Work, Potential Energy, Conservation of Energy. the electric forces are conservative: ur r
|
|
- Constance Cameron
- 6 years ago
- Views:
Transcription
1
2 Wok, Potentil Enegy, Consevtion of Enegy the electic foces e consevtive: u Fd =
3 Wok, Potentil Enegy, Consevtion of Enegy b b W = u b b Fdl = F()[ d + $ $ dl ] = F() d u Fdl = the electic foces e consevtive
4 Wok, Potentil Enegy, Consevtion of Enegy u W = K K = F dl b b W b = U Ub U U = K K U + K = U + K b b b b b q q U qq qq = U = 4 b πεb nottions : u F q q foce with which chge q cts on q ; W b wok pefomed when chge is moved fom point to point b
5 Wok, Potentil Enegy, infinity s efeence point U qq qq = U = 4 b πεb u U U F dl b = b q q U potentil enegy hs to be mesued with espect to W = b U Ub qq qq = U = 4 b πεb u = = = U U Fq qdl u = = = Ub U b Fq qdl b qq qq b Hee, potentil enegy is mesued with espect to n infinitely emote point U = wok which ws done by chge q when n electic chge q o is deliveed to infinity = wok which ws done (by ) to bing n electic chge q fom infinity
6 Wok, Potentil Enegy, Consevtion of Enegy U = qq U = wok which ws done to bing n electic chge q o fom infinity = the kinetic enegy which this chge will cquie t infinity if it will be elesed u F u el. field = Fq q U U Fextenldl u = = = u u Fel. fielddl = Fel. fielddl = qq Hee, potentil enegy is mesued with espect to n infinitely emote point
7 Hee (!!!), potentil enegy is mesued with espect to n infinitely emote point U qq qq = U = 4 b πεb
8 Potentil Enegy U qq qq = U = 4 b πεb Positon q=e, lph pticle Q=2e
9 Electic Potentil Enegy with Sevel Chges U= wok which ws done to bing n Electic Chge q o fom infinity u N u N u N U = Edl = dl E = E dl = U j j i i j i long long i= 1 i= 1 long i= 1 contou contou contou
10 Electic Potentil Enegy with Sevel Chges U= wok which ws done to bing n electic chge q o fom infinity = the kinetic enegy which this chge will cquie t infinity if it will be elesed
11 potentil enegy U(x,y,z) nd foce F: gdient of the potentil enegy is equl to minus foce U() U() U() Fx = Fy = Fz = x y z u u F = U() 2 u U U Fdl 2 1 = 1 u = i / x + j / y+ k / z The opetion is clled tking gdient ; esult gdient. u F U() U() U() = i + j + k x y z diffeentition of this integl with woks like diffeentition of n odiny integl: u u u Fdl = u F( ) u u u Fdl = F( )
12 u F u E Electic Potentil U() V () = V() U() q u F u E u = U() u = V() Electic field is equl to electic foce pe unit chge Electic potentil is equl to potentil enegy pe unit chge V q () = q CAUTION: potentil enegy hs to be mesued with espect to hee, electic potentil is mesued with espect to n infinitely emote point, while is the distnce fom the loction of chge q
13 Electic Potentil u F u E U ( ) V ( ) V ( ) = U ( ) q u F u E u = U ( ) u = V ( ) Electic potentil is equl to potentil enegy pe unit chge of pobe chge U() qv() q = = q V () = q
14 Units: 1V=1 volt= 1 J/C= 1 joule/coulomb 1V=1 (N/C)m electic field is, thus, mesued in volts pe mete: V/m V ( ) = 1 q ε = C/(Vm) E = Const V ( x) = E 1 9 = 9 1 Vm C x
15 Similly to U(x,y,z), one cn intoduce the electic potentil V(x,y,z) such tht: u V V Edl V() V() V() Ex = Ey = Ez = x y z u u u u u Edl Edl E( ) u u E = V() In electosttics, the electic fields e consevtive: Ed = This implies tht the diffeence between the electic potentils of two points does not depend on the tjectoy connecting these points: 2 1 Ed = [ V ( ) V ( )] 2 1 = = = = u u E = V() if V(x,y,z) is known one cn find the components of the electic field nd vice ves
16 Elements of mth u u E = V() x V() V() V() Ex = Ey = Ez = x y z u V() V() V() E = i+ j+ k x y z ptil deivtive (conside y nd z s constnts) u dv () useful fomul: V ( ) = $ d u dv ()( l ) V( ) = dl is unit vecto long ( )
17
18 Moving though potentil diffeence: chnge of the potentil enegy of chge q is equl to the poduct of the electic potentil chnge nd the chge 1eV = (14) 1 17 J
19 Moving though potentil diffeence 1V =1J/C
20 Thee is no electic field inside conducto Net chge cn only eside on the sufce of conducto Any extenl electic field lines e pependicul to the sufce (thee is no component of electic field tht is tngent to the sufce). The electic potentil within conducto is constnt (vlid only in the bsence of cuents)
21 Definition: Owing to the fct tht E = inside the conducto, the diffeence Fo ny two points 1 nd 2 inside the conducto 2 V V Edl 2 1 = the electic potentil diffeence is equl to n integl long line connecting the two points (ny line!) V V = V 1 V = In equilibium (i.e., without cuents): 1) the conducto s sufce is equipotentil. 2) the whole body of the conducto is equipotentil.
22 Equipotentil Sufces An equipotentil sufce is sufce on which the electic potentil V is the sme t evey point. Convesely, the electic field cn do no wok on chge moving long n equipotentil sufce. Electic field must be pependicul to the sufce t evey point so tht the electic foce is lwys pependicul to the displcement of chge moving on the sufce. Field lines nd equipotentil sufces e lwys mutully pependicul.
23 Gdient of the electic potentil is equl to minus electic field. It mens tht long the electic field line the electic potentil goes DOWN, down, down! u u u E() = V() Edl = E( ) 1 Edl= [ V ( ) V ( )] = lim [ U ( ) U ( )] q q So, wht is the use in one moe quntity? Becuse to dw mp of scl quntity epesenting the shpe (elief) of the potentil enegy is much esie thn to dw mp of the vecto field. Look on topogphic mp of mountins! It is mp of the gvittionl potentil: Question: wht is n nlogue of lke?
24 V ( ) = 1 q Exmples E = Const V ( x) = E x + const
25 2 1 Ed = 1 lim [ U( 2) U( 1)] = [ V ( 2) V ( 1)] q q Fo point chge t the oigin (we ledy know the nswe): [ V( ) V( )] Ed q d q = = = V ( ) = dv () V() = $ d u dv () useful fomul: V ( ) = $ d u dv ()( l ) V( ) = dl ( ) is unit vecto pointing fom towd
26 Exmple Nonconducting with homogenously distibuted chge Q 1 VV( ) Q R?
27 R Nonconducting sphee of dius A with homogenously distibuted chge Q E > R E = 1 Q 2 < R E = Q R 3 VV( ) 1 Q R? R V() =? beceful! with espect to wht point? V() V( ) = E( y) dy
28 R Nonconducting sphee of dius A with homogenously distibuted chge Q E > < R R E( ) = E () = 1 Q 2 Q R 3 VV( ) 1 Q R? R V() V( ) = E( y) dy < R V() V( ) = + > R 1 Q V() V( ) = 2 1 Q 3 2 R 2R 2
29 R Nonconducting sphee of dius A with homogenously distibuted chge Q E > < R R E E = = 1 Q 2 Q R 3 R V()V() is like V()V( ), but is shifted down on constnt 3 Q equl to 8πε R V() is pbolic t smll V () V() = E( ydy ) = E( ydy ) 2 < R V() V() = 8πε 1 Q R 3 > R V() V() = R 2 Q R 3
30 Potentil of chged plte E = Const V ( x) = E x + const V ( ) =??? V ( ) =??? +σ Fo point chge t the oigin: [ V( ) V( )] = Ed = q d q = = x
31 Potentil of chged plte +σ σ + S S E = Const V ( x) = E x + const x Cution: it is convenient to plce the initil point, i.e., b, to infinity nd set V(b)=. Neve do it when you del with infinitely lge/extended objects!
32 Two pllel conducting pltes σ S σ V ( ed ) = E x + σ + σ + V ( blue) = E x Field between the pltes is constnt, potentil is line E = σ /ε x
33 Two pllel conducting pltes σ S σ σ +σ σ +σ + Field between the pltes is constnt, potentil is line x E = σ /ε Potentil t infinity is not zeo!
34 V ( ) =??? [ V( ) V( )] = Ed = λ dy λ = = 2πε y 2πε ln y??? V( ) = λ dy λ V( )] = Ed = = ln y 2πε y 2πε Agin the poblem with infinity Resolution: only potentil diffeence mttes, mesue potentil diffeence with espect to n bity point = λ 2πε ln
35 Method of imges: Wht is foce on the point chge ne conducting plte? Equipotentil sufce The tick with imging cn be done not only with flt sufce
36 Additionl mteil 1
37 Physics of Lightings Benjmin Fnklin
38 ioniztion nd coon dischge Thee is mximum potentil to which conducto in i cn be ised becuse of ioniztion. 6 E m = 3 1 V m V m = R E m
39 A lightning od hs shp end so tht lightning bolts will pss though conducting pth in the i tht leds to the od; conducting wie leds fom the lightning od to the gound. The metl mst t the top of the Empie Stte Building cts s lightning od. It is stuck by lightning s mny s 5 times ech ye. Even eltively smll potentils pplied to shp points in i poduce sufficiently high fields just outside the point to ionize the suounding i. Cution: wht cn be misleding in the bove quottion?
40 Even eltively smll potentils pplied to shp points in i poduce sufficiently high fields just outside the point to ionize the suounding i. Thee is mximum potentil to which conducto in i cn be ised becuse of ioniztion. 6 E m = 3 1 V m ioniztion nd coon dischge A shp edge leds to the defomtion of equipotentil lines nd, hence, to high electic field ne the edge The tlle the edge the stonge defomtion of the lines nd, theefoe, the moe pobble dischge by lighting
41 Method of imges: Wht is foce on the point chge ne conducting plte? Equipotentil sufce The tick with imging cn be done not only with flt sufce
42 The foce cting on the positive chge is exctly the sme s it would be with the negtive imge chge insted of the plte. The point chge feels foce towds the plte with mgnitude: F = 1 q 2 2 (2) Cution: 2 the thn!
43 Additionl mteil 2
44 2 1 1 Edl= [ V ( ) V ( )] = lim [ U ( ) U ( )] q q electic potentil V=U/q is simil to U/m in the cse of the Eth gvittion 2 1 u u 1 Fdl= [ U ( 2) U ( 1)] g = U() m u
45 b Edl = [ V( ) V( )] = V( ) + V( b) u b u b dl it is convenient to plce the initil point, i.e., b, to infinity nd set V(b)=. Wht potentil V() cetes positive chge Q? It is positive! Note tht when positive chge is ppoched, is diected ginst, nd theefoe Edl is genelly negtive. Hence V() is positive. E dl V( ) = ( E) dl = [ V( ) V( )] >
46 qv() is equl to wok W needed to pefom in ode to delive the chge q fom infinity to the point. The wok W is done ginst the electic field. u qv ( ) = ( qe) dl = q[ V ( ) V ( )] dl Fo pointlike chge: If qq> this wok is positive; W>. Coespondingly V()=W/q is positive if Q>.
47 u dl qv ( ) = ( qe) dl = q[ V ( ) V ( )] = qedl Now qv() cn be intepeted s wok W which pefoms the electic field of the chge Q when the chge q is moving wy. In this fomultion, the wok W is done by the electic field. Fo pointlike chge. We e moving wy fom positive chge: now nd E hve common diection towd infinity. Hence Edl >, If qq>, this wok is positive; W>. Coespondingly V()=W/q is positive when Q>. dl u dl
48 Additionl mteil 3
49 Clcultion of the electic field ne the conducting sufce is possible, but not the shotest wy, to find foce cting on the metllic plne y u F =? E E + 1 = 1 = q ( + y ) q ( + y ) E = 1 q 2 πε ( + y )
50 Clcultion of the electic field ne the conducting sufce is possible, but not the shotest wy, to find foce cting on the metllic plne E 1 q = 2 πε ( + y ) E π 1 q df = ()! σ2πydy = ydy ε 2 π ( y ) q 1 q 1 q F = ydy = = 4 πε ( + y ) 4 4 πε (2 ) y σ = ε E u F =?
51 Additionl mteil 4
52 In ecoded lectue fom 1961, Richd Feynmn explined to his students why physicists use electon volts to mesue enegy insted of some multiple of the joule: A single tom is such smll thing tht to tlk bout its enegy in joules would be inconvenient. But insted of tking definite unit in the sme system, like 1 2 J, [physicists] hve unfotuntely chosen, bitily, funny unit clled n electonvolt (ev)... I m soy tht we do tht, but tht's the wy it is fo the physicists.
53 Electonvolt The electonvolt (ev) is unit of enegy. By definition, it is equl to the mount of enegy gined by single unbound electon when it cceletes though n electosttic potentil diffeence of one volt. 1 ev = (14) 1 19 J. So n electon volt is 1 volt (1 joule divided by 1 coulomb) multiplied by the electon chge ( (14) 1 19 coulomb). The electonvolt is now ccepted within SI.
54 1eV = (14) 1 19 J 1eV = (14) 1 17 J
55
56
U>, and is negative. Electric Potential Energy
Electic Potentil Enegy Think of gvittionl potentil enegy. When the lock is moved veticlly up ginst gvity, the gvittionl foce does negtive wok (you do positive wok), nd the potentil enegy (U) inceses. When
More informationElectric Potential. and Equipotentials
Electic Potentil nd Euipotentils U Electicl Potentil Review: W wok done y foce in going fom to long pth. l d E dl F W dl F θ Δ l d E W U U U Δ Δ l d E W U U U U potentil enegy electic potentil Potentil
More informationAnswers to test yourself questions
Answes to test youself questions opic Descibing fields Gm Gm Gm Gm he net field t is: g ( d / ) ( 4d / ) d d Gm Gm Gm Gm Gm Gm b he net potentil t is: V d / 4d / d 4d d d V e 4 7 9 49 J kg 7 7 Gm d b E
More informationElectricity & Magnetism Lecture 6: Electric Potential
Electicity & Mgnetism Lectue 6: Electic Potentil Tody s Concept: Electic Potenl (Defined in tems of Pth Integl of Electic Field) Electicity & Mgnesm Lectue 6, Slide Stuff you sked bout:! Explin moe why
More informationPhysics 11b Lecture #11
Physics 11b Lectue #11 Mgnetic Fields Souces of the Mgnetic Field S&J Chpte 9, 3 Wht We Did Lst Time Mgnetic fields e simil to electic fields Only diffeence: no single mgnetic pole Loentz foce Moving chge
More informationPhysics 1502: Lecture 2 Today s Agenda
1 Lectue 1 Phsics 1502: Lectue 2 Tod s Agend Announcements: Lectues posted on: www.phs.uconn.edu/~cote/ HW ssignments, solutions etc. Homewok #1: On Mstephsics this Fid Homewoks posted on Msteingphsics
More informationChapter 2: Electric Field
P 6 Genel Phsics II Lectue Outline. The Definition of lectic ield. lectic ield Lines 3. The lectic ield Due to Point Chges 4. The lectic ield Due to Continuous Chge Distibutions 5. The oce on Chges in
More information6. Gravitation. 6.1 Newton's law of Gravitation
Gvittion / 1 6.1 Newton's lw of Gvittion 6. Gvittion Newton's lw of gvittion sttes tht evey body in this univese ttcts evey othe body with foce, which is diectly popotionl to the poduct of thei msses nd
More informationGeneral Physics II. number of field lines/area. for whole surface: for continuous surface is a whole surface
Genel Physics II Chpte 3: Guss w We now wnt to quickly discuss one of the moe useful tools fo clculting the electic field, nmely Guss lw. In ode to undestnd Guss s lw, it seems we need to know the concept
More informationPX3008 Problem Sheet 1
PX38 Poblem Sheet 1 1) A sphee of dius (m) contins chge of unifom density ρ (Cm -3 ). Using Guss' theoem, obtin expessions fo the mgnitude of the electic field (t distnce fom the cente of the sphee) in
More informationChapter 21: Electric Charge and Electric Field
Chpte 1: Electic Chge nd Electic Field Electic Chge Ancient Gees ~ 600 BC Sttic electicit: electic chge vi fiction (see lso fig 1.1) (Attempted) pith bll demonsttion: inds of popeties objects with sme
More informationπ,π is the angle FROM a! TO b
Mth 151: 1.2 The Dot Poduct We hve scled vectos (o, multiplied vectos y el nume clled scl) nd dded vectos (in ectngul component fom). Cn we multiply vectos togethe? The nswe is YES! In fct, thee e two
More informationLecture 11: Potential Gradient and Capacitor Review:
Lectue 11: Potentil Gdient nd Cpcito Review: Two wys to find t ny point in spce: Sum o Integte ove chges: q 1 1 q 2 2 3 P i 1 q i i dq q 3 P 1 dq xmple of integting ove distiution: line of chge ing of
More informationGeneral Physics (PHY 2140)
Genel Physics (PHY 40) Lightning Review Lectue 3 Electosttics Lst lectue:. Flux. Guss s s lw. simplifies computtion of electic fields Q Φ net Ecosθ ε o Electicl enegy potentil diffeence nd electic potentil
More informationSchool of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007
School of Electicl nd Compute Engineeing, Conell Univesity ECE 303: Electomgnetic Fields nd Wves Fll 007 Homewok 4 Due on Sep. 1, 007 by 5:00 PM Reding Assignments: i) Review the lectue notes. ii) Relevnt
More informationUnit 1. Electrostatics of point charges
Unit 1 Electosttics of point chges 1.1 Intoduction 1. Electic chge 1.3 Electosttic foces. Coulomb s lw 1.4 Electic field. Field lines 1.5 Flux of the electic field. Guss s lw 1.6 Wok of the foces of electic
More informationELECTROSTATICS. 4πε0. E dr. The electric field is along the direction where the potential decreases at the maximum rate. 5. Electric Potential Energy:
LCTROSTATICS. Quntiztion of Chge: Any chged body, big o smll, hs totl chge which is n integl multile of e, i.e. = ± ne, whee n is n intege hving vlues,, etc, e is the chge of electon which is eul to.6
More informationChapter 28 Sources of Magnetic Field
Chpte 8 Souces of Mgnetic Field - Mgnetic Field of Moving Chge - Mgnetic Field of Cuent Element - Mgnetic Field of Stight Cuent-Cying Conducto - Foce Between Pllel Conductos - Mgnetic Field of Cicul Cuent
More informationPhysics 604 Problem Set 1 Due Sept 16, 2010
Physics 64 Polem et 1 Due ept 16 1 1) ) Inside good conducto the electic field is eo (electons in the conducto ecuse they e fee to move move in wy to cncel ny electic field impessed on the conducto inside
More informationThis chapter is about energy associated with electrical interactions. Every
23 ELECTRIC PTENTIAL whee d l is n infinitesiml displcement long the pticle s pth nd f is the ngle etween F nd d l t ech point long the pth. econd, if the foce F is consevtive, s we defined the tem in
More informationSolutions to Midterm Physics 201
Solutions to Midtem Physics. We cn conside this sitution s supeposition of unifomly chged sphee of chge density ρ nd dius R, nd second unifomly chged sphee of chge density ρ nd dius R t the position of
More informationr = (0.250 m) + (0.250 m) r = m = = ( N m / C )
ELECTIC POTENTIAL IDENTIFY: Apply Eq() to clculte the wok The electic potentil enegy of pi of point chges is given y Eq(9) SET UP: Let the initil position of q e point nd the finl position e point, s shown
More informationThe Formulas of Vector Calculus John Cullinan
The Fomuls of Vecto lculus John ullinn Anlytic Geomety A vecto v is n n-tuple of el numbes: v = (v 1,..., v n ). Given two vectos v, w n, ddition nd multipliction with scl t e defined by Hee is bief list
More informationFI 2201 Electromagnetism
FI 1 Electomgnetism Alexnde A. Isknd, Ph.D. Physics of Mgnetism nd Photonics Resech Goup Electosttics ELECTRIC PTENTIALS 1 Recll tht we e inteested to clculte the electic field of some chge distiution.
More information= ΔW a b. U 1 r m 1 + K 2
Chpite 3 Potentiel électiue [18 u 3 mi] DEVOIR : 31, 316, 354, 361, 35 Le potentiel électiue est le tvil p unité de chge (en J/C, ou volt) Ce concept est donc utile dns les polèmes de consevtion d énegie
More informationAlgebra Based Physics. Gravitational Force. PSI Honors universal gravitation presentation Update Fall 2016.notebookNovember 10, 2016
Newton's Lw of Univesl Gvittion Gvittionl Foce lick on the topic to go to tht section Gvittionl Field lgeb sed Physics Newton's Lw of Univesl Gvittion Sufce Gvity Gvittionl Field in Spce Keple's Thid Lw
More informationRadial geodesics in Schwarzschild spacetime
Rdil geodesics in Schwzschild spcetime Spheiclly symmetic solutions to the Einstein eqution tke the fom ds dt d dθ sin θdϕ whee is constnt. We lso hve the connection components, which now tke the fom using
More informationFriedmannien equations
..6 Fiedmnnien equtions FLRW metic is : ds c The metic intevl is: dt ( t) d ( ) hee f ( ) is function which detemines globl geometic l popety of D spce. f d sin d One cn put it in the Einstein equtions
More informationSURFACE TENSION. e-edge Education Classes 1 of 7 website: , ,
SURFACE TENSION Definition Sufce tension is popety of liquid by which the fee sufce of liquid behves like stetched elstic membne, hving contctive tendency. The sufce tension is mesued by the foce cting
More informationElectric Field F E. q Q R Q. ˆ 4 r r - - Electric field intensity depends on the medium! origin
1 1 Electic Field + + q F Q R oigin E 0 0 F E ˆ E 4 4 R q Q R Q - - Electic field intensity depends on the medium! Electic Flux Density We intoduce new vecto field D independent of medium. D E So, electic
More informationdx was area under f ( x ) if ( ) 0
13. Line Integls Line integls e simil to single integl, f ( x) dx ws e unde f ( x ) if ( ) 0 Insted of integting ove n intevl [, ] (, ) f xy ds f x., we integte ove cuve, (in the xy-plne). **Figue - get
More information3.1 Magnetic Fields. Oersted and Ampere
3.1 Mgnetic Fields Oested nd Ampee The definition of mgnetic induction, B Fields of smll loop (dipole) Mgnetic fields in mtte: ) feomgnetism ) mgnetiztion, (M ) c) mgnetic susceptiility, m d) mgnetic field,
More informationSchool of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007
School of Electicl nd Compute Engineeing, Conell Univesity ECE 303: Electomgnetic Fields nd Wves Fll 007 Homewok 3 Due on Sep. 14, 007 by 5:00 PM Reding Assignments: i) Review the lectue notes. ii) Relevnt
More informationChapter 25: Current, Resistance and Electromotive Force. Charge carrier motion in a conductor in two parts
Chpte 5: Cuent, esistnce nd Electomotive Foce Chge cie motion in conducto in two pts Constnt Acceletion F m qe ndomizing Collisions (momentum, enegy) =>esulting Motion Avege motion = Dift elocity = v d
More informationCHAPTER 18: ELECTRIC CHARGE AND ELECTRIC FIELD
ollege Physics Student s Mnul hpte 8 HAPTR 8: LTRI HARG AD LTRI ILD 8. STATI LTRIITY AD HARG: OSRVATIO O HARG. ommon sttic electicity involves chges nging fom nnocoulombs to micocoulombs. () How mny electons
More informationCHAPTER 2 ELECTROSTATIC POTENTIAL
1 CHAPTER ELECTROSTATIC POTENTIAL 1 Intoduction Imgine tht some egion of spce, such s the oom you e sitting in, is pemeted by n electic field (Pehps thee e ll sots of electiclly chged bodies outside the
More informationCHAPTER 25 ELECTRIC POTENTIAL
CHPTE 5 ELECTIC POTENTIL Potential Diffeence and Electic Potential Conside a chaged paticle of chage in a egion of an electic field E. This filed exets an electic foce on the paticle given by F=E. When
More informationChapter 25: Current, Resistance and Electromotive Force. ~10-4 m/s Typical speeds ~ 10 6 m/s
Chpte 5: Cuent, esistnce nd lectomotive Foce Chge cie motion in conducto in two pts Constnt Acceletion F m q ndomizing Collisions (momentum, enegy) >esulting Motion http://phys3p.sl.psu.edu/phys_nim/m/ndom_wlk.vi
More informationContinuous Charge Distributions
Continuous Chge Distibutions Review Wht if we hve distibution of chge? ˆ Q chge of distibution. Q dq element of chge. d contibution to due to dq. Cn wite dq = ρ dv; ρ is the chge density. = 1 4πε 0 qi
More informationFluids & Bernoulli s Equation. Group Problems 9
Goup Poblems 9 Fluids & Benoulli s Eqution Nme This is moe tutoil-like thn poblem nd leds you though conceptul development of Benoulli s eqution using the ides of Newton s 2 nd lw nd enegy. You e going
More information1 Using Integration to Find Arc Lengths and Surface Areas
Novembe 9, 8 MAT86 Week Justin Ko Using Integtion to Find Ac Lengths nd Sufce Aes. Ac Length Fomul: If f () is continuous on [, b], then the c length of the cuve = f() on the intevl [, b] is given b s
More information(A) 6.32 (B) 9.49 (C) (D) (E) 18.97
Univesity of Bhin Physics 10 Finl Exm Key Fll 004 Deptment of Physics 13/1/005 8:30 10:30 e =1.610 19 C, m e =9.1110 31 Kg, m p =1.6710 7 Kg k=910 9 Nm /C, ε 0 =8.8410 1 C /Nm, µ 0 =4π10 7 T.m/A Pt : 10
More informationELECTROSTATICS. Syllabus : Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road PE 1
PE ELECTOSTATICS Syllbus : Electic chges : Consevtion of chge, Coulumb s lw-foces between two point chges, foces between multiple chges; supeposition pinciple nd continuous chge distibution. Electic field
More informationMAGNETIC EFFECT OF CURRENT & MAGNETISM
TODUCTO MAGETC EFFECT OF CUET & MAGETM The molecul theo of mgnetism ws given b Webe nd modified lte b Ewing. Oested, in 18 obseved tht mgnetic field is ssocited with n electic cuent. ince, cuent is due
More information( ) ( ) ( ) ( ) ( ) # B x ( ˆ i ) ( ) # B y ( ˆ j ) ( ) # B y ("ˆ ( ) ( ) ( (( ) # ("ˆ ( ) ( ) ( ) # B ˆ z ( k )
Emple 1: A positie chge with elocit is moing though unifom mgnetic field s shown in the figues below. Use the ight-hnd ule to detemine the diection of the mgnetic foce on the chge. Emple 1 ˆ i = ˆ ˆ i
More information10 m, so the distance from the Sun to the Moon during a solar eclipse is. The mass of the Sun, Earth, and Moon are = =
Chpte 1 nivesl Gvittion 11 *P1. () The un-th distnce is 1.4 nd the th-moon 8 distnce is.84, so the distnce fom the un to the Moon duing sol eclipse is 11 8 11 1.4.84 = 1.4 The mss of the un, th, nd Moon
More informationEnergy Dissipation Gravitational Potential Energy Power
Lectue 4 Chpte 8 Physics I 0.8.03 negy Dissiption Gvittionl Potentil negy Powe Couse wesite: http://fculty.uml.edu/andiy_dnylov/teching/physicsi Lectue Cptue: http://echo360.uml.edu/dnylov03/physicsfll.html
More informationPHYS 2421 Fields and Waves
PHYS 242 Felds nd Wves Instucto: Joge A. López Offce: PSCI 29 A, Phone: 747-7528 Textook: Unvesty Physcs e, Young nd Feedmn 23. Electc potentl enegy 23.2 Electc potentl 23.3 Clcultng electc potentl 23.4
More informationThis immediately suggests an inverse-square law for a "piece" of current along the line.
Electomgnetic Theoy (EMT) Pof Rui, UNC Asheville, doctophys on YouTube Chpte T Notes The iot-svt Lw T nvese-sque Lw fo Mgnetism Compe the mgnitude of the electic field t distnce wy fom n infinite line
More informationPhysics 111. Uniform circular motion. Ch 6. v = constant. v constant. Wednesday, 8-9 pm in NSC 128/119 Sunday, 6:30-8 pm in CCLIR 468
ics Announcements dy, embe 28, 2004 Ch 6: Cicul Motion - centipetl cceletion Fiction Tension - the mssless sting Help this week: Wednesdy, 8-9 pm in NSC 128/119 Sundy, 6:30-8 pm in CCLIR 468 Announcements
More informationChapter 25 Electric Potential
Chpte 5 lectic Potentil consevtive foces -> potentil enegy - Wht is consevtive foce? lectic potentil = U / : the potentil enegy U pe unit chge is function of the position in spce Gol:. estblish the eltionship
More informationChapter 4. Energy and Potential
Chpte 4. Enegy nd Ptentil Hyt; 0/5/009; 4-4. Enegy Expended in Mving Pint Chge in n Electic Field The electic field intensity is defined s the fce n unit test chge. The fce exeted y the electic field n
More informationElectric Potential Energy
Electic Ptentil Enegy Ty Cnsevtive Fces n Enegy Cnsevtin Ttl enegy is cnstnt n is sum f kinetic n ptentil Electic Ptentil Enegy Electic Ptentil Cnsevtin f Enegy f pticle fm Phys 7 Kinetic Enegy (K) nn-eltivistic
More informationEELE 3331 Electromagnetic I Chapter 4. Electrostatic fields. Islamic University of Gaza Electrical Engineering Department Dr.
EELE 3331 Electomagnetic I Chapte 4 Electostatic fields Islamic Univesity of Gaza Electical Engineeing Depatment D. Talal Skaik 212 1 Electic Potential The Gavitational Analogy Moving an object upwad against
More informationEinstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,
PE ELECTOSTATICS C Popeties of chges : (i) (ii) (iii) (iv) (v) (vi) Two kinds of chges eist in ntue, positive nd negtive with the popety tht unlike chges ttct ech othe nd like chges epel ech othe. Ecess
More informationOptimization. x = 22 corresponds to local maximum by second derivative test
Optimiztion Lectue 17 discussed the exteme vlues of functions. This lectue will pply the lesson fom Lectue 17 to wod poblems. In this section, it is impotnt to emembe we e in Clculus I nd e deling one-vible
More informationMath 4318 : Real Analysis II Mid-Term Exam 1 14 February 2013
Mth 4318 : Rel Anlysis II Mid-Tem Exm 1 14 Febuy 2013 Nme: Definitions: Tue/Flse: Poofs: 1. 2. 3. 4. 5. 6. Totl: Definitions nd Sttements of Theoems 1. (2 points) Fo function f(x) defined on (, b) nd fo
More informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
More informationELECTRO - MAGNETIC INDUCTION
NTRODUCTON LCTRO - MAGNTC NDUCTON Whenee mgnetic flu linked with cicuit chnges, n e.m.f. is induced in the cicuit. f the cicuit is closed, cuent is lso induced in it. The e.m.f. nd cuent poduced lsts s
More informationCh 26 - Capacitance! What s Next! Review! Lab this week!
Ch 26 - Cpcitnce! Wht s Next! Cpcitnce" One week unit tht hs oth theoeticl n pcticl pplictions! Cuent & Resistnce" Moving chges, finlly!! Diect Cuent Cicuits! Pcticl pplictions of ll the stuff tht we ve
More informationChapter 4 Kinematics in Two Dimensions
D Kinemtic Quntities Position nd Velocit Acceletion Applictions Pojectile Motion Motion in Cicle Unifom Cicul Motion Chpte 4 Kinemtics in Two Dimensions D Motion Pemble In this chpte, we ll tnsplnt the
More informationPreviously. Extensions to backstepping controller designs. Tracking using backstepping Suppose we consider the general system
436-459 Advnced contol nd utomtion Extensions to bckstepping contolle designs Tcking Obseves (nonline dmping) Peviously Lst lectue we looked t designing nonline contolles using the bckstepping technique
More informationOn the Eötvös effect
On the Eötvös effect Mugu B. Răuţ The im of this ppe is to popose new theoy bout the Eötvös effect. We develop mthemticl model which loud us bette undestnding of this effect. Fom the eqution of motion
More informationClass Summary. be functions and f( D) , we define the composition of f with g, denoted g f by
Clss Summy.5 Eponentil Functions.6 Invese Functions nd Logithms A function f is ule tht ssigns to ech element D ectly one element, clled f( ), in. Fo emple : function not function Given functions f, g:
More informationUnit 6. Magnetic forces
Unit 6 Mgnetic foces 6.1 ntoduction. Mgnetic field 6. Mgnetic foces on moving electic chges 6. oce on conducto with cuent. 6.4 Action of unifom mgnetic field on flt cuent-cying loop. Mgnetic moment. Electic
More informationChapter 23 Electrical Potential
hpte Electicl Potentil onceptul Polems [SSM] A poton is moved to the left in unifom electic field tht points to the ight. Is the poton moving in the diection of incesing o decesing electic potentil? Is
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 informationSection 35 SHM and Circular Motion
Section 35 SHM nd Cicul Motion Phsics 204A Clss Notes Wht do objects do? nd Wh do the do it? Objects sometimes oscillte in simple hmonic motion. In the lst section we looed t mss ibting t the end of sping.
More informationChapter 4 Two-Dimensional Motion
D Kinemtic Quntities Position nd Velocit Acceletion Applictions Pojectile Motion Motion in Cicle Unifom Cicul Motion Chpte 4 Two-Dimensionl Motion D Motion Pemble In this chpte, we ll tnsplnt the conceptul
More information2.2 This is the Nearest One Head (Optional) Experimental Verification of Gauss s Law and Coulomb s Law
2.2 This is the Neest One Hed 743 P U Z Z L R Some ilwy compnies e plnning to cot the windows of thei commute tins with vey thin lye of metl. (The coting is so thin you cn see though it.) They e doing
More informationCollection of Formulas
Collection of Fomuls Electomgnetic Fields EITF8 Deptment of Electicl nd Infomtion Technology Lund Univesity, Sweden August 8 / ELECTOSTATICS field point '' ' Oigin ' Souce point Coulomb s Lw The foce F
More informationReview: Electrostatics and Magnetostatics
Review: Electostatics and Magnetostatics In the static egime, electomagnetic quantities do not vay as a function of time. We have two main cases: ELECTROSTATICS The electic chages do not change postion
More informationRELATIVE KINEMATICS. q 2 R 12. u 1 O 2 S 2 S 1. r 1 O 1. Figure 1
RELAIVE KINEMAICS he equtions of motion fo point P will be nlyzed in two diffeent efeence systems. One efeence system is inetil, fixed to the gound, the second system is moving in the physicl spce nd the
More informationTopics for Review for Final Exam in Calculus 16A
Topics fo Review fo Finl Em in Clculus 16A Instucto: Zvezdelin Stnkov Contents 1. Definitions 1. Theoems nd Poblem Solving Techniques 1 3. Eecises to Review 5 4. Chet Sheet 5 1. Definitions Undestnd the
More informationr a + r b a + ( r b + r c)
AP Phsics C Unit 2 2.1 Nme Vectos Vectos e used to epesent quntities tht e chcteized b mgnitude ( numeicl vlue with ppopite units) nd diection. The usul emple is the displcement vecto. A quntit with onl
More informationProblem Set 3 SOLUTIONS
Univesity of Albm Deptment of Physics nd Astonomy PH 10- / LeCli Sping 008 Poblem Set 3 SOLUTIONS 1. 10 points. Remembe #7 on lst week s homewok? Clculte the potentil enegy of tht system of thee chges,
More informationChapter 6 Thermoelasticity
Chpte 6 Themoelsticity Intoduction When theml enegy is dded to n elstic mteil it expnds. Fo the simple unidimensionl cse of b of length L, initilly t unifom tempetue T 0 which is then heted to nonunifom
More informationExample 2: ( ) 2. $ s ' 9.11" 10 *31 kg ( )( 1" 10 *10 m) ( e)
Emple 1: Two point chge e locted on the i, q 1 = e t = 0 nd q 2 = e t =.. Find the wok tht mut be done b n etenl foce to bing thid point chge q 3 = e fom infinit to = 2. b. Find the totl potentil eneg
More informationPotential Energy. The change U in the potential energy. is defined to equal to the negative of the work. done by a conservative force
Potential negy The change U in the potential enegy is defined to equal to the negative of the wok done by a consevative foce duing the shift fom an initial to a final state. U = U U = W F c = F c d Potential
More information10 Statistical Distributions Solutions
Communictions Engineeing MSc - Peliminy Reding 1 Sttisticl Distiutions Solutions 1) Pove tht the vince of unifom distiution with minimum vlue nd mximum vlue ( is ) 1. The vince is the men of the sques
More information2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum
2. Electostatics D. Rakhesh Singh Kshetimayum 1 2.1 Intoduction In this chapte, we will study how to find the electostatic fields fo vaious cases? fo symmetic known chage distibution fo un-symmetic known
More informationProf. Anchordoqui Problems set # 12 Physics 169 May 12, 2015
Pof. Anchodoqui Poblems set # 12 Physics 169 My 12, 2015 1. Two concentic conducting sphees of inne nd oute dii nd b, espectively, cy chges ±Q. The empty spce between the sphees is hlf-filled by hemispheicl
More information( ) ( ) Physics 111. Lecture 13 (Walker: Ch ) Connected Objects Circular Motion Centripetal Acceleration Centripetal Force Sept.
Physics Lectue 3 (Wlke: Ch. 6.4-5) Connected Objects Cicul Motion Centipetl Acceletion Centipetl Foce Sept. 30, 009 Exmple: Connected Blocks Block of mss m slides on fictionless tbletop. It is connected
More informationChapter 24. Gauss s Law
Chpte 24 Guss s Lw CHAPTR OUTLIN 24.1 lectic Flux 24.2 Guss s Lw 24.3 Appliction of Guss s Lw to Vious Chge Distibutions 24.4 Conductos in lectosttic uilibium 24.5 Foml Deivtion of Guss s Lw In tble-top
More information13.5. Torsion of a curve Tangential and Normal Components of Acceleration
13.5 osion of cuve ngentil nd oml Components of Acceletion Recll: Length of cuve '( t) Ac length function s( t) b t u du '( t) Ac length pmetiztion ( s) with '( s) 1 '( t) Unit tngent vecto '( t) Cuvtue:
More informationDEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING FLUID MECHANICS III Solutions to Problem Sheet 3
DEPATMENT OF CIVIL AND ENVIONMENTAL ENGINEEING FLID MECHANICS III Solutions to Poblem Sheet 3 1. An tmospheic vote is moelle s combintion of viscous coe otting s soli boy with ngul velocity Ω n n iottionl
More informationDYNAMICS. Kinetics of Particles: Newton s Second Law VECTOR MECHANICS FOR ENGINEERS: Ninth Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr.
Ninth E CHPTER VECTOR MECHNICS OR ENGINEERS: DYNMICS edinnd P. ee E. Russell Johnston, J. Lectue Notes: J. Wlt Ole Texs Tech Univesity Kinetics of Pticles: Newton s Second Lw The McGw-Hill Copnies, Inc.
More informationForce and Work: Reminder
Electic Potential Foce and Wok: Reminde Displacement d a: initial point b: final point Reminde fom Mechanics: Foce F if thee is a foce acting on an object (e.g. electic foce), this foce may do some wok
More informationChapter 22 The Electric Field II: Continuous Charge Distributions
Chpte The lectic Field II: Continuous Chge Distibutions Conceptul Poblems [SSM] Figue -7 shows n L-shped object tht hs sides which e equl in length. Positive chge is distibuted unifomly long the length
More informationSPA7010U/SPA7010P: THE GALAXY. Solutions for Coursework 1. Questions distributed on: 25 January 2018.
SPA7U/SPA7P: THE GALAXY Solutions fo Cousewok Questions distibuted on: 25 Jnuy 28. Solution. Assessed question] We e told tht this is fint glxy, so essentilly we hve to ty to clssify it bsed on its spectl
More informationElectrostatics. 3) positive object: lack of electrons negative object: excess of electrons
Electostatics IB 12 1) electic chage: 2 types of electic chage: positive and negative 2) chaging by fiction: tansfe of electons fom one object to anothe 3) positive object: lack of electons negative object:
More informationCharge in a Cavity of Conductor
Tdy s Pln Electic Ptentil Enegy (mesued in Jules Electic Ptentil Ptentil Enegy pe unit Chge (mesued in Vlts). Recll tht the electic field E is fce F pe unit chge. Cpcitnce BB Chge in Cvity f Cnduct A pticle
More informationELECTROSTATICS. JEE-Physics ELECTRIC CHARGE
J-Physics LCTIC CHAG LCTOSTATICS Chge is the popety ssocited with mtte due to which it poduces nd epeiences electicl nd mgnetic effects. The ecess o deficiency of electons in body gives the concept of
More informationCourse Updates. Reminders: 1) Assignment #8 available. 2) Chapter 28 this week.
Couse Updtes http://www.phys.hwii.edu/~vne/phys7-sp1/physics7.html Remindes: 1) Assignment #8 vilble ) Chpte 8 this week Lectue 3 iot-svt s Lw (Continued) θ d θ P R R θ R d θ d Mgnetic Fields fom long
More informationof Technology: MIT OpenCourseWare). (accessed MM DD, YYYY). License: Creative Commons Attribution- Noncommercial-Share Alike.
MIT OpenouseWe http://ocw.mit.edu 6.1/ESD.1J Electomgnetics nd pplictions, Fll 25 Plese use the following cittion fomt: Mkus Zhn, Eich Ippen, nd Dvid Stelin, 6.1/ESD.1J Electomgnetics nd pplictions, Fll
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationHomework 3 MAE 118C Problems 2, 5, 7, 10, 14, 15, 18, 23, 30, 31 from Chapter 5, Lamarsh & Baratta. The flux for a point source is:
. Homewok 3 MAE 8C Poblems, 5, 7, 0, 4, 5, 8, 3, 30, 3 fom Chpte 5, msh & Btt Point souces emit nuetons/sec t points,,, n 3 fin the flux cuent hlf wy between one sie of the tingle (blck ot). The flux fo
More informationWeek 8. Topic 2 Properties of Logarithms
Week 8 Topic 2 Popeties of Logithms 1 Week 8 Topic 2 Popeties of Logithms Intoduction Since the esult of ithm is n eponent, we hve mny popeties of ithms tht e elted to the popeties of eponents. They e
More informationChapter 2. Review of Newton's Laws, Units and Dimensions, and Basic Physics
Chpte. Review of Newton's Lws, Units nd Diensions, nd Bsic Physics You e ll fili with these ipotnt lws. But which e bsed on expeients nd which e ttes of definition? FIRST LAW n object oves unifoly (o eins
More informationGet Solution of These Packages & Learn by Video Tutorials on EXERCISE-1
FEE Downlod Study Pckge fom website: www.tekoclsses.com & www.mthsbysuhg.com Get Solution of These Pckges & Len by Video Tutoils on www.mthsbysuhg.com EXECISE- * MAK IS MOE THAN ONE COECT QUESTIONS. SECTION
More information