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22 Pages

### Lecture_22_Part_2

Course: PHYS 272, Winter 2007
School: Purdue
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Word Count: 1149

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Flux Flux Electric through small area: r ^ flux ~ E &quot; n!A Definition of electric flux on a surface: r ^ ! E # n&quot;A surface surface r ^ ! E # n&quot;A r ^ ! E &quot; ndA r r ! E &quot; dA r dA r r electric flux on a closed surface = &quot; E ! dA Gauss's Law surface r ^ &quot; E \$ n#A = &quot;q inside !0 r ^ ! E # ndA = &quot;q inside \$0 Can derive one...

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Flux Flux Electric through small area: r ^ flux ~ E " n!A Definition of electric flux on a surface: r ^ ! E # n"A surface surface r ^ ! E # n"A r ^ ! E " ndA r r ! E " dA r dA r r electric flux on a closed surface = " E ! dA Gauss's Law surface r ^ " E \$ n#A = "q inside !0 r ^ ! E # ndA = "q inside \$0 Can derive one from another Gauss's law is more universal: works at relativistic speeds 1 Q E= 4!" 0 r 2 Gauss's law: If we know the field distribution on closed surface we can tell what is inside. 1. Knowing E can conclude what is inside 2. Knowing charges inside can conclude what is E Gauss's Law for Magnetism Dipoles: Electric field: `+' and `' charges can be separated Magnetic field: no monopoles Suppose magnetic dipole consists of two magnetic monopoles, each producing a magnetic field similar to the electric field. One cannot separate them total magnetic `charge' is zero. surface r ^ " E \$ n#A = "q inside !0 surface r ^ # B " n!A = 0 or r ^ # B " n!A = 0 Gauss's law for magnetism Ampre's Law r r " B # dl = 0 ! I inside _ path All the currents in the universe contribute to B but only the ones inside the path result in nonzero path integral Ampere's law is almost equivalent to the Biot-Savart law: but Ampere's law is relativistically correct Inside the Path Ampere's law r r " B # dl = 0 ! I inside _ path 1. 2. 3. 4. Choose the closed path Imagine surface (`soap film') over the path r r Walk counterclockwise around the path adding up B " dl ! Count upward currents as positive, inward going as negative !I inside _ path = I1 " I 2 !I inside _ path = I up = I up ! I down + I up Ampere's Law: A Long Thick Wire r r " B # dl = 0 ! I inside _ path Can B have an out of plane component? Is it always parallel to the path? r r # B " dl = B2!r B 2!r = 0 I for thick wire: 0 2 I B= 4! r (the same as for thin wire) Would be hard to derive using Biot-Savart law Ampere's Law: A Solenoid r r " B # dl = 0 ! I inside _ path Number of wires: (N/L)d r r What is B ! dl on sides? B outside is very small r r " B ! dl = Bd 0 IN (solenoid) Bd = 0 I ( N / L ) d L Uniform: same B no matter where is the path B= Ampere's Law: A Toroid r r " B # dl = 0 ! I inside _ path Symmetry: B || path B 2!r = 0 IN 0 2 NI B= 4! r Is magnetic field constant across the toroid? Maxwell's Equations Three equations: Gauss's law for electricity r ^ ! E # ndA = "q inside r Gauss's law for magnetism ^ " B ! ndA = 0 r r Ampere's law for magnetism " B # dl = 0 ! I inside _ path (incomplete) \$0 Is anything missing? r r `Ampere's law for electricity' E ! dl = 0 " Maxwell's Equations (incomplete) r ^ ! E # ndA = "q inside \$0 Gauss's law for electricity Gauss's law for magnetism Incomplete version of Faraday's law Ampere's law (Incomplete Ampere-Maxwell law) r ^ " B ! ndA = 0 r r " E ! dl = 0 r r " B # dl = 0 ! I inside _ path First two: integrals over a surface Second two: integrals along a path Incomplete: no time dependence Is Gauss's Law Relativistically Correct? r ^ ! E # ndA = "q inside \$0 Gauss's law for electricity Problem: If charges inside move, changes in E will be delayed. To avoid retardation problem: find a property of E at (x,y,z,t) relate it to source charges at (x,y,z,t) Divergence r ^ ! E # ndA = "q inside \$0 Gauss's law for electricity , where is density of charges r %"V ^ ! E # ndA = \$ 0 r ^ % \$ E ndA = " #V !0 `divergence': r div( E ) # lim scalar! r ^ \$ E " ndA !V !V %0 r ! div E = "0 ( ) Physical Meaning of Divergence r ^ ! E # ndA = "q inside \$0 Electric field directions `diverge' from a point r ! div E = "0 ( ) Electric field directions `anti-diverge' from a point Electric field flows through Is Divergence Relativistically Correct? r ^ ! E # ndA = "q inside \$0 Differential form of Gauss's law r ! div E = "0 ( ) Provides local relationship between charge and electric field E and are at the same location at the same time; that avoids the problem of relativistic retardation! Integral Form of Gauss's Law Is integral form of Gauss's relativistically correct? Divergence theorem: r relativistically ! r r div E = ^ correct "0 ! div E dV = ! E " ndA ( ) ( ) r ! div E dV = ( ) " ! # 0 dV Used divergence theorem (pure math) Got Gauss's law r # "dV ^ # E \$ ndA = ! 0 r Q ^ E " ndA = inside # !0 Gauss's law is relativistically correct (unlike Coulomb's law) Divergence as Partial Derivative r div( E ) # lim r ^ \$ E " ndA !V !V %0 r E2 !y!z " E1!y!z div( E ) = lim !V #0 !x!y!z r E2 " E1 !E div( E ) = lim = !V #0 !x !x r !E x !E y !E z + + If E has x, y and z components: div( E ) = !x !y !z r !Ex !Ey !Ez " div E = + + = !x !y !z # 0 ( ) differential form of Gauss's law Electric Potential and Gauss's Law r !Ex !Ey !Ez " + + = div E = !x !y !z # 0 ( ) !V Ex = " , !x !V Ey = " , !y !V Ez = " !z A boundary value problem \$ 2V \$ 2V \$ 2V " + 2 + 2 =# 2 \$x \$y \$z !0 Laplace's equation: ! 2V ! 2V ! 2V + 2 + 2 =0 2 !x !y !z (empty space) Example Problem In a certain region of space: V (x, y, z) = V0 + a x 2 + y 2 + z 2 What is the charge density in this region? Solution: ( ) \$ 2V \$ 2V \$ 2V " + 2 + 2 =# 2 \$x \$y \$z !0 !V = 2 !x 2 ! 2 ax 2 !x 2 " 6a = # !0 6a "=# !0 ( ) = ! ( 2ax ) = 2a !x Standard Notation r ! ! ! ^ Introduce del operator: " = ^ + ^ + k i j !x !y !z r r " #E x #E y #E z " 1) + + = \$#E = #x #y #z ! 0 !0 " ! ^ ! ^ ! ^% ^ E ^ E k = !Ex + !Ey + !Ez i+ j+ k ' Ex i+ y j+ z ^ \$ !x !y !z & # !x !y !z ( ) 2) r & 'V ^ 'V ^ 'V E = (\$ \$ 'x i + 'y j+ 'z % \$ 2V \$ 2V \$ 2V " + 2 + 2 =# 2 \$x \$y \$z !0 ^# k! ! " 3) divergence of a gradient (Laplacian) r r E = "! V gradient r r \$ ! " !V = # %0 \$ 2 ! V = "V = # %0 ( ) Is Ampere's Law Relativistically Correct? r r " B # dl = 0 ! I inside _ path Problem: If currents inside change, changes in B will be delayed. To avoid retardation problem: find a property of B at (x,y,z,t) relate it to current at (x,y,z,t) Differential Form of Ampere's Law r r " B # dl = 0 ! I inside _ path r ^ Current passing through tiny area: J " n!A r r r ^ # B " dl = 0 J " n!A r r r % B ! dl ^ curl B ! n " lim #A\$0 #A Differential form r r different of Ampere's law: curl B = 0 J notation A J n ( ) ( ) r r r " ! B = 0 J Note: it is a vector equation it is relativistically correct Gauss's Law r ^ ! E # ndA = "q inside \$0 Divergence theorem: r r ^ ! div E dV = ! E " ndA ( ) r ! div E = "0 ( ) Differential form of Gauss's law Provides local relationship between charge and electric field E and are at the same location at the same time; that avoids the problem of relativistic retardation! Divergence: r div( E ) # lim r ^ \$ E " ndA !V !V %0 r !Ex !Ey !Ez " div E = + + = !x !y !z # 0 ( )
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