Electoagetic Theoy Pof Ruiz, UNC Asheville, doctophys o YouTube Chapte M Notes Laplace's Equatio M Review of Necessay Foe Mateial The Electic Potetial Recall i you study of echaics the usefuless of the potetial Hee is how it aises i the siplest case of costat gavity Pic a ass up ad ove it upwad a distace h The the wo that the lifte ust apply is h W = gdz = gh Holdig the ass up thee, we have potetial eegy gh If you dop the ball ad let it fall though h, gavity does the wo ow taslatig the eegy ito ietic eegy gh = v The potetial is the potetial eegy pe uit ass: V = gh We ca expess this potetial as a fuctio of z: V ( z = gz The foce of gavity is dow, so we wite Note that the gavitatioal field F = g d( gz dv ( z g = g = = dz dz This is egative the gadiet of the potetial I geeal we ca wite g = V Aalogous to this we defie the electic potetial V such that The fist Maxwell Equatio E ρ E = ε = V Michael J Ruiz, Ceative Coos Attibutio-NoCoecial-ShaeAlie 3 Upoted Licese leads us to ρ V = This is called ε Poisso's equatio If the desity is zeo, this educes to Laplace's equatio
Eule's Foula (This is Ou Jewel Physicist Richad Feya, i his faous lectues o physics deliveed at Caltech ad available i boo fo, states that the Eule Foula is the ost eaable foula i atheatics! He states "This is ou jewel" Stat with z = cosθ + i siθ, whee i = PM (Pactice Poble Show that ultiplyig the value of ay poit o the cicufeece of the cicle i the figue taes you to a poit 9 couteclocwise Note by efeece to the figue that This leads to e iθ, which gives the "jewel" e iθ dz iz dθ = = cosθ + i siθ Now coes the secet to coectig i, π ad e It is though ou study of the uit cicle i the coplex plae (above figue Let θ = π i the Eule Foula e iθ = cosθ + i siθ : i e π = This is equivalet to the followig agic foula, called Eule's Idetity i e π + = This is ou opal Michael J Ruiz, Ceative Coos Attibutio-NoCoecial-ShaeAlie 3 Upoted Licese
The Rhoades-Robiso Gigebead House (4 To see the full Gigebead Albu, go to: http://wwwjtuizco/gigebeadphp You ight ejoy eadig the explaatios I pepaed thee fo the sat high school studet PM (Pactice Poble Fo show that e iθ = cosθ + i siθ iθ iθ e + e cosθ = ad siθ e iθ e i iθ = Michael J Ruiz, Ceative Coos Attibutio-NoCoecial-ShaeAlie 3 Upoted Licese
3 Two Tig Itegals We Will Need I this sectio ad ae ubes,, 3, etc π π π dz I = si ( θ dθ = si ( z = si ( z dz Itegatig ove a whole ube of π itevals, it does't atte if we ae itegatig the cosie o sie Note: This is tue i ou case due to the itegal ultiple of π π π I = cos z si z dz dz + = = π = π Now fo ou secod itegal coside the followig whee π = I si( θ si( θ dθ Sice iθ iθ iθ iθ We use si( si( e e e e θ θ = i i, all itegals have the fo whee p below is a ozeo itege π ipθ π π ipθ e e dx = = [ cos( pθ + i si( pθ ] = ip ip The cosie pat gives cos( π p cos( = = sice p is a whole ube ad p is a eve ube This taes us aoud the cicle p ties, always givig fo the cosie Fo the sie, si( π p = always Theefoe, π si( θ si( θ dθ = πδ This is efeed to as the othogoality of sie fuctios Michael J Ruiz, Ceative Coos Attibutio-NoCoecial-ShaeAlie 3 Upoted Licese
M Laplace's Equatio i Cylidical Coodiates This poble is oe of the ichest oes you will eve ecoute i you physics couses Thee is a aazig aout of atheatical physics hee The souce fo this avelous poble coes fo the supeb text Matheatical Physics by Eugee Butov (Addiso-Wesley, Readig, MA, 968 Hee ae igediets i ou aalysis Laplace's Equatio Sepaatio of Vaiables Solvig a Secod-Ode Diffeetial Equatio The Method of Fobeius (Powe Seies Solutio Bouday Coditios (Bouday-Value Poble Othogoality of Fuctios A Supise Edig! Coside a "Log Hollow Cylide Shell" with Potetial V ( R, θ, z such that o the shell ( = R, V ( R, θ, z = Vo fo < θ < π ad V ( R, θ, z = Vo fo π < θ < Iagie a eaby cicuit, whee we coect a wie at 9 volts to the top pat of the cylidical shell ad -9 volts to the botto Note the sall isulatio ods Michael J Ruiz, Ceative Coos Attibutio-NoCoecial-ShaeAlie 3 Upoted Licese
We wish to solve Laplace's equatio sice thee is o chage desity iside V V V V = + + = ( θ z Sice we have cylidical syety, ou potetial will ot be a fuctio of z Theefoe we have V (, θ ad the z-deivative is zeo This leaves us with the siple V V ( + = θ We use the ethod called sepaatio of vaiables We set ou solutio to a poduct of two fuctios, oe fo each of the idepedet vaiables V (, θ = f ( g( θ [ f ( g( θ ] [ f ( g( θ ] ( + = θ d df d g g f d + = d dθ The tic ow is to divide by fg so we ca stat to sepaate "-stuff" fo "theta-stuff" d df d g f d + = d g dθ Now clea the so that each pat has oly oe vaiable d df d g f d + = d g dθ Each of these pieces ust be a costat sice that is the oly way to get zeo all the tie Reebe that ad θ ae vaiables that ae idepedet of each othe d g g dθ = λ Michael J Ruiz, Ceative Coos Attibutio-NoCoecial-ShaeAlie 3 Upoted Licese
d g dθ = λg λ >, the solutios ae expoetials: e λθ ad e λθ Fo But this ca't be sice the bouday coditios equie that if we ove alog the agle a full cicle we have to coe bac to whee we stated This coditio is stated atheatically as g( θ + π = g( θ λ < This bouday coditio equies sies ad cosies, which we ca get if It is coveiet to defie λ = so that ou diffeetial equatio becoes d g g dθ = with solutio g( θ a cos( θ b si( θ = + The bouday coditio g( θ + π = g( θ equies =,,, 3, The adial equatio is d f d df + λ = d, ie, d f d df d = Wite this as d d df = d f Fist coside the = case The, d d df = d, which gives df d + = d d f, usig the poduct ule fo diffeetiatio Woig with the fist deivative f ' df d =, this diffeetial equatio becoes Michael J Ruiz, Ceative Coos Attibutio-NoCoecial-ShaeAlie 3 Upoted Licese
df ' f ' + = d Sice is abitay, what's iside the bacets ust vaish: f df ' ' + = d Theefoe we have the followig df ' f d = ' df f ' ' d = Itegatig, we fid l f ' = l + l C, whee we wite ou costat as log of a costat The above equatio ca be expessed as l f ' = l C ad fially as f ' C = PM3 (Pactice Poble Show that f = C l + D, whee d is a costat This solutio is uacceptable sice we have a blow-up at = as the l( us off to egative ifiity Theefoe, thee is o physical solutio fo = This leaves us with d d df = d f ad g( θ = a cos( θ + b si( θ whee =,, 3, Michael J Ruiz, Ceative Coos Attibutio-NoCoecial-ShaeAlie 3 Upoted Licese
Usig the poduct ule of diffeetiatio i the adial equatio leads to df + f = d d d f, ie, d f df + f = d d This is Eule's diffeetial equatio " + ' = f f f The classic "Method of Fobeius" is to loo fo a solutio i tes of a powe seies f ( Plug ito you equatio f(, f'(, ad f"( = = c = ad = f '( c f "( = ( c = Note that oe deivative ills the fist te (the costat ad two deivatives ill off the fist two tes So we stat at ad at i the above equatios But otice that we ca stat the with = ayway sice the factos give zeo fo us ayway The esult is the followig = = = ( c + c c = We ae vey lucy ad aive at a vey siple esult below This does ot happe i geeal as the powes of ae usually diffeet i the vaious sus ( c + c c = = = = c = ( + = Michael J Ruiz, Ceative Coos Attibutio-NoCoecial-ShaeAlie 3 Upoted Licese
Sice is abitay ad ca be chose at will, the pat iside the bacets ust vaish to ae the equatio tue at all ties ( + = This leads to + =, which gives =, =, ad = ± Thee ae oly two values fo This leads us to solutios ad The secod solutio blows up at = So we eep the fist oe The geeal solutio ca be costucted fo the poduct of the fuctios f ( = c ad g( θ = a cos( θ + b si( θ We ca absob the c costat ito a ad b, witig [ ] f ( g( θ = a cos( θ + b si( θ Reebe that =,, 3, Theefoe, the geeal solutio is [ ] V (, θ = a cos( θ + b si( θ = But we have ot exploited all the bouday coditios As appoaches R, we appoach +V fo agles fo to π ad V fo agles fo π to π Theefoe V (, θ = V (, θ, ie, the potetial is a odd fuctio Theefoe we oly have the sie fuctios I othe wods a = = = V (, θ b si( θ Michael J Ruiz, Ceative Coos Attibutio-NoCoecial-ShaeAlie 3 Upoted Licese
What eais is to fid the b coefficiets i We do this by focusig ou attetio o ou solutio at the shell, ie, = R = = V ( R, θ R b si( θ We ultiply both sides by si( x ad itegate π π si( θ V ( R, θ dθ = si( θ R b si( θ dθ We wo o both sides as we go Sice V is a odd fuctio ad sie is odd, the poduct i the itegad o the left side is eve So we ca itegate ove half the agle age ad iclude a facto of Fo the ight side, ou othogoality elatio ics i = π π = = si( θ V dθ R b si( θ si( θ dθ π = = V si( θ dθ R b πδ π cos( θ V = R b π Note that cos( oddπ = ad cos( eveπ = + Theefoe, oly the odd cases ae ozeo We fid We wat to solve fo the b coefficiets ( V = R b π, whee is odd 4V = R b π Michael J Ruiz, Ceative Coos Attibutio-NoCoecial-ShaeAlie 3 Upoted Licese
b = R 4V π Substitutig the b coefficiets i V (, θ = b si( θ, we get the followig =,3,5, 4V V (, θ = si( θ =,3,5, R π V (, θ = 4V si( θ π =,3,5, R Ad ow we each the supise The liitig case = R gives us V ( R, θ 4V si( θ = π =,3,5, What's this? It's the Fouie Seies fo a Squae Wave! Chec out the gaph of ou potetial o the cylidical shell (see left figue It's a squae wave V R θ 4V si(3 θ si(5 θ θ π 3 5 (, = si + + + With sleight of had, ou solutio leads to the Fouie seies of a squae wave as a bypoduct This is such a ich poble! PM4 (Pactice Tae the gadiet to fid the electic field: E = V (, θ Michael J Ruiz, Ceative Coos Attibutio-NoCoecial-ShaeAlie 3 Upoted Licese