# Module 4: General Formulation of Electric Circuit Theory

Size: px
Start display at page:

## Transcription

1 Mdule 4: General Frmulatin f Electric Circuit Thery

2 4. General Frmulatin f Electric Circuit Thery All electrmagnetic phenmena are described at a fundamental level by Maxwell's equatins and the assciated auxiliary relatinships. Fr certain classes f prblems, such as representing the behavir f electric circuits driven at lw frequency, applicatin f these relatinships may be cumbersme. As a result, apprximate techniques fr the analysis f lw-frequency circuits have been develped. These specializatins are used t describe the "ideal" behavir f cmmn circuit elements such as wires, resistrs, capacitrs, and inductrs. Hwever, when devices are perated in a regime, r an envirnment, which lies utside the range f validity f such apprximatins, a mre fundamental descriptin f electrical systems is required. When viewed in this mre general cntext, what may initially appear t be unexpected behavir f a circuit element ften reveals itself t be nrmal peratin under a mre cmplex set f rules. An understanding f this lies at the cre f electrmagnetically cmpatible designs. In this sectin, electric circuit thery will be presented in a general frm, and the relatinship between circuit thery and electrmagnetic principles will be examined. The apprximatins assciated with circuit thery and a discussin f the range f validity f these apprximatins will be included. It will be seen that effects due t radiatin and inductin are always present in systems immersed in time-varying fields, althugh under certain cnditins these effects may be ignred. 4. Limitatins f Kirchff's laws The behavir f electric circuits is typically described thrugh Kirchff's vltage and current laws. Kirchff's vltage law states that the sum f the vltages arund any clsed circuit path is zer M n V n 0 and Kirchff's current law states that the sum f the currents flwing ut f a circuit nde is zer N M n I n 0 It is thrugh applicatin f these relatinships that mst descriptins f electric circuits prceed. Hwever, bth f these relatinships are nly valid under certain cnditins: - The structures under cnsideratin must be electrically small. At 60 Hz, the wavelength f a wave prpagating thrugh free space is 5 millin meters, while at 300 MHz a wavelength in free space is m lng. Radiatin and inductin effects arise when the current amplitude and phase vary at pints alng the cnductr. - N variatin exists alng uninterrupted cnductrs. 4-

3 - N delay time exists between surces and the rest f the circuit. Als all cnductrs are equiptential surfaces. - The lss f energy frm the circuit, ther than dissipatin, is neglected. In reality, lsses due t radiatin may becme significant at high frequency. In chapter, a time-dependent generalizatin f KVL was presented v(t) R surce R i(t) L di(t) dt. Althugh this expressin is valid fr time-changing fields, it is assumed that the circuit elements are lumped, i.e, the resistance and inductance are cncentrated in relatively small regins. This assumptin begins t break dwn at frequencies where the circuit elements are a significant fractin f a wavelength lng. In this regime, circuits must be described in terms f distributed parameters. Every part f the circuit has a certain impedance per unit length assciated with it. This impedance may be bth real (resistive) and imaginary (reactive). Als, interactins which ccur in ne part f the circuit may affect interactins which ccur everywhere else in the circuit. In additin, the presence f ther external circuits will affect interactins within a different circuit. Thus at high frequency, a circuit must be viewed as a single entity, nt a cllectin f individual cmpnents, and multiple circuits must be viewed as cmpsing a single, cupled system. 4. General frmulatin fr a single RLC circuit The general frmulatin f electric circuit thery will begin with an analysis f a single circuit cnstructed with a cnducting wire f radius a that may include a cil (inductr), a capacitr, and a resistr. It will be assumed that the radius f the wire is much smaller than a wavelength at the frequency f peratin a/ << r a << where Œ/. A current flws arund the circuit. The tangential cmpnent f electric current 4-3

4 J s ŝ 3J ŝ( 3E) E s is driven by the tangential cmpnent f electric field E s alng the wire, which is prduced by charge and current in the circuit. Currents in the circuit are supprted by a generatr. The generatr is a surce regin where a nn-cnservative (meaning that the ptential rises in the directin f current flw) impressed electric field 3E e maintains a charge separatin. This impressed field is due t an electrchemical r ther type f frce, and is assumed t be independent f current and charge in the circuit. The charge separatin supprts an electric field 3E (Culmb field) within and external t the surce regin which gives rise t the current flwing in the circuit. Figure. Generalized electric circuit. In the regins external t the surce, the impressed field 3E e des nt exist, hwever within the surce bth fields exist, therefre by Ohm's law the current density present at any pint in the circuit is given by 3J ( 3E 3E e ) where varies frm pint t pint. Frm this it is apparent that in rder t drive the current density 3J against the electric field 3E which ppses the charge separatin in the surce regin, the impressed electric field must be such that 3E e > 3E. 4-4

5 Psitins alng the circuit are measured using a displacement variable s, having it's rigin at the center f the generatr. The unit vectr parallel t the wire axis at any pint alng the circuit is ŝ. The tangential cmpnent f electric field alng the cnductr is therefre E s ŝ 3E and the assciated axial cmpnent f current density is J s ŝ 3J ŝ( 3E) E s. The ttal current flwing thrugh the cnductr crss-sectin is then I s Pc.s. J s ds. bundary cnditins at the surface f the wire circuit Accrding t the bundary cnditin ˆ t ( 3E 3E ) 0 the tangential cmpnent f electric field is cntinuus acrss an interface between materials. Applicatin f this bundary cnditin at the surface f the wire cnductr leads t E s (r a ) E s (r a ), where E inside s (s) E s (ra ) is the field just inside the cnductr at psitin s alng the circuit, and E utside s (s) E s (ra ) represents the field maintained at psitin s just utside the surface f the cnductr by the current and charge in the circuit. Therefre the fundamental bundary cnditin emplyed in an 4-5

6 electrmagnetic descriptin f circuit thery is E inside s (ra ) E utside s (ra ). determinatin f E inside s (s) In rder t apply the bundary cnditin abve, the tangential electric field that exits at pints just inside the surface f the circuit must be determined. This is nt an easy task, because the impedance may differ in the varius regins f the circuit. At any pint alng the cnducting wire, including cils and resistrs, the electric field is in general E inside s (s) I s (s) where E inside s (s) I s (s) is the internal impedance per unit length f the regin. - surce regin In the surce regin, bth the impressed and induced electric fields exist, therefre the current density is J s e (E s E e s ) where e is the cnductivity f the material in the surce regin, E s is the tangential cmpnent f electric field in the surce regin maintained by charge and current in the circuit, and apparent that E e s is the tangential cmpnent f impressed electric field. Frm this, it is 4-6

7 E s J s E e e s J s S e E e e S e S I s E e e S e s where S e is the crss-sectinal area f the surce regin. In an ideal generatr, the material in the surce regin is perfectly cnducting ( e ), and has zer internal impedance. Thus E s E e s in a gd surce generatr. In general, in the surce regin E s (s) e I s (s) E e s (s) where e e S e is the internal impedance per unit length f the surce regin. - capacitr The tangential cmpnent f electric field at the edge f the capacitr flwing t the capacitr lie in the same directin. Therefre, within the capacitr E s and the current E s (s) c I s (s) where c is the internal impedance per unit thickness f the dielectric material cntained in the capacitr. The ttal ptential difference acrss the capacitr is the line integral f the nrmal cmpnent f electric field which exists between the capacitr plates r B A A V AB V B V a E P s ds E P s ds P A A V AB I s P B B c ds B c I s (s)ds if it is assumed that a cnstant current I s flws t the capacitr. 4-7

8 Figure. Capacitr. The time-harmnic cntinuity equatin states / 3J j!. Vlume integratin f bth sides f this expressin, and applicatin f the divergence therem yields Q S ( ˆn 3J )ds j P V!dv resulting in I s jq where Q is the ttal charge cntained n the psitive capacitr plate. The ttal ptential difference between the capacitr plates is then V AB jq P B A c ds 4-8

9 Figure 3. Single capacitr plate. but by the definitin f capacitance C Q V AB therefre B Q C jq P A c ds. Frm this cmes the expected expressin fr the impedance f a capacitr A P B c ds jc jx c. - arbitrary pint alng the surface f the circuit By cmbining the results fr the three cases abve, a general expressin fr the tangential cmpnent f electric field residing just inside the surface f the cnductr at any pint alng the circuit is determined 4-9

10 E inside s (s) (s)i s (s) E e s (s) where (s) is the internal impedance per unit length which is different fr the varius cmpnents f the circuit, and E e s (s) is the impressed electric field which is zer everywhere utside f the surce regin. determinatin f E utside s (s) In Chapter it was shwn that an electric field may be represented in terms f scalar and vectr ptentials. Thus at any pint in space utside the electric circuit the electric field is 3E /- j 3A where - is the scalar ptential maintained at the surface f the circuit by the charge present in the circuit, and 3A is the vectr ptential maintained at the surface f the circuit by the current flwing in the circuit. The well knwn Lrentz cnditin states that / 3A jk - 0 where k µ0 j 0. In the free space utside the circuit µ µ, 0, and 0 thus 0 k µ 0. Applying this t the Lrentz cnditin yields - j / 3A which, upn substitutin int the expressin fr electric field utside the circuit gives 4-0

11 3E j /(/ 3A) 3 A. The cmpnent f electric field tangent t the surface f the circuit is then given by E utside s (s) ŝ 3E ŝ/- j(ŝ 3A) j ŝ /(/ 3A) 3 A r E utside s (s) 0-0s ja s j 0 0s (/ 3A) A s. satisfactin f the fundamental bundary cnditin The bundary cnditin at the surface f the circuit states that the tangential cmpnent f electric field must be cntinuus, r E inside s (s) E utside (s) s therefre the basic equatin fr circuit thery is E e s (s) (s)i s (s) 0 0s -(s) ja s (s) j 0 0s / 3A(s) A s (s). pen and clsed circuit expressins Frm the develpment abve, the expressin fr the impressed electric field is E e s (s) 0 0s -(s) ja s (s) (s)i s (s). 4-

12 Figure 4. General circuit. Integrating alng a path C n the inner surface f the circuit frm a pint s t a pint s, which represent the ends f an pen circuit, gives s E e P s s s s S (s)ds 0 P 0s -(s)ds A j P s (s)ds (s)i P s (s)ds s s s but, because the impressed electric field exists nly in the surce regin s E e P s s (s)ds P B E e A s (s)ds where is the driving vltage. Nw it can be seen that s P 0s -(s)ds d- -(s P ) -(s ) s s 0 s and thus the equatin fr an pen circuit can be expressed 4-

13 s s 0 -(S ) -(s ) A j P s (s)ds (s)i P s (s)ds. s s If the circuit is clsed, then s =s and -(s )--(s )=0. In this case, the circuit equatin becmes 0 QC (s)i s (s)ds jq C A s (s)ds. 4.3 General equatins fr cupled circuits Nw the cncepts develped abve are extended t the case f tw cupled circuits, each cntaining a generatr, a resistr, a cil (inductr), and a capacitr. Circuit will be referred t as the primary circuit, and circuit will be referred t as the secndary circuit. This case is represented by a pair f cupled general circuit equatins 0 Q C (s )I s (s )ds j Q C 3A (s ) 3A (s ) 3 ds 0 Q C (s )I s (s )ds j Q C 3A (s ) 3A (s ) 3 ds. Here 3A and 3A are the magnetic vectr ptentials at the surface f the primary circuit maintained by the currents and in the primary and secndary circuits, given by I s I s and 3A (s ) µ C I s (s ) e j R R (s, s ds ) 3 3A (s ) µ C I s (s ) e j R R (s, s ds ) 3 and 3A and 3A are the vectr ptentials at the surface f the secndary circuit maintained by the currents and in the primary and secndary circuits, given by I s I s 4-3

14 Figure 5. Generalized cupled circuits. 3A (s ) µ C I s (s ) e j R R (s, s ds ) 3 and 3A (s ) µ C I s (s ) e j R R (s, s ). Substituting these expressins fr the varius magnetic vectr ptentials int the general circuit equatins abve leads t 0 (s )I s (s )ds jµ C C C I s (s ) e j R R (s ds ) 3 jµ C C I s (s ) e j R R (s ds ) 3 4-4

15 0 (s )I s (s )ds jµ C C C I s (s ) e j R R (s ds ) 3 jµ C C I s (s ) e j R R (s ). Nte that C and C, lie alng the inner periphery f the circuit while C and C lie alng the centerline. When the circuit dimensins and 0, 0 are specified, the equatins abve becme a pair f cupled simultaneus integral equatins fr the unknwn currents I s (s ) and I s (s ) in the primary and secndary circuits. These equatins are in general t cmplicated t be slved exactly. self and mutual impedances f electric circuits Reference currents I 0 and I 0 are chsen at the lcatins f the generatrs in the primary and secndary circuits, i.e., I 0 I s (s 0) at the center f the primary circuit generatr, and I 0 I s (s 0) at the center f the secndary circuit generatr. Nw let the currents be represented by I s (s ) I 0 f (s ) I s (s ) I 0 f (s ) where f (0) f (0).0, and f, f are cmplex distributin functins. The general circuit equatins frmulated abve may be expressed in terms f the reference currents as 0 I 0 Z I 0 Z 0 I 0 Z I 0 Z 4-5

16 where Z = self-impedance f the primary circuit referenced t I 0 Z = self-impedance f the secndary circuit referenced t I 0 Z = mutual-impedance f the primary circuit referenced t I 0 Z = mutual-impedance f the secndary circuit referenced t I 0 and Z Z i Z e Z Z i Z e. Here Z i is referred t as the internal self-impedance f the primary and secndary circuits. This term depends primarily upn the internal impedance per unit length f the cnductrs present in the circuits, and includes effects due t capacitance and resistance. Z e is referred t as the external self-impedance f the primary and secndary circuits. This term depends entirely upn the interactin between currents in varius parts f the circuit, and includes effects due t inductance. The varius impedance terms are expressed as Z i Q C (s ) f (s ) ds Z i Q C (s ) f (s ) ds Z e jµ C C f (s ) e j R R (s ds ) 3 Z e jµ C C f (s ) e j R R (s ds ) 3 4-6

17 Z jµ C C f (s ) e j R R (s ds ) 3 Z jµ C C f (s ) e j R R (s ). It is nted that all f the circuit impedances depend in general n the current distributin functins f (s ) and f (s ). driving pint impedance, cupling cefficient, and induced vltage Cnsider the case where nly the primary circuit is driven by a generatr excitatin, i.e., 0 0 In this case the general circuit equatins becme 0 I 0 Z I 0 Z 0 I 0 Z I 0 Z. This leads t a pair f equatins I 0 I 0 Z Z 0 I 0 Z Z Z Z which can be slved t yield 4-7

18 I 0 0 Z Z Z Z and I 0 0 Z Z Z Z which are the reference currents in the generatr regins. Frm these, it can be seen that the driving pint impedance f the primary circuit is Z in 0 I 0 Z Z Z Z Z Z Z Z Z. Fr the case f a lsely cupled electric circuit having Z Z Z Z «, Z and Z are bth negligibly small, and therefre (Z ) in Z. Fr this case the expressins 0 I 0 Z I 0 Z 0 I 0 Z I 0 Z. becme 0 V i I 0 Z V i I 0 Z where V i I 0 Z is the vltage induced in the primary circuit by the current in the secndary circuit, and V i I 0 Z is the vltage induced in the secndary circuit by the current in the primary circuit. Because the circuits are cnsidered t be lsely cupled 0» V i 4-8

19 and therefre the general circuit equatins becme 0 x I 0 Z V i I 0 Z I 0 Z. near zne electric circuit Fr cases in which a circuit is perated such that the circuit dimensins are much less than a wavelength, then the phase f the current des nt change appreciably as it travels arund the circuit. The assciated EM fields are such that the circuit is cnfined t the near- r inductinznes. Mst electric circuits used at pwer and lw radi frequencies may be mdeled this way. In this type f cnventinal circuit R «, R «, R «, R «and therefre the fllwing apprximatin may be made e j R ij R ij! 4 R 4 ij 4!... j R ij R ij 3! 4 R 4 ij... x 5! fr i=, and j=,. Because the circuit dimensins are small cmpared t a wavelength f (s ) x f (s ) x and thus f i (s i ) e j R ij (s i j ) R ij (s i j ) x R ij (s i j ) fr i=, and j=,. Substituting the apprximatins stated abve int the expressins fr the varius circuit impedances leads t Z i Q C (s ) ds Z i Q C (s ) ds 4-9

20 Z e jx e jµ C C R (s ) Z e jx e jµ C C R (s ) Z jx jµ C C R (s ) Z jx jµ C C R (s. ) Since the integrals abve are frequency independent and functins nly f the gemetry f the circuit, the self and mutual inductances are defined as fllws: L e, where X e L e µ C C R (s ) is the external self-inductance f the primary circuit; L e, where X e L e µ C C R (s ) is the external self-inductance f the secndary circuit; 4-0

21 X L, where L µ C C 3 ds R (s ) is the mutual inductance between the primary and secndary circuit; X L, where L µ C C 3 ds R (s ) is the mutual inductance between the primary and secndary circuit. It is seen by inspectin that L L. radiating electric circuit At high, ultrahigh, and lw micrwave frequencies, the assumptins fr cnventinal nearzne electric circuits are nt valid since is nt usually satisfied. In a quasi-cnventinal R «r radiating circuit, the less restrictive dimensinal requirement f R «is assumed t be valid fr the frequencies f interest, therefre R «, R «, R «, R «. In this case e! j " R ij R ij! 4 R 4 ij 4! x j R ij R ij 3!... j R ij R ij 3! 4 R 4 ij... 5! fr i=, and j=,. In the apprximatin made abve, the higher rder term retained. Fr a quasi-cnventinal circuit R ij 6 is 4-

22 f (s ) x f (s ) x is bserved experimentally t remain apprximately valid. Thus f i (s i ) e j\$ R ij (s i % j ) R ij (s i j ) x R ij (s i j ) j R ij (s i j ) 6 fr i=, and j=,. Frm this expressin it is apparent why the higher rder term R ij 6 in the expnential series was retained. The leading term in the imaginary part f the expnential series integrates t zer in the impedance expressins, i.e., Q j i j i 0 Q Ci Ci fr i=,. The impedance expressins fr the quasi-cnventinal circuit are thus given by Z e jµ C C % R (s ) j 6 C % R (s ) Z e jµ C C % R (s ) j 6 C % R (s ) Z jµ C C % R (s ) j 6 C % R (s ) Z jµ C C % R (s ) j 6 C % R (s ) and the expressins fr the internal impedances f the primary and secndary circuits remain unchanged. It is seen that the impedance expressins abve cnsist f bth real and imaginary parts 4-

23 Z e R e jx e Z e R e jx e and Z Z R jx. Nw, recalling that v p jµ 3 j 6 µ v p Œ therefre R e 4 5 C C ' R (s ( ) ( L e, where X e L e µ C C ' ( R (s ( ) R e 4 5 C C ' R (s ( ) ( X e L e, where 4-3

24 L e µ C C ) * R (s * ) 4 R 5 C C ) R (s * ) * and X L, where L µ C C ) * R (s *. ) It is nted that the inductances, and L fr the quasi-cnventinal circuit are the same as thse fr the cnventinal near-zne circuit. In the quasi-cnventinal circuit, hwever, impedances Z e, Z e and Z becme cmplex due t the presence f R e, R e and R. The resistive cmpnents R e and R e d nt represent dissipatin lsses in the circuit (dissipatin lsses are included in the internal impedance terms Z i and Z i ), but instead indicate a pwer lss frm the circuit due t radiatin f EM energy t space. R e and R e are therefre radiatin resistances which describe the pwer lss frm a quasi-cnventinal circuit due t EM radiatin. L e L e References. R.W.P. King, and S. Prasad, Fundamental Electrmagnetic Thery and Applicatins, Prentice- Hall, R.W.P. King, Fundamental Electrmagnetic Thery, Dver Publicatins,

### Chapter 30. Inductance

Chapter 30 nductance 30. Self-nductance Cnsider a lp f wire at rest. f we establish a current arund the lp, it will prduce a magnetic field. Sme f the magnetic field lines pass thrugh the lp. et! be the

### (1.1) V which contains charges. If a charge density ρ, is defined as the limit of the ratio of the charge contained. 0, and if a force density f

1.0 Review f Electrmagnetic Field Thery Selected aspects f electrmagnetic thery are reviewed in this sectin, with emphasis n cncepts which are useful in understanding magnet design. Detailed, rigrus treatments

### PHYS College Physics II Final Examination Review

PHYS 1402- Cllege Physics II Final Examinatin Review The final examinatin will be based n the fllwing Chapters/Sectins and will cnsist f tw parts. Part 1, cnsisting f Multiple Chice questins, will accunt

### Revision: August 19, E Main Suite D Pullman, WA (509) Voice and Fax

.7.4: Direct frequency dmain circuit analysis Revisin: August 9, 00 5 E Main Suite D Pullman, WA 9963 (509) 334 6306 ice and Fax Overview n chapter.7., we determined the steadystate respnse f electrical

### Electric Current and Resistance

Electric Current and Resistance Electric Current Electric current is the rate f flw f charge thrugh sme regin f space The SI unit f current is the ampere (A) 1 A = 1 C / s The symbl fr electric current

### Physics 2B Chapter 23 Notes - Faraday s Law & Inductors Spring 2018

Michael Faraday lived in the Lndn area frm 1791 t 1867. He was 29 years ld when Hand Oersted, in 1820, accidentally discvered that electric current creates magnetic field. Thrugh empirical bservatin and

### Schedule. Time Varying electromagnetic fields (1 Week) 6.1 Overview 6.2 Faraday s law (6.2.1 only) 6.3 Maxwell s equations

chedule Time Varying electrmagnetic fields (1 Week) 6.1 Overview 6.2 Faraday s law (6.2.1 nly) 6.3 Maxwell s equatins Wave quatin (3 Week) 6.5 Time-Harmnic fields 7.1 Overview 7.2 Plane Waves in Lssless

### Sections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic.

Tpic : AC Fundamentals, Sinusidal Wavefrm, and Phasrs Sectins 5. t 5., 6. and 6. f the textbk (Rbbins-Miller) cver the materials required fr this tpic.. Wavefrms in electrical systems are current r vltage

### Supplementary Course Notes Adding and Subtracting AC Voltages and Currents

Supplementary Curse Ntes Adding and Subtracting AC Vltages and Currents As mentined previusly, when cmbining DC vltages r currents, we nly need t knw the plarity (vltage) and directin (current). In the

### Honors Physics Final Review Summary

Hnrs Physics Final Review Summary Wrk Dne By A Cnstant Frce: Wrk describes a frce s tendency t change the speed f an bject. Wrk is dne nly when an bject mves in respnse t a frce, and a cmpnent f the frce

### Q1. In figure 1, Q = 60 µc, q = 20 µc, a = 3.0 m, and b = 4.0 m. Calculate the total electric force on q due to the other 2 charges.

Phys10 Secnd Majr-08 Zer Versin Crdinatr: Dr. I. M. Nasser Saturday, May 3, 009 Page: 1 Q1. In figure 1, Q = 60 µc, q = 0 µc, a = 3.0 m, and b = 4.0 m. Calculate the ttal electric frce n q due t the ther

### Surface and Contact Stress

Surface and Cntact Stress The cncept f the frce is fundamental t mechanics and many imprtant prblems can be cast in terms f frces nly, fr example the prblems cnsidered in Chapter. Hwever, mre sphisticated

### Supplementary Course Notes Adding and Subtracting AC Voltages and Currents

Supplementary Curse Ntes Adding and Subtracting AC Vltages and Currents As mentined previusly, when cmbining DC vltages r currents, we nly need t knw the plarity (vltage) and directin (current). In the

### Q1. A) 48 m/s B) 17 m/s C) 22 m/s D) 66 m/s E) 53 m/s. Ans: = 84.0 Q2.

Phys10 Final-133 Zer Versin Crdinatr: A.A.Naqvi Wednesday, August 13, 014 Page: 1 Q1. A string, f length 0.75 m and fixed at bth ends, is vibrating in its fundamental mde. The maximum transverse speed

### Chapter VII Electrodynamics

Chapter VII Electrdynamics Recmmended prblems: 7.1, 7., 7.4, 7.5, 7.7, 7.8, 7.10, 7.11, 7.1, 7.13, 7.15, 7.17, 7.18, 7.0, 7.1, 7., 7.5, 7.6, 7.7, 7.9, 7.31, 7.38, 7.40, 7.45, 7.50.. Ohm s Law T make a

### FIELD QUALITY IN ACCELERATOR MAGNETS

FIELD QUALITY IN ACCELERATOR MAGNETS S. Russenschuck CERN, 1211 Geneva 23, Switzerland Abstract The field quality in the supercnducting magnets is expressed in terms f the cefficients f the Furier series

### Q1. A string of length L is fixed at both ends. Which one of the following is NOT a possible wavelength for standing waves on this string?

Term: 111 Thursday, January 05, 2012 Page: 1 Q1. A string f length L is fixed at bth ends. Which ne f the fllwing is NOT a pssible wavelength fr standing waves n this string? Q2. λ n = 2L n = A) 4L B)

### February 28, 2013 COMMENTS ON DIFFUSION, DIFFUSIVITY AND DERIVATION OF HYPERBOLIC EQUATIONS DESCRIBING THE DIFFUSION PHENOMENA

February 28, 2013 COMMENTS ON DIFFUSION, DIFFUSIVITY AND DERIVATION OF HYPERBOLIC EQUATIONS DESCRIBING THE DIFFUSION PHENOMENA Mental Experiment regarding 1D randm walk Cnsider a cntainer f gas in thermal

### 37 Maxwell s Equations

37 Maxwell s quatins In this chapter, the plan is t summarize much f what we knw abut electricity and magnetism in a manner similar t the way in which James Clerk Maxwell summarized what was knwn abut

### ( ) ( ) ( ) ( ) ( z) ( )

EE433-08 Planer Micrwave Circuit Design Ntes Returning t the incremental sectin, we will nw slve fr V and I using circuit laws. We will assume time-harmnic excitatin. v( z,t ) = v(z)cs( ωt ) jωt { s }

### GAUSS' LAW E. A. surface

Prf. Dr. I. M. A. Nasser GAUSS' LAW 08.11.017 GAUSS' LAW Intrductin: The electric field f a given charge distributin can in principle be calculated using Culmb's law. The examples discussed in electric

### Chapter 2 GAUSS LAW Recommended Problems:

Chapter GAUSS LAW Recmmended Prblems: 1,4,5,6,7,9,11,13,15,18,19,1,7,9,31,35,37,39,41,43,45,47,49,51,55,57,61,6,69. LCTRIC FLUX lectric flux is a measure f the number f electric filed lines penetrating

### ECE 546 Lecture 02 Review of Electromagnetics

C 546 Lecture 0 Review f lectrmagnetics Spring 018 Jse. Schutt-Aine lectrical & Cmputer ngineering University f Illinis jesa@illinis.edu C 546 Jse Schutt Aine 1 Printed Circuit Bard C 546 Jse Schutt Aine

### PHY 2054C Review guide Fall 2018 Chapter 17 Wave optics

PHY 2054C Review guide Fall 2018 Chapter 17 Wave ptics Light acts as a wave, ray, particle, and phtn. Refractive index n = c/v Light waves travel with speed c in a vacuum they slw dwn when they pass thrugh

### Chapter 6. Dielectrics and Capacitance

Chapter 6. Dielectrics and Capacitance Hayt; //009; 6- Dielectrics are insulating materials with n free charges. All charges are bund at mlecules by Culmb frce. An applied electric field displaces charges

### 22.54 Neutron Interactions and Applications (Spring 2004) Chapter 11 (3/11/04) Neutron Diffusion

.54 Neutrn Interactins and Applicatins (Spring 004) Chapter (3//04) Neutrn Diffusin References -- J. R. Lamarsh, Intrductin t Nuclear Reactr Thery (Addisn-Wesley, Reading, 966) T study neutrn diffusin

### Coupled Inductors and Transformers

Cupled nductrs and Transfrmers Self-nductance When current i flws thrugh the cil, a magnetic flux is prduced arund it. d d di di v= = = dt di dt dt nductance: = d di This inductance is cmmnly called self-inductance,

### Phys102 Final-061 Zero Version Coordinator: Nasser Wednesday, January 24, 2007 Page: 1

Crdinatr: Nasser Wednesday, January 4, 007 Page: 1 Q1. Tw transmitters, S 1 and S shwn in the figure, emit identical sund waves f wavelength λ. The transmitters are separated by a distance λ /. Cnsider

### AQA GCSE Physics. Topic 7: Magnetism and Electromagnetism. Notes. (Content in bold is for Higher Tier only)

AQA GCSE Physics Tpic 7: Magnetism and Electrmagnetism Ntes (Cntent in bld is fr Higher Tier nly) Magnets - Nrth and Suth Ples - Same Ples repel - Oppsite ples attract Permanent Magnets - Always magnetic,

### Lab 11 LRC Circuits, Damped Forced Harmonic Motion

Physics 6 ab ab 11 ircuits, Damped Frced Harmnic Mtin What Yu Need T Knw: The Physics OK this is basically a recap f what yu ve dne s far with circuits and circuits. Nw we get t put everything tgether

### 4) What is the magnitude of the net electric field at the center of the square?

Fur charges are n the fur crners f a square. Q = +5C, Q = -0C, Q 3 = +5C, Q 4 = -0C. The side length f each side f the square is 3 m. Q Q ) What is the directin f the frce n Q due t ONLY Q 4? (a) up (b)

### Interference is when two (or more) sets of waves meet and combine to produce a new pattern.

Interference Interference is when tw (r mre) sets f waves meet and cmbine t prduce a new pattern. This pattern can vary depending n the riginal wave directin, wavelength, amplitude, etc. The tw mst extreme

### Synchronous Motor V-Curves

Synchrnus Mtr V-Curves 1 Synchrnus Mtr V-Curves Intrductin Synchrnus mtrs are used in applicatins such as textile mills where cnstant speed peratin is critical. Mst small synchrnus mtrs cntain squirrel

### Computational modeling techniques

Cmputatinal mdeling techniques Lecture 4: Mdel checing fr ODE mdels In Petre Department f IT, Åb Aademi http://www.users.ab.fi/ipetre/cmpmd/ Cntent Stichimetric matrix Calculating the mass cnservatin relatins

### NGSS High School Physics Domain Model

NGSS High Schl Physics Dmain Mdel Mtin and Stability: Frces and Interactins HS-PS2-1: Students will be able t analyze data t supprt the claim that Newtn s secnd law f mtin describes the mathematical relatinship

### ECE 2100 Circuit Analysis

ECE 2100 Circuit Analysis Lessn 25 Chapter 9 & App B: Passive circuit elements in the phasr representatin Daniel M. Litynski, Ph.D. http://hmepages.wmich.edu/~dlitynsk/ ECE 2100 Circuit Analysis Lessn

### Lecture 02 CSE 40547/60547 Computing at the Nanoscale

PN Junctin Ntes: Lecture 02 CSE 40547/60547 Cmputing at the Nanscale Letʼs start with a (very) shrt review f semi-cnducting materials: - N-type material: Obtained by adding impurity with 5 valence elements

### ENGI 4430 Parametric Vector Functions Page 2-01

ENGI 4430 Parametric Vectr Functins Page -01. Parametric Vectr Functins (cntinued) Any nn-zer vectr r can be decmpsed int its magnitude r and its directin: r rrˆ, where r r 0 Tangent Vectr: dx dy dz dr

### Edexcel GCSE Physics

Edexcel GCSE Physics Tpic 10: Electricity and circuits Ntes (Cntent in bld is fr Higher Tier nly) www.pmt.educatin The Structure f the Atm Psitively charged nucleus surrunded by negatively charged electrns

### Applying Kirchoff s law on the primary circuit. V = - e1 V+ e1 = 0 V.D. e.m.f. From the secondary circuit e2 = v2. K e. Equivalent circuit :

TRANSFORMERS Definitin : Transfrmers can be defined as a static electric machine which cnverts electric energy frm ne ptential t anther at the same frequency. It can als be defined as cnsists f tw electric

### 1. Transformer A transformer is used to obtain the approximate output voltage of the power supply. The output of the transformer is still AC.

PHYSIS 536 Experiment 4: D Pwer Supply I. Intrductin The prcess f changing A t D is investigated in this experiment. An integrated circuit regulatr makes it easy t cnstruct a high-perfrmance vltage surce

### Three charges, all with a charge of 10 C are situated as shown (each grid line is separated by 1 meter).

Three charges, all with a charge f 0 are situated as shwn (each grid line is separated by meter). ) What is the net wrk needed t assemble this charge distributin? a) +0.5 J b) +0.8 J c) 0 J d) -0.8 J e)

### Information for Physics 1201 Midterm I Wednesday, February 20

My lecture slides are psted at http://www.physics.hi-state.edu/~humanic/ Infrmatin fr Physics 1201 Midterm I Wednesday, February 20 1) Frmat: 10 multiple chice questins (each wrth 5 pints) and tw shw-wrk

### ELECTROSTATIC FIELDS IN MATERIAL MEDIA

MF LCTROSTATIC FILDS IN MATRIAL MDIA 3/4/07 LCTURS Materials media may be classified in terms f their cnductivity σ (S/m) as: Cnductrs The cnductivity usually depends n temperature and frequency A material

### TOPPER SAMPLE PAPER 2 Class XII- Physics

TOPPER SAMPLE PAPER 2 Class XII- Physics Time: Three Hurs Maximum Marks: 70 General Instructins (a) All questins are cmpulsry. (b) There are 30 questins in ttal. Questins 1 t 8 carry ne mark each, questins

### Chapter 16. Capacitance. Capacitance, cont. Parallel-Plate Capacitor, Example 1/20/2011. Electric Energy and Capacitance

summary C = ε A / d = πε L / ln( b / a ) ab C = 4πε 4πε a b a b >> a Chapter 16 Electric Energy and Capacitance Capacitance Q=CV Parallel plates, caxial cables, Earth Series and parallel 1 1 1 = + +..

### Chapter 32. Maxwell s Equations and Electromagnetic Waves

Chapter 32 Maxwell s Equatins and Electrmagnetic Waves Maxwell s Equatins and EM Waves Maxwell s Displacement Current Maxwell s Equatins The EM Wave Equatin Electrmagnetic Radiatin MFMcGraw-PHY 2426 Chap32-Maxwell's

### CBSE Board Class XII Physics Set 1 Board Paper 2008 (Solution)

CBSE Bard Class XII Physics Set 1 Bard Paper 2008 (Slutin) 1. The frce is given by F qv B This frce is at right angles t &. 2. Micrwaves. It is used in radar & cmmunicatin purpses. 3. Or As m e e m S,

### Phys102 Second Major-102 Zero Version Coordinator: Al-Shukri Thursday, May 05, 2011 Page: 1

Crdinatr: Al-Shukri Thursday, May 05, 2011 Page: 1 1. Particles A and B are electrically neutral and are separated by 5.0 μm. If 5.0 x 10 6 electrns are transferred frm particle A t particle B, the magnitude

### W V. (d) W. (3) Which one is used to determine the internal resistance of a cell

[CHAPT-13 CUNT LCTICITY] www.prfaminz.cm MULTIPL CHOIC QUSTIONS (1) In carbn resistr the gld band indicates tlerance f (a) 5% (b) % 0% (d) 10% () The wrk dne t mve a psitive charge frm ne pint t anther

### I. Analytical Potential and Field of a Uniform Rod. V E d. The definition of electric potential difference is

Length L>>a,b,c Phys 232 Lab 4 Ch 17 Electric Ptential Difference Materials: whitebards & pens, cmputers with VPythn, pwer supply & cables, multimeter, crkbard, thumbtacks, individual prbes and jined prbes,

### Chapter 23 Electromagnetic Waves Lecture 14

Chapter 23 Electrmagnetic Waves Lecture 14 23.1 The Discvery f Electrmagnetic Waves 23.2 Prperties f Electrmagnetic Waves 23.3 Electrmagnetic Waves Carry Energy and Mmentum 23.4 Types f Electrmagnetic

### 4F-5 : Performance of an Ideal Gas Cycle 10 pts

4F-5 : Perfrmance f an Cycle 0 pts An ideal gas, initially at 0 C and 00 kpa, underges an internally reversible, cyclic prcess in a clsed system. The gas is first cmpressed adiabatically t 500 kpa, then

### Time varying fields and Maxwell's equations Chapter 9

Tie varying fields and Maxwell's equatins hapter 9 Dr. Naser Abu-Zaid Page 9/7/202 FARADAY LAW OF ELETROMAGNETI INDUTION A tie varying agnetic field prduces (induces) a current in a clsed lp f wire. The

### Kinetics of Particles. Chapter 3

Kinetics f Particles Chapter 3 1 Kinetics f Particles It is the study f the relatins existing between the frces acting n bdy, the mass f the bdy, and the mtin f the bdy. It is the study f the relatin between

### ECE 2100 Circuit Analysis

ECE 00 Circuit Analysis Lessn 6 Chapter 4 Sec 4., 4.5, 4.7 Series LC Circuit C Lw Pass Filter Daniel M. Litynski, Ph.D. http://hmepages.wmich.edu/~dlitynsk/ ECE 00 Circuit Analysis Lessn 5 Chapter 9 &

### BASIC DIRECT-CURRENT MEASUREMENTS

Brwn University Physics 0040 Intrductin BASIC DIRECT-CURRENT MEASUREMENTS The measurements described here illustrate the peratin f resistrs and capacitrs in electric circuits, and the use f sme standard

### A Novel Isolated Buck-Boost Converter

vel slated uck-st Cnverter S-Sek Kim *,WOO-J JG,JOOG-HO SOG, Ok-K Kang, and Hee-Jn Kim ept. f Electrical Eng., Seul atinal University f Technlgy, Krea Schl f Electrical and Cmputer Eng., Hanyang University,

### Study Group Report: Plate-fin Heat Exchangers: AEA Technology

Study Grup Reprt: Plate-fin Heat Exchangers: AEA Technlgy The prblem under study cncerned the apparent discrepancy between a series f experiments using a plate fin heat exchanger and the classical thery

### Plan o o. I(t) Divide problem into sub-problems Modify schematic and coordinate system (if needed) Write general equations

STAPLE Physics 201 Name Final Exam May 14, 2013 This is a clsed bk examinatin but during the exam yu may refer t a 5 x7 nte card with wrds f wisdm yu have written n it. There is extra scratch paper available.

### Kinematic transformation of mechanical behavior Neville Hogan

inematic transfrmatin f mechanical behavir Neville Hgan Generalized crdinates are fundamental If we assume that a linkage may accurately be described as a cllectin f linked rigid bdies, their generalized

### Building to Transformations on Coordinate Axis Grade 5: Geometry Graph points on the coordinate plane to solve real-world and mathematical problems.

Building t Transfrmatins n Crdinate Axis Grade 5: Gemetry Graph pints n the crdinate plane t slve real-wrld and mathematical prblems. 5.G.1. Use a pair f perpendicular number lines, called axes, t define

### JIRI GALAS Czech Technical University, Engineering, Prague.

Magnetic Separatin News, Vl. 2, pp. 119-136 Reprints available directly frm the publisher Phtcpying permitted by license nly 1988 Grdn and Breach, Science Publishers, Inc. Printed in the United Kingdm

### Bootstrap Method > # Purpose: understand how bootstrap method works > obs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(obs) >

Btstrap Methd > # Purpse: understand hw btstrap methd wrks > bs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(bs) > mean(bs) [1] 21.64625 > # estimate f lambda > lambda = 1/mean(bs);

### Examiner: Dr. Mohamed Elsharnoby Time: 180 min. Attempt all the following questions Solve the following five questions, and assume any missing data

Benha University Cllege f Engineering at Banha Department f Mechanical Eng. First Year Mechanical Subject : Fluid Mechanics M111 Date:4/5/016 Questins Fr Final Crrective Examinatin Examiner: Dr. Mhamed

### 11. DUAL NATURE OF RADIATION AND MATTER

11. DUAL NATURE OF RADIATION AND MATTER Very shrt answer and shrt answer questins 1. Define wrk functin f a metal? The minimum energy required fr an electrn t escape frm the metal surface is called the

### Chapter 3. AC Machinery Fundamentals. Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 3 AC Machinery Fundamentals 1 The Vltage Induced in a Rtating Lp e v B ind v = velcity f the cnductr B = Magnetic Flux Density vectr l = Length f the Cnductr Figure 3-1 A simple rtating lp in a

### Series and Parallel Resonances

Series and Parallel esnances Series esnance Cnsider the series circuit shwn in the frequency dmain. The input impedance is Z Vs jl jl I jc C H s esnance ccurs when the imaginary part f the transfer functin

### Chem 115 POGIL Worksheet - Week 8 Thermochemistry (Continued), Electromagnetic Radiation, and Line Spectra

Chem 115 POGIL Wrksheet - Week 8 Thermchemistry (Cntinued), Electrmagnetic Radiatin, and Line Spectra Why? As we saw last week, enthalpy and internal energy are state functins, which means that the sum

### Introduction to Smith Charts

Intrductin t Smith Charts Dr. Russell P. Jedlicka Klipsch Schl f Electrical and Cmputer Engineering New Mexic State University as Cruces, NM 88003 September 2002 EE521 ecture 3 08/22/02 Smith Chart Summary

### ELECTRON CYCLOTRON HEATING OF AN ANISOTROPIC PLASMA. December 4, PLP No. 322

ELECTRON CYCLOTRON HEATING OF AN ANISOTROPIC PLASMA by J. C. SPROTT December 4, 1969 PLP N. 3 These PLP Reprts are infrmal and preliminary and as such may cntain errrs nt yet eliminated. They are fr private

### ChE 471: LECTURE 4 Fall 2003

ChE 47: LECTURE 4 Fall 003 IDEL RECTORS One f the key gals f chemical reactin engineering is t quantify the relatinship between prductin rate, reactr size, reactin kinetics and selected perating cnditins.

### ANALYTICAL SOLUTIONS TO THE PROBLEM OF EDDY CURRENT PROBES

ANALYTICAL SOLUTIONS TO THE PROBLEM OF EDDY CURRENT PROBES CONSISTING OF LONG PARALLEL CONDUCTORS B. de Halleux, O. Lesage, C. Mertes and A. Ptchelintsev Mechanical Engineering Department Cathlic University

### Thermodynamics and Equilibrium

Thermdynamics and Equilibrium Thermdynamics Thermdynamics is the study f the relatinship between heat and ther frms f energy in a chemical r physical prcess. We intrduced the thermdynamic prperty f enthalpy,

### Dispersion Ref Feynman Vol-I, Ch-31

Dispersin Ref Feynman Vl-I, Ch-31 n () = 1 + q N q /m 2 2 2 0 i ( b/m) We have learned that the index f refractin is nt just a simple number, but a quantity that varies with the frequency f the light.

### Bicycle Generator Dump Load Control Circuit: An Op Amp Comparator with Hysteresis

Bicycle Generatr Dump Lad Cntrl Circuit: An Op Amp Cmparatr with Hysteresis Sustainable Technlgy Educatin Prject University f Waterl http://www.step.uwaterl.ca December 1, 2009 1 Summary This dcument describes

### GENERAL FORMULAS FOR FLAT-TOPPED WAVEFORMS. J.e. Sprott. Plasma Studies. University of Wisconsin

GENERAL FORMULAS FOR FLAT-TOPPED WAVEFORMS J.e. Sprtt PLP 924 September 1984 Plasma Studies University f Wiscnsin These PLP Reprts are infrmal and preliminary and as such may cntain errrs nt yet eliminated.

### Materials Engineering 272-C Fall 2001, Lecture 7 & 8 Fundamentals of Diffusion

Materials Engineering 272-C Fall 2001, Lecture 7 & 8 Fundamentals f Diffusin Diffusin: Transprt in a slid, liquid, r gas driven by a cncentratin gradient (r, in the case f mass transprt, a chemical ptential

### Transduction Based on Changes in the Energy Stored in an Electrical Field

Lecture 6-3 Transductin Based n Changes in the Energy Stred in an Electrical ield Department f Mechanical Engineering Example:Capacitive Pressure Sensr Pressure sensitive capacitive device With separatin

### Chapter 9: Quantization of Light

Chapter 9: Quantizatin Light 9.1 Planck s Quantum Thery 9.1.1 Distinguish between Planck s quantum thery and classical thery energy The undatin the Planck s quantum thery is a thery black bdy radiatin.

### Problem 1 Known: Dimensions and materials of the composition wall, 10 studs each with 2.5m high

Prblem Knwn: Dimensins and materials f the cmpsitin wall, 0 studs each with.5m high Unknwn:. Thermal resistance assciate with wall when surfaces nrmal t the directin f heat flw are isthermal. Thermal resistance

### Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System

Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed

### EXPERIMENTAL STUDY ON DISCHARGE COEFFICIENT OF OUTFLOW OPENING FOR PREDICTING CROSS-VENTILATION FLOW RATE

EXPERIMENTAL STUD ON DISCHARGE COEFFICIENT OF OUTFLOW OPENING FOR PREDICTING CROSS-VENTILATION FLOW RATE Tmnbu Gt, Masaaki Ohba, Takashi Kurabuchi 2, Tmyuki End 3, shihik Akamine 4, and Tshihir Nnaka 2

### CANKAYA UNIVERSITY FACULTY OF ENGINEERING MECHANICAL ENGINEERING DEPARTMENT ME 313 HEAT TRANSFER

CANKAYA UNIVERSITY FACUTY OF ENGINEERING MECHANICA ENGINEERING DEPARTMENT ME 313 HEAT TRANSFER CHAPTER-3 EXAMPES 1) Cnsider a slab f thicness as illustrated in figure belw. A fluid at temperature T 1 with

### 20 Faraday s Law and Maxwell s Extension to Ampere s Law

Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law 20 Faraday s Law and Maxwell s Extensin t Ampere s Law Cnsider the case f a charged particle that is ming in the icinity f a ming bar magnet

### Lecture 13: Electrochemical Equilibria

3.012 Fundamentals f Materials Science Fall 2005 Lecture 13: 10.21.05 Electrchemical Equilibria Tday: LAST TIME...2 An example calculatin...3 THE ELECTROCHEMICAL POTENTIAL...4 Electrstatic energy cntributins

### EDA Engineering Design & Analysis Ltd

EDA Engineering Design & Analysis Ltd THE FINITE ELEMENT METHOD A shrt tutrial giving an verview f the histry, thery and applicatin f the finite element methd. Intrductin Value f FEM Applicatins Elements

### Chapter 9 Vector Differential Calculus, Grad, Div, Curl

Chapter 9 Vectr Differential Calculus, Grad, Div, Curl 9.1 Vectrs in 2-Space and 3-Space 9.2 Inner Prduct (Dt Prduct) 9.3 Vectr Prduct (Crss Prduct, Outer Prduct) 9.4 Vectr and Scalar Functins and Fields

DT, Kevin t. lectric Circuit Thery DT87/ Tw-Prt netwrk parameters ummary We have seen previusly that a tw-prt netwrk has a pair f input terminals and a pair f utput terminals figure. These circuits were

### ECEN 4872/5827 Lecture Notes

ECEN 4872/5827 Lecture Ntes Lecture #5 Objectives fr lecture #5: 1. Analysis f precisin current reference 2. Appraches fr evaluating tlerances 3. Temperature Cefficients evaluatin technique 4. Fundamentals

### Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System

Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed

### Pressure And Entropy Variations Across The Weak Shock Wave Due To Viscosity Effects

Pressure And Entrpy Variatins Acrss The Weak Shck Wave Due T Viscsity Effects OSTAFA A. A. AHOUD Department f athematics Faculty f Science Benha University 13518 Benha EGYPT Abstract:-The nnlinear differential

### THE TOPOLOGY OF SURFACE SKIN FRICTION AND VORTICITY FIELDS IN WALL-BOUNDED FLOWS

THE TOPOLOGY OF SURFACE SKIN FRICTION AND VORTICITY FIELDS IN WALL-BOUNDED FLOWS M.S. Chng Department f Mechanical Engineering The University f Melburne Victria 3010 AUSTRALIA min@unimelb.edu.au J.P. Mnty

### 1 The limitations of Hartree Fock approximation

Chapter: Pst-Hartree Fck Methds - I The limitatins f Hartree Fck apprximatin The n electrn single determinant Hartree Fck wave functin is the variatinal best amng all pssible n electrn single determinants

### Equilibrium of Stress

Equilibrium f Stress Cnsider tw perpendicular planes passing thrugh a pint p. The stress cmpnents acting n these planes are as shwn in ig. 3.4.1a. These stresses are usuall shwn tgether acting n a small

### Phy 213: General Physics III 6/14/2007 Chapter 28 Worksheet 1

Ph 13: General Phsics III 6/14/007 Chapter 8 Wrksheet 1 Magnetic Fields & Frce 1. A pint charge, q= 510 C and m=110-3 m kg, travels with a velcit f: v = 30 ˆ s i then enters a magnetic field: = 110 T ˆj.

### 3. Design of Channels General Definition of some terms CHAPTER THREE

CHAPTER THREE. Design f Channels.. General The success f the irrigatin system depends n the design f the netwrk f canals. The canals may be excavated thrugh the difference types f sils such as alluvial

### STUDENT NAME: STUDENT id #: WORK ONLY 5 QUESTIONS

GENERAL PHYSICS PH -A (MIROV) Exam 3 (03/31/15) STUDENT NAME: STUDENT i #: ------------------------------------------------------------------------------------------------------------------------------------------