte (RC trnsmission line eprtment of EECS
16 MOS Cpcitor Gte (n poly Boy (p-type substrte ε s 11.7ε 0 0 x Oxie (SiO 2 ε ox 3.9ε 0 ery Thin! t ox ~ 1nm MOS Metl Oxie Silicon Snwich of conuctors seprte by n insultor Metl is more commonly hevily ope polysilicon lyer n or p lyer MOS p-type substrte, PMOS n-type substrte eprtment of EECS
17 P-I- Junction Gte (n poly Boy (p-type substrte Uner therml equilibrium, the n-type poly gte is t higher potentil thn the p-type substrte kt p ln 550m n q n i o current cn flow becuse of the insultor but this potentil ifference is ccompnie with n electric fiel Fiels terminte on chrge! eprtment of EECS
18 Fiels n Chrge t Equilibrium ox B X 0 Boy (p-type substrte E ox At equilibrium there is n electric fiel from the gte to the boy. The chrges on the gte re positive. The negtive chrges in the boy come from epletion region eprtment of EECS
19 Goo Plce to Sleep: Flt Bn FB < 0 Q ( 0 G GB FB Boy (p-type substrte If we pply s, we cn compenste for this built-in potentil ( FB In this cse the chrge on the gte goes to zero n the epletion region isppers In soli-stte physics lingo, the energy bns re flt uner this conition n p eprtment of EECS
20 Accumultion Q C ( G ox GB FB GB < FB Q B Q G Boy (p-type substrte If we further ecrese the potentil beyon the flt-bn conition, we essentilly hve prllel plte cpcitor Plenty of holes n electrons re vilble to chrge up the pltes egtive s ttrcts holes uner gte eprtment of EECS
21 epletion GB > FB Boy (p-type substrte Q ( Q G GB B Q q X ( B GB Similr to equilibrium, the potentil in the gte is higher thn the boy Boy chrge is me up of the epletion region ions Potentil rop cross the boy n epletion region eprtment of EECS
22 Inversion GB T s Boy (p-type substrte As we further increse the gte voltge, eventully the surfce potentil increses to point where the electron ensity t the surfce equls the bckgroun ion ensity qs kt s i n ne s p At this point, the epletion region stops growing n the extr chrge is provie by the inversion chrge t surfce eprtment of EECS
23 Threshol oltge The threshol voltge is efine s the gte-boy voltge tht cuses the surfce to chnge from p- type to n-type For this conition, the surfce potentil hs to equl the negtive of the p-type potentil We ll erive tht this voltge is equl to: 1 Tn FB 2 p 2 qε s( 2 p C ox eprtment of EECS
24 Inversion Stops epletion A simple pproximtion is to ssume tht once inversion hppens, the epletion region stops growing This is goo ssumption since the inversion chrge is n exponentil function of the surfce potentil Uner this conition: Q ( Q G Tn B,mx Q ( C ( Q G GB ox GB Tn B,mx eprtment of EECS
25 Q- Curve for MOS Cpcitor Q G epletion inversion Q B,mx Q ( GB ccumultion FB Tn ( GB In ccumultion, the chrge is simply proportionl to the pplies gte-boy s In inversion, the sme is true In epletion, the chrge grows slower since the voltge is pplie over epletion region eprtment of EECS
26 umericl Exmple MOS Cpcitor with p-type substrte: t ox 20nm Clculte flt-bn: Clculte threshol voltge: 5 10 cm 16 3 ( (550 ( FB n C ox p ε t ox ox F/cm cm 1 Tn FB 2 p 2 qε s ( 2 p C ox Tn ( C ox eprtment of EECS
27 um Exmple: Electric Fiel in Oxie Apply gte-to-boy voltge: GB 2.5 < FB evice is in ccumultion The entire voltge rop is cross the oxie: E ox GB ox n p ( t t ox ox cm The chrge in the substrte (boy consist of holes: Q C ( C/cm B ox GB FB 7 2 eprtment of EECS
28 umericl Exmple: epletion Region In inversion, wht s the epletion region with n chrge? B,mx s p p p 2p 0.8 B,mx 1 q 2 ε s X 2,mx X,mx 2ε s B,mx q 144nm Q,mx q X,mx C/cm B 7 2 eprtment of EECS
Physics Lecture 14: MON 29 SEP
Physics 2113 Physics 2113 Lecture 14: MON 29 SEP CH25: Cpcitnce Von Kleist ws le to store electricity in the jr. Unknowingly, he h ctully invente novel evice to store potentil ifference. The wter in the
More informationHomework 4 , (1) 1+( NA +N D , (2)
Homework 4. Problem. Find the resistivity ρ (in ohm-cm) for piece of Si doped with both cceptors (N A = 9 cm 3 ) nd donors (N D = 6 cm 3 ). Since the electron nd hole mobilities depend on the concentrtion
More informationPhysics 2135 Exam 1 September 23, 2014
Exm Totl Physics 2135 Exm 1 September 23, 2014 Key Printe Nme: 200 / 200 N/A Rec. Sec. Letter: Five multiple choice questions, 8 points ech. Choose the best or most nerly correct nswer. B 1. Object A hs
More informationCAPACITORS AND DIELECTRICS
Importnt Definitions nd Units Cpcitnce: CAPACITORS AND DIELECTRICS The property of system of electricl conductors nd insultors which enbles it to store electric chrge when potentil difference exists between
More informationA, Electromagnetic Fields Final Exam December 14, 2001 Solution
304-351, Electrognetic Fiels Finl Ex Deceer 14, 2001 Solution 1. e9.8. In chpter9.proles.extr.two loops, e of thin wire crry equl n opposite currents s shown in the figure elow. The rius of ech loop is
More informationHomework Assignment 5 Solution Set
Homework Assignment 5 Solution Set PHYCS 44 3 Februry, 4 Problem Griffiths 3.8 The first imge chrge gurntees potentil of zero on the surfce. The secon imge chrge won t chnge the contribution to the potentil
More informationExam 1 September 21, 2012 Instructor: Timothy Martin
PHY 232 Exm 1 Sept 21, 212 Exm 1 September 21, 212 Instructor: Timothy Mrtin Stuent Informtion Nme n section: UK Stuent ID: Set #: Instructions Answer the questions in the spce provie. On the long form
More informationLast Time emphasis on E-field. Potential of spherical conductor. Quick quiz. Connected spheres. Varying E-fields on conductor.
Lst Time emphsis on Efiel Electric flux through surfce Guss lw: Totl electric flux through close surfce proportionl to chrge enclose Q " E = E = 4$k e Q % o Chrge istribution on conuctors Chrge ccumultes
More informationPHYS102 - Electric Energy - Capacitors
PHYS102 - lectric nerg - Cpcitors Dr. Suess Februr 14, 2007 Plcing Chrges on Conuctors................................................. 2 Plcing Chrges on Conuctors II................................................
More informationDIRECT CURRENT CIRCUITS
DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through
More informationPhysics 1402: Lecture 7 Today s Agenda
1 Physics 1402: Lecture 7 Tody s gend nnouncements: Lectures posted on: www.phys.uconn.edu/~rcote/ HW ssignments, solutions etc. Homework #2: On Msterphysics tody: due Fridy Go to msteringphysics.com Ls:
More informationProblems for HW X. C. Gwinn. November 30, 2009
Problems for HW X C. Gwinn November 30, 2009 These problems will not be grded. 1 HWX Problem 1 Suppose thn n object is composed of liner dielectric mteril, with constnt reltive permittivity ɛ r. The object
More informationPhys 201 Midterm 1 S. Nergiz, E.Oğuz, C. Saçlıoğlu T. Turgut Fall '01 No :. Name :. Total Grade :. Grade :. y=a. x=-a +q. x=a -q +q. +Q r.
Phs 0 Miterm S. Nergiz, E.ğuz, C. Sçlıoğlu T. Turgut Fll '0 No :. Nme :. Totl Gre :. Gre :. Question : Three point chrges re plce s shown in figure. ) Clculte the coulom force on ech chrge. ) Fin the electrosttic
More informationCh. 24 Molecular Reaction Dynamics 1. Collision Theory 2. Diffusion-Controlled Reaction
Ch. 4 Moleculr Rection Dynmics 1. Collision Theory. Diffusion-Controlle Rection Lecture 17 3. The Mteril Blnce Eqution 4. Trnsition Stte Theory: The Eyring Eqution 5. Trnsition Stte Theory: Thermoynmic
More informationPhysics 1B schedule Winter Instructor: D.N. Basov Ch18: 1,3,5,7,13,17,21,26,31,33,35,
Week 1:Jn 5 : Jn 1 Physics 1B scheule Winter 009. Instructor: D.N. Bsov bsov@ucs.eu Mon We Friy Lecture: the Electric fiel Lecture: Intro, 15.1-15.3 Lecture: The oulomb lw, Lecture: Electric Flux & Guss
More information200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes
PHYSICS 132 Smple Finl 200 points 5 Problems on 4 Pges nd 20 Multiple Choice/Short Answer Questions on 5 pges 1 hour, 48 minutes Student Nme: Recittion Instructor (circle one): nme1 nme2 nme3 nme4 Write
More informationPhys102 General Physics II
Phys1 Generl Physics II pcitnce pcitnce pcitnce definition nd exmples. Dischrge cpcitor irculr prllel plte cpcitior ylindricl cpcitor oncentric sphericl cpcitor Dielectric Sls 1 pcitnce Definition of cpcitnce
More informationPhysics 2135 Exam 1 February 14, 2017
Exm Totl / 200 Physics 215 Exm 1 Ferury 14, 2017 Printed Nme: Rec. Sec. Letter: Five multiple choice questions, 8 points ech. Choose the est or most nerly correct nswer. 1. Two chrges 1 nd 2 re seprted
More informationReading from Young & Freedman: For this topic, read the introduction to chapter 24 and sections 24.1 to 24.5.
PHY1 Electricity Topic 5 (Lectures 7 & 8) pcitors nd Dielectrics In this topic, we will cover: 1) pcitors nd pcitnce ) omintions of pcitors Series nd Prllel 3) The energy stored in cpcitor 4) Dielectrics
More informationDesigning Information Devices and Systems I Discussion 8B
Lst Updted: 2018-10-17 19:40 1 EECS 16A Fll 2018 Designing Informtion Devices nd Systems I Discussion 8B 1. Why Bother With Thévenin Anywy? () Find Thévenin eqiuvlent for the circuit shown elow. 2kΩ 5V
More informationLecture 5 Capacitance Ch. 25
Lecture 5 pcitnce h. 5 rtoon - pcitnce definition nd exmples. Opening Demo - Dischrge cpcitor Wrm-up prolem Physlet Topics pcitnce Prllel Plte pcitor Dielectrics nd induced dipoles oxil cle, oncentric
More informationPhysics 3323, Fall 2016 Problem Set 7 due Oct 14, 2016
Physics 333, Fll 16 Problem Set 7 due Oct 14, 16 Reding: Griffiths 4.1 through 4.4.1 1. Electric dipole An electric dipole with p = p ẑ is locted t the origin nd is sitting in n otherwise uniform electric
More informationSummary of equations chapters 7. To make current flow you have to push on the charges. For most materials:
Summry of equtions chpters 7. To mke current flow you hve to push on the chrges. For most mterils: J E E [] The resistivity is prmeter tht vries more thn 4 orders of mgnitude between silver (.6E-8 Ohm.m)
More informationSolid State Device Fundamentals. MOS Capacitor
Solid Stte Device Fundmentl MOS pcitor Solid Stte Device Fundmentl Metl-Oxide-Semiconductor cpcitor MOS cpcitor i the centrl prt of emiconductor field effect trnitor. Gte metl Source te d Drin SiO 2 N
More informationPhysics 202, Lecture 10. Basic Circuit Components
Physics 202, Lecture 10 Tody s Topics DC Circuits (Chpter 26) Circuit components Kirchhoff s Rules RC Circuits Bsic Circuit Components Component del ttery, emf Resistor Relistic Bttery (del) wire Cpcitor
More informationExam 1 Solutions (1) C, D, A, B (2) C, A, D, B (3) C, B, D, A (4) A, C, D, B (5) D, C, A, B
PHY 249, Fll 216 Exm 1 Solutions nswer 1 is correct for ll problems. 1. Two uniformly chrged spheres, nd B, re plced t lrge distnce from ech other, with their centers on the x xis. The chrge on sphere
More informationSample Exam 5 - Skip Problems 1-3
Smple Exm 5 - Skip Problems 1-3 Physics 121 Common Exm 2: Fll 2010 Nme (Print): 4 igit I: Section: Honors Code Pledge: As n NJIT student I, pledge to comply with the provisions of the NJIT Acdemic Honor
More informationLecture 1: Electrostatic Fields
Lecture 1: Electrosttic Fields Instructor: Dr. Vhid Nyyeri Contct: nyyeri@iust.c.ir Clss web site: http://webpges.iust.c. ir/nyyeri/courses/bee 1.1. Coulomb s Lw Something known from the ncient time (here
More information13.4 Work done by Constant Forces
13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push
More informationSome parameters of varicaps with gradient base area based on Shottky barrier
ISSN: 35-38 Vol. 4, Issue, December 7 Some prmeters of vricps with grdient bse re bsed on Shottky brrier Mmtkrimov O.O., KuchkrovB.Kh. Rector, Nmngn engineering-technology institute, Kosonsoy str.,7, Nmngn,
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationTIME VARYING MAGNETIC FIELDS AND MAXWELL S EQUATIONS
TIME VARYING MAGNETIC FIED AND MAXWE EQUATION Introuction Electrosttic fiels re usull prouce b sttic electric chrges wheres mgnetosttic fiels re ue to motion of electric chrges with uniform velocit (irect
More informationHomework Assignment 9 Solution Set
Homework Assignment 9 Solution Set PHYCS 44 3 Mrch, 4 Problem (Griffiths 77) The mgnitude of the current in the loop is loop = ε induced = Φ B = A B = π = π µ n (µ n) = π µ nk According to Lense s Lw this
More informationAl C-V Photoresponse Mg 3. Workfunction in vacuum (ev)
6.3. MOS nlyi 6.3.. ltbn voltge clcultion If there i no chrge preent in the ie or t the ie-emiconuctor interfce, the flt bn voltge imply equl the ifference between the gte metl workfunction, Φ M, n the
More informationECE321 Electronics I
ECE321 Electronics I Lecture 4: Physics of Semiconductor iodes Payman Zarkesh-Ha Office: ECE Bldg. 230B Office hours: Tuesday 2:00-3:00PM or by appointment E-mail: pzarkesh.unm.edu Slide: 1 Review of Last
More informationChapter E - Problems
Chpter E - Prolems Blinn College - Physics 2426 - Terry Honn Prolem E.1 A wire with dimeter d feeds current to cpcitor. The chrge on the cpcitor vries with time s QHtL = Q 0 sin w t. Wht re the current
More informationI1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
More informationDesigning Information Devices and Systems I Spring 2018 Homework 8
EECS 16A Designing Informtion Devices nd Systems I Spring 2018 Homework 8 This homework is due Mrch 19, 2018, t 23:59. Self-grdes re due Mrch 22, 2018, t 23:59. Sumission Formt Your homework sumission
More informationPhysics 121 Sample Common Exam 1 NOTE: ANSWERS ARE ON PAGE 8. Instructions:
Physics 121 Smple Common Exm 1 NOTE: ANSWERS ARE ON PAGE 8 Nme (Print): 4 Digit ID: Section: Instructions: Answer ll questions. uestions 1 through 16 re multiple choice questions worth 5 points ech. You
More information7.6 The Use of Definite Integrals in Physics and Engineering
Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationUniversity of Alabama Department of Physics and Astronomy. PH126: Exam 1
University of Albm Deprtment of Physics nd Astronomy PH 16 LeClir Fll 011 Instructions: PH16: Exm 1 1. Answer four of the five questions below. All problems hve equl weight.. You must show your work for
More information4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve
Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions
More informationELETROSTATICS Part II: BASICS
GROWING WITH ONPTS: Physics LTROSTTIS Prt II: SIS Presence of chrge on ny oject cretes n electrosttic fiel roun it n in turn n electricl potentil is experience roun the oject. This phenomenon hs foun ppliction
More information5.4, 6.1, 6.2 Handout. As we ve discussed, the integral is in some way the opposite of taking a derivative. The exact relationship
5.4, 6.1, 6.2 Hnout As we ve iscusse, the integrl is in some wy the opposite of tking erivtive. The exct reltionship is given by the Funmentl Theorem of Clculus: The Funmentl Theorem of Clculus: If f is
More informationConservation Law. Chapter Goal. 6.2 Theory
Chpter 6 Conservtion Lw 6.1 Gol Our long term gol is to unerstn how mthemticl moels re erive. Here, we will stuy how certin quntity chnges with time in given region (sptil omin). We then first erive the
More information4.5 THE FUNDAMENTAL THEOREM OF CALCULUS
4.5 The Funmentl Theorem of Clculus Contemporry Clculus 4.5 THE FUNDAMENTAL THEOREM OF CALCULUS This section contins the most importnt n most use theorem of clculus, THE Funmentl Theorem of Clculus. Discovere
More informationLecture 13 - Linking E, ϕ, and ρ
Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on
More informationChapter 6 Electrostatic Boundary Value Problems. Dr. Talal Skaik
Chpter 6 Electrosttic Boundry lue Problems Dr. Tll Skik 1 1 Introduction In previous chpters, E ws determined by coulombs lw or Guss lw when chrge distribution is known, or potentil is known throughout
More informationElectric Potential. Electric Potential Video: Section 1 4. Electric Fields and WORK 9/3/2014. IB Physics SL (Year Two) Wednesday, September 3, 2014
9/3/014 lectric Potentil IB Physics SL (Yer Two) Wenesy, Septemer 3, 014 lectric Potentil Vieo: Section 1 4 lectric Fiels n WORK In orer to rin two like chres ner ech other work must e one. In orer to
More informationThe Velocity Factor of an Insulated Two-Wire Transmission Line
The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the
More informationPhysics Jonathan Dowling. Lecture 9 FIRST MIDTERM REVIEW
Physics 10 Jonthn Dowling Physics 10 ecture 9 FIRST MIDTERM REVIEW A few concepts: electric force, field nd potentil Electric force: Wht is the force on chrge produced by other chrges? Wht is the force
More informationPhysics 24 Exam 1 February 18, 2014
Exm Totl / 200 Physics 24 Exm 1 Februry 18, 2014 Printed Nme: Rec. Sec. Letter: Five multiple choice questions, 8 points ech. Choose the best or most nerly correct nswer. 1. The totl electric flux pssing
More informationRead section 3.3, 3.4 Announcements:
Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f
More informationPH 102 Exam I Solutions
PH 102 Exm I Solutions 1. Three ienticl chrges of = 5.0 µc lie long circle of rius 2.0 m t ngles of 30, 150, n 270 s shown below. Wht is the resultnt electric fiel t the center of the circle? By symmetry,
More informationCLASS XII PHYSICS. (a) 30 cm, 60 cm (b) 20 cm, 30 cm (c) 15 cm, 20 cm (d) 12 cm, 15 cm. where
PHYSICS combintion o two thin lenses with ocl lengths n respectively orms n imge o istnt object t istnce cm when lenses re in contct. The position o this imge shits by cm towrs the combintion when two
More informationMath 211A Homework. Edward Burkard. = tan (2x + z)
Mth A Homework Ewr Burkr Eercises 5-C Eercise 8 Show tht the utonomous system: 5 Plne Autonomous Systems = e sin 3y + sin cos + e z, y = sin ( + 3y, z = tn ( + z hs n unstble criticl point t = y = z =
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationELE B7 Power System Engineering. Unbalanced Fault Analysis
Power System Engineering Unblnced Fult Anlysis Anlysis of Unblnced Systems Except for the blnced three-phse fult, fults result in n unblnced system. The most common types of fults re single lineground
More informationJackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationUNIVERSITY OF CALIFORNIA, BERKELEY College of Engineering Department of Electrical Engineering and Computer Sciences
UNIVERSITY OF CALIFORNIA, BERKELEY College of Engineering Deprtment of Electricl Engineering nd Computer Sciences Eld Alon Homework #3 Solutions EECS4 PROBLEM : CMOS Logic ) Implement the logic function
More informationHints for Exercise 1 on: Current and Resistance
Hints for Exercise 1 on: Current nd Resistnce Review the concepts of: electric current, conventionl current flow direction, current density, crrier drift velocity, crrier numer density, Ohm s lw, electric
More informationBasic Derivative Properties
Bsic Derivtive Properties Let s strt this section by remining ourselves tht the erivtive is the slope of function Wht is the slope of constnt function? c FACT 2 Let f () =c, where c is constnt Then f 0
More information196 Circuit Analysis with PSpice: A Simplified Approach
196 Circuit Anlysis with PSpice: A Simplified Approch i, A v L t, min i SRC 5 μf v C FIGURE P7.3 () the energy stored in the inductor, nd (c) the instntneous power input to the inductor. (Dul of Prolem
More informationIntro to Nuclear and Particle Physics (5110)
Intro to Nucler nd Prticle Physics (5110) Feb, 009 The Nucler Mss Spectrum The Liquid Drop Model //009 1 E(MeV) n n(n-1)/ E/[ n(n-1)/] (MeV/pir) 1 C 16 O 0 Ne 4 Mg 7.7 14.44 19.17 8.48 4 5 6 6 10 15.4.41
More informationMAC-solutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationIMPORTANT. Read these directions carefully:
Physics 208: Electricity nd Mgnetism Finl Exm, Secs. 506 510. 7 My. 2004 Instructor: Dr. George R. Welch, 415 Engineering-Physics, 845-7737 Print your nme netly: Lst nme: First nme: Sign your nme: Plese
More informationLesson 8. Thermomechanical Measurements for Energy Systems (MENR) Measurements for Mechanical Systems and Production (MMER)
Lesson 8 Thermomechnicl Mesurements for Energy Systems (MEN) Mesurements for Mechnicl Systems nd Production (MME) A.Y. 205-6 Zccri (ino ) Del Prete Mesurement of Mechnicl STAIN Strin mesurements re perhps
More informationSection 20.1 Electric Charge and Static Electricity (pages )
Nme Clss Dte Section 20.1 Electric Chrge nd Sttic (pges 600 603) This section explins how electric chrge is creted nd how positive nd negtive chrges ffect ech other. It lso discusses the different wys
More information1.1 Functions. 0.1 Lines. 1.2 Linear Functions. 1.3 Rates of change. 0.2 Fractions. 0.3 Rules of exponents. 1.4 Applications of Functions to Economics
0.1 Lines Definition. Here re two forms of the eqution of line: y = mx + b y = m(x x 0 ) + y 0 ( m = slope, b = y-intercept, (x 0, y 0 ) = some given point ) slope-intercept point-slope There re two importnt
More informationQuantum Physics II (8.05) Fall 2013 Assignment 2
Quntum Physics II (8.05) Fll 2013 Assignment 2 Msschusetts Institute of Technology Physics Deprtment Due Fridy September 20, 2013 September 13, 2013 3:00 pm Suggested Reding Continued from lst week: 1.
More informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationMath 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
More informationName Solutions to Test 3 November 8, 2017
Nme Solutions to Test 3 November 8, 07 This test consists of three prts. Plese note tht in prts II nd III, you cn skip one question of those offered. Some possibly useful formuls cn be found below. Brrier
More informationMeasuring Electron Work Function in Metal
n experiment of the Electron topic Mesuring Electron Work Function in Metl Instructor: 梁生 Office: 7-318 Emil: shling@bjtu.edu.cn Purposes 1. To understnd the concept of electron work function in metl nd
More informationArea Under the Torque vs. RPM Curve: Average Power
Are Uner the orque vs. RM Curve: Averge ower Wht is torque? Some Bsics Consier wrench on nut, the torque bout the nut is Force, F F θ r rf sinθ orque, If F is t right ngle to moment rm r then rf How oes
More informationFlow in porous media
Red: Ch 2. nd 2.2 PART 4 Flow in porous medi Drcy s lw Imgine point (A) in column of wter (figure below); the point hs following chrcteristics: () elevtion z (2) pressure p (3) velocity v (4) density ρ
More information1. Weak acids. For a weak acid HA, there is less than 100% dissociation to ions. The B-L equilibrium is:
th 9 Homework: Reding, M&F, ch. 15, pp. 584-598, 602-605 (clcultions of ph, etc., for wek cids, wek bses, polyprotic cids, nd slts; fctors ffecting cid strength). Problems: Nkon, ch. 18, #1-10, 16-18,
More informationPhysics 712 Electricity and Magnetism Solutions to Final Exam, Spring 2016
Physics 7 Electricity nd Mgnetism Solutions to Finl Em, Spring 6 Plese note tht some possibly helpful formuls pper on the second pge The number of points on ech problem nd prt is mrked in squre brckets
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationMath 4310 Solutions to homework 1 Due 9/1/16
Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1
More informationragsdale (zdr82) HW2 ditmire (58335) 1
rgsdle (zdr82) HW2 ditmire (58335) This print-out should hve 22 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. 00 0.0 points A chrge of 8. µc
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationLecture 3: Curves in Calculus. Table of contents
Mth 348 Fll 7 Lecture 3: Curves in Clculus Disclimer. As we hve textook, this lecture note is for guidnce nd supplement only. It should not e relied on when prepring for exms. In this lecture we set up
More informationReference. Vector Analysis Chapter 2
Reference Vector nlsis Chpter Sttic Electric Fields (3 Weeks) Chpter 3.3 Coulomb s Lw Chpter 3.4 Guss s Lw nd pplictions Chpter 3.5 Electric Potentil Chpter 3.6 Mteril Medi in Sttic Electric Field Chpter
More information#6A&B Magnetic Field Mapping
#6A& Mgnetic Field Mpping Gol y performing this lb experiment, you will: 1. use mgnetic field mesurement technique bsed on Frdy s Lw (see the previous experiment),. study the mgnetic fields generted by
More informationData Structures and Algorithms CMPSC 465
Dt Structures nd Algorithms CMPSC 465 LECTURE 10 Solving recurrences Mster theorem Adm Smith S. Rskhodnikov nd A. Smith; bsed on slides by E. Demine nd C. Leiserson Review questions Guess the solution
More informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend
More information10/25/2005 Section 5_2 Conductors empty.doc 1/ Conductors. We have been studying the electrostatics of freespace (i.e., a vacuum).
10/25/2005 Section 5_2 Conductors empty.doc 1/3 5-2 Conductors Reding Assignment: pp. 122-132 We hve been studying the electrosttics of freespce (i.e., vcuum). But, the universe is full of stuff! Q: Does
More informationCHAPTER 08: MONOPROTIC ACID-BASE EQUILIBRIA
Hrris: Quntittive Chemicl Anlysis, Eight Edition CHAPTER 08: MONOPROTIC ACIDBASE EQUILIBRIA CHAPTER 08: Opener A CHAPTER 08: Opener B CHAPTER 08: Opener C CHAPTER 08: Opener D CHAPTER 08: Opener E Chpter
More information( ) 2. ( ) is the Fourier transform of! ( x). ( ) ( ) ( ) = Ae i kx"#t ( ) = 1 2" ( )"( x,t) PC 3101 Quantum Mechanics Section 1
1. 1D Schrödinger Eqution G chpters 3-4. 1.1 the Free Prticle V 0 "( x,t) i = 2 t 2m x,t = Ae i kxt "( x,t) x 2 where = k 2 2m. Normliztion must hppen: 2 x,t = 1 Here, however: " A 2 dx " " As this integrl
More informationDensity of Energy Stored in the Electric Field
Density of Energy Stored in the Electric Field Deprtment of Physics, Cornell University c Tomás A. Aris October 14, 01 Figure 1: Digrm of Crtesin vortices from René Descrtes Principi philosophie, published
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationOverview of Calculus
Overview of Clculus June 6, 2016 1 Limits Clculus begins with the notion of limit. In symbols, lim f(x) = L x c In wors, however close you emn tht the function f evlute t x, f(x), to be to the limit L
More informationDesigning Information Devices and Systems I Spring 2018 Homework 7
EECS 16A Designing Informtion Devices nd Systems I Spring 2018 omework 7 This homework is due Mrch 12, 2018, t 23:59. Self-grdes re due Mrch 15, 2018, t 23:59. Sumission Formt Your homework sumission should
More informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
More information