Electromagnetism Physics 15b
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1 Electromagnetism Physics 15b Lecture #18 Maxwell s Equations Electromagnetic Waves Purcell What We Did Last Time Impedance of R,, and L Z R = V R = R Z I = V = 1 Z R I iω L = V L = iωl I L Generally a frequency-dependent complex number Low- and high-pass filters ut-off frequency ω cutoff = 1 or R ω = R cutoff L 1 Resonance of an RL circuit ω 0 = urrent amplitude peaks L Phase between voltage and current changes by 180º Quality factor Q = narrowness (steepness) of the resonance Q = ω 0 L R 1
2 Today s Goals Introduce displacement current The last element of Maxwell s equations omplete Maxwell s equations tudy Maxwell s eqns. in vacuum Derive wave equations Find a solution Electromagnetic waves James lerk Maxwell ( ) Incomplete Equations In Lecture #14, we got this set of equations There is a small problem E = 4πρ B = 0 E = 1 c B B = 4π c J B = 4π means div J must be zero c J harge conservation tells us div J = ρ ρ/ may not be zero if the system is time-dependent The above equations work only for stationary charge distributions 2
3 Fixing the Inconsistency omething must be done to B = 4π c J Try adding a vector F to the rhs What is F? Remember that div of the lhs is 0 4π c J + F = 0 Take time derivative of Gauss s law E = 4πρ ompare the rhs 1 c time deriv. F = 4π c J = 4π c ( E) = F F = 1 c ( E) = 4π ρ ρ We ve found the missing piece! Displacement urrent New-and-improved Ampère s Law: econd term can be seen as an additional current J d 1 B = 4π c J + J d 4π current The (obscure) name is historical Displacement current J d is not a real current It does not describe charges flowing through some region... but it acts like a real current Let s see how it fits in an example B = 4π c J + 1 c ( ) where Displacement 3
4 Displacement urrent onsider a charging capacitor I Apply Ampère to loop Fine, but this is supposed to hold for any that s bounded by hoose that intersects the capacitor J = 0 on Naive Ampère fails because +Q Q B ds J da = 0 I = dq = 4π c I = 4π c J da Displacement urrent E field between the plates is increasing with time I If the capacitor has an area A, Displacement current is J d = 1 = I 4π A Extended Ampère works E +Q Q E = 4πQ A B ds = 4π c I = dq de I J d da = A da = I = 4π A dq (J + J d ) da = 4πI A = 4π c I 4
5 More Generally For Ampère s Law to work consistently, must depend only on the border of Using charge conservation J da J d a Gauss s Law tells us E da E d a = 4πQ in J da J d a = 1 d 4π 4π c J + 1 4π da = J + 1 4π d a (J + J d ) da = dq in da da ( E da E d a ) omplete Maxwell s Equations In differential forms: E = 4πρ G B = 0 E = 1 B c B = 4π c J + 1 c E = ρ I ε 0 B = 0 E = B B = µ 0 J + µ 0 ε 0 Typo in Purcell 9.3 (15 ) Maxwell introduced the last term (in 1861) based purely on the argument of mathematical consistency 5
6 Integral Forms First two equations apply to any volume V enclosed by surface econd two apply to any surface bounded by a contour Φ E = 4πQ Φ B = 0 E ds = 1 dφ B c B ds = 4π c I + 1 c E ds G dφ E where Φ E Φ B Q I in the third equation is the emf on the loop V E da B da ρ dv J da Maxwell in Vacuum In vacuum, where ρ = 0 and J = 0, E = 4πρ E = 0 B = 0 B = 0 E = 1 B E = 1 B c c B = 4π c J + 1 B = 1 c c No source No field? If E and B are both changing with time, they can create each other It s like pulling yourself up with bootstraps? Let s try to solve these equations together lean and symmetric between E and B 6
7 olving Maxwell in Vacuum trategy: Decouple E and M E = 1 B c E Faraday ( ) = 1 c =0 in vacuum How about B? B ame except for 1/c ( E) = 1 2 E c 2 2 ( ) 2 E = 1 E 2 E c 2 2 B = 1 c 1 c ( B) = Ampère An E-only differential equation olving Maxwell in Vacuum Repeat, but this time try to eliminate E B = 1 c B Ampère ( ) = 1 c always 0 Now we must solve ame except for 1/c ( B) = 1 2 B c 2 2 ( ) 2 B = 1 B = 0 E = 1 B c 1 c ( E) = 2 B c 2 2 Faraday B A B-only differential equation 2 E = 1 2 E c 2 and 2 2 B = 1 2 B c 2 2 7
8 1-D Wave olutions uppose E(x,y,z,t) = E(x,t), i.e. no y and z dependence 2 E = 1 2 E c E x = 1 2 E 2 c 2 2 This can be satisfied if E(x,t) = f(x ± ct) lhs = f (x ± ct) = rhs = 1 c 2 (±c)2 f (x ± ct) What is f(x ± ct)? At t = 0, f(x) is an arbitrary vector function of x As t increases, f(x ± ct) moves along the x axis f(x + ct) moves toward negative x with velocity c f(x ct) moves toward positive x with velocity +c Waves propagating with the speed of light Electromagnetic Waves olutions to Maxwell s equations in vacuum are waves of E and B fields = electromagnetic waves Maxwell, based on the experimental data of the day, found the speed was m/s We can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena Electricity and magnetism were unified with optics In G, the speed comes out to be c This is because of the 1/c in Faraday and Ampère We checked that 1/c came naturally out of pecial Relativity NB: we only used oulomb + Relativity there 8
9 ummary omplete Maxwell s equations E = 4πρ B = 0 E = 1 c Displacement current consistency Electromagnetic waves Maxwell s eqns. in vacuum needed for mathematical olutions are waves propagating with speed of light which is light itself B J d = 1 4π B = 4π c J + 1 c 2 E = 1 2 E 2 B = 1 2 B c 2 2 c 2 2 9
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