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 E is negve of delt V! I don't like tht we wee told to use gdients when we hven't even done them in mth yet.! So we just need to supeimpose the dil field lines which e found by tking the negve of the gdient of the electic potenl in 3d ctesin/spheicl/ cylindicl coodinte system nd e pependicul to euipotenls, the locus point of ll point with the sme potenl diffeence. Simple enough... We'll just do tht!.... Ohhhhh wit... WHAT?.! so electic potenl is the ndeivve of electic field is tht coect? in othe wods the e unde electic field funcon gives the electic potenl?! Wht is the diffeence between n integl of dot poduct nd integl of simple poduct?! Cn you plese explin the Gdient in diffeent Coodinte systems? Electicity & Mgnesm Lectue 6, Slide
Big Ide Lst me we defined the electic potenl enegy of chge in n electic field: b ΔU b = F dl b = E dl The only menon of the pcle ws though its chge. We cn obtin new unty, the electic potenl, which is PROPERTY OF THE SPACE, s the potenl enegy pe unit chge. ΔV b ΔU b Note the simility to the definion of nothe unty which is lso PROPERTY OF THE SPACE, the electic field. v E F = b E d l Electicity & Mgnesm Lectue 6, Slide 3
Electic Potentil fom E field Conside the thee points A, B, nd C locted in egion of constnt electic field s shown. D Δx Wht is the sign of ΔV AC = V C - V A? A) ΔV AC < B) ΔV AC = C) ΔV AC > Remembe the definion: ΔV A C = Choose pth (ny will do!) ΔV A C = D A E d l C D E d l C A E d l ΔV A C = C E d l = EΔx < D Electicity & Mgnesm Lectue 6, Slide
CheckPoint: Zeo Electic Field Suppose the electic field is zeo in cetin egion of spce. Which of the following sttements best descibes the electic potenl in this egion? A. The electic potenl is zeo eveywhee in this egion. B. The electic potenl is zeo t lest one point in this egion. C. The electic potenl is constnt eveywhee in this egion. D. Thee is not enough infomon given to disnguish which of the bove nswes is coect. Remembe the definion ΔV A B = B A E d l E = ΔV = A B V is constnt! Electicity & Mgnesm Lectue 6, Slide 5
E fom V If we cn get the potenl by integng the electic field: ΔV b = b E d l We should be ble to get the electic field by diffeenng the potenl? In Ctesin coodintes: E x = dv dx E y = dv dy E z = dv dz v E = V Electicity & Mgnesm Lectue 6, Slide 6
CheckPoint: Sptil Dependence of Potentil The electic potenl in cetin egion is ploded in the following gph At which point is the mgnitude of the E-FIELD getest? o A o B o C o D How do we get E fom V? v E = V E x = V dx Look t slopes! Electicity & Mgnesm Lectue 6, Slide 7
CheckPoint: Sptil Dependence of Potentil The electic potenl in cetin egion is ploded in the following gph At which point is the diecon of the E- field long the negve x-xis? o A o B o C o D At B, the slope is decesing (-) so the diection of the E field is negtive E is negtive when the slope of V is positive (E=-dV/dx). Theefoe E is diected long the x-xis t point C. How do we get E fom V? v E = V E x = dv dx Look t slopes! Electicity & Mgnesm Lectue 6, Slide 8
Euipotentils Euipotenls e the locus of points hving the sme potenl. Euipotentils poduced by point chge Euipotenls e ALWAYS pependicul to the electic field lines. The SPACING of the euipotenls indictes The STRENGTH of the electic field. Electicity & Mgnesm Lectue 6, Slide 9
Contou Lines on Topogphic Mps Electicity & Mgnesm Lectue 6, Slide
Visulizing the Potentil of Point Chge Electicity & Mgnesm Lectue 6, Slide
CheckPoint: Electic Field Lines The field-line epesenton of the E-field in cetin egion in spce is shown below. The dshed lines epesent euipotenl lines. At which point in spce is the E-field the wekest? o A o B o C o D The electic field lines e the lest dense t D Fom wht I know, the nswe should be D D is whee the electic field lines e the lest dense I m petty sue the electic field lines e the lest dense t D I d guess D Electicity & Mgnesm Lectue 6, Slide
CheckPoint: Electic Field Lines The field-line epesenton of the E-field in cetin egion in spce is shown below. The dshed lines epesent euipotenl lines. Compe the wok done moving negve chge fom A to B nd fom C to D. Which one euies moe wok? A. Moe wok is euied to move negve chge fom A to B thn fom C to D B. Moe wok is euied to move negve chge fom C to D thn fom A to B C. The sme mount of wok is euied to move negve chge fom A to B s to move it fom C to D D. Cnnot detemine without pefoming the clculon Electicity & Mgnesm Lectue 6, Slide 3
Clicke Question: Electonic Field Wht e these? ELECTRIC FIELD LINES! Wht e these? EQUIPOTENTIALS! Wht is the sign of W AC = wok done by E field to move negve chge fom A to C? A) W AC < B) W AC = C) W AC > A nd C e on the sme euipotenl W AC = Euipotenls e pependicul to the E field: No wok is done long n euipotenl Electicity & Mgnesm Lectue 6, Slide
CheckPoint Results: Electic Field Lines The field-line epesenton of the E-field in cetin egion in spce is shown below. The dshed lines epesent euipotenl lines. Compe the wok done moving negve chge fom A to B nd fom C to D. Which one euies moe wok? A. Moe wok is euied to move negve chge fom A to B thn fom C to D B. Moe wok is euied to move negve chge fom C to D thn fom A to B C. The sme mount of wok is euied to move negve chge fom A to B s to move it fom C to D D. Cnnot detemine without pefoming the clculon! A nd C e on the sme euipotenl! B nd D e on the sme euipotenl! Theefoe the potenl diffeence between A nd B is the SAME s the potenl between C nd D Electicity & Mgnesm Lectue 6, Slide 5
CheckPoint: Electic Field Lines 3 The field-line epesenton of the E-field in cetin egion in spce is shown below. The dshed lines epesent euipotenl lines. Compe the wok done moving negve chge fom A to B nd fom A to D. Which one euies moe wok? A. Moe wok is euied to move negve chge fom A to B thn fom A to D B. Moe wok is euied to move negve chge fom A to D thn fom A to B C. The sme mount of wok is euied to move negve chge fom A to B s to move it fom A to D D. Cnnot detemine without pefoming the clculon Electicity & Mgnesm Lectue 6, Slide 6
3 coss-secon Clcultion fo Potentil Q Point chge t cente of concentic conducng spheicl shells of dii,, 3, nd. The inne shell is unchged, but the oute shell cies chge Q. metl Wht is V s funcon of? metl Conceptul Anlysis:! Chges nd Q will cete n E field thoughout spce! V () = E d l Sttegic Anlysis:! Spheicl symmety: Use Guss Lw to clculte E eveywhee! Integte E to get V Electicity & Mgnesm Lectue 6, Slide 7
3 coss-secon Clcultion: Quntittive Anlysis 5 metl metl Q > : Wht is E()? A) B) C) πε Q D) E) πε Q πε πε Q Q Why? Guss lw: E d A = Q enclosed ε Eπ E Q = ε = πε Q Electicity & Mgnesm Lectue 6, Slide 8
3 Clcultion: Quntittive Anlysis coss-secon Q 3 < < : Wht is E()? A) B) πε C) πε metl D) πε E) πε Q metl Applying Guss lw, wht is Q enclosed fo ed sphee shown? A) B) - C) How is this possible? - must be induced t = 3 sufce chge t = sufce = Q σ 3 Q = = π 3 π σ Electicity & Mgnesm Lectue 6, Slide 9
3 Clcultion: Quntittive Anlysis coss-secon metl metl Q Connue on in < < 3 : E = πε < < : E = πε To find V: ) Choose such tht V( ) = (usul: = infinity) ) Integte! > : Q V = πε 3 < < : A) < : E = B) C) V = Q V = = ΔV πε Q V = πε 3 ( ) Electicity & Mgnesm Lectue 6, Slide
> : 3 < < : Clcultion: Quntittive Anlysis metl metl Q 3 coss-secon Electicity & Mgnesm Lectue 6, Slide Q V = πε Q V = πε < < 3 : ) ( ) ( ) ( 3 V V V Δ = Δ!! " # $ $ % & = 3 ) ( Q V πε πε V () = πε Q 3 $ % & ' ( ) < < :!! " # $ $ % & = 3 ) ( Q V πε < < :!! " # $ $ % & = 3 ) ( Q V πε