n Power transmission, X rays, lightning protection n Solid-state Electronics: resistors, capacitors, FET n Computer peripherals: touch pads, LCD, CRT

Transcription

1 .. Cu-Pl, INE 45- Electmagnetics I Electstatic fields anda Cu-Pl, Ph.. INE 45 ch 4 ECE UPM Maagüe, P me applicatins n Pwe tansmissin, X as, lightning ptectin n lid-state Electnics: esists, capacits, FET n Cmpute peipheals: tuch pads, C, CT n Medicine: electcadigams, electephalgams, mniting ee actiit n gicultue: seed sting, mistue cntent mniting, spinning cttn, n t: spa painting n We will stud Electic chages: n Culmb's aw- n Use when chage distibutin is knwn n Gauss s aw Ψ k F n Use when chage distibutin is smmetical n Electic Ptential (uses scala, nt ects) n Use when ptential V is knwn d V ( ) Culmb s aw (785) n Fce ne chage eets n anthe whee k 9 9 k / ε k F *upepsitin applies Pint chages Fce that eets n Fce with diectin F Nte: Obseatin pint ges Fist Eample Eample: Pint chages 5nC and -nc ae lcated at (,,4) and (-,,5), espectiel. a) Find the fce n a nc pint chage,, lcated at (,-,7) ppl supepsitin: F ( ) ( ) 5(,, ) F 9 ( 9 9) ( 4,, ) 56. (.4,.85,.998)nN Electical Engineeing, UPM (please pint n OTH sides f pape)

2 .. Cu-Pl, INE 45- Electmagnetics I Electic field intensit If we hae man chages n Is the fce pe unit chage when placed in the E field Eample: ame pint chages 5nC and -nc ae lcated at (,,4) and (-,,5), espectiel. b) Find the E field at (,-,7). E F E E F E ine chage densit, uface chage densit Vlume chage densit C/m C/m C/m ( ) ( ) E (.,.9,.4)V / m dl d d E pint chage The ttal E-field intensit is E E E Me Chage distibutins ( dl) ( d) ( d) Find E fm n Pint chage (we just saw this ne) n ine chage n uface chage n Vlume chage ine chage heet chage esults Peiew E πε We will deie these cases. Using Culmb E n ε. Using Gauss Vlume Chage E Find E fm INE chage n ine chage w/unifm chage densit, *use clindical cdinates (,, ) T dl (,,) de dl ' OT tan dl d' d' E sec cs sin cs sin Electical Engineeing, UPM (please pint n OTH sides f pape)

3 .. Cu-Pl, INE 45- Electmagnetics I efining angles and imagina pependicula line with the back imagina pependicula line with the fnt dl T de INE chage n ubstituting in: dl T d' E ' OT tan d' [ sec ] d sec cs sin E [ sec ]d [cs sec sin ] finite ine Chage: E [ (sin sin ) (cs cs ) ] infinite ine Chage ( E πε, 9 ) Me Chage distibutins n Pint chage n ine chage n uface chage n Vlume chage Find E fm uface chage n heet f chage w/ unifm densit d d Element f aea is: d dφd Obseatin pint is at -ais: dφd de ( ) h d de [ h ] [ h ] n ue t YMMETY the cmpnent cancels ut. UFCE chage E π E ε n E infinite uface Chage : hd π dφ φ h h [ h ] Me Chage distibutins n Pint chage n ine chage n uface chage n Vlume chage Electical Engineeing, UPM (please pint n OTH sides f pape)

4 .. Cu-Pl, INE 45- Electmagnetics I Find E fm Vlume chage n phee f chage w/ unifm densit, de φ θ P(,,) (,θ,φ ) d d d (Eq. *) iffeentiating (Eq. *) suies. aw f csines: ' 'csθ ' ' cs d ' d θ ' ' ue t smmet de de cs ' d de ' dθ ' dφ' d' nl Find E fm Vlume chage n ubstituting de θ φ P(,,) e dnde salen ls limites de? (,θ,φ ) d de d π a ' ˆ cs a ' dθ ' dφ' d' d ' d θ ' ' ' ' cs d ' E ' ' d' 4 dφ 4 πε E ' φ ' ' πa P.E. 4.5 n squae plate at plane and ±, ± caies a chage mc/m. Find the ttal chage n the plate and the electic field intensit at (,,). d d d d () d 4 () 9mC E s d ' sd -- 4 πε ' ' (,,) ( ', ',) ( ', ',) E Cnt dd (,,) ( ) / 6 ( ) ( ) 8 / / ( ) dd ue t smmet nl E suies: E 6.5 MV / m sheet f chage dd dd 6 8 ( ) / dd / n Culmb's aw- Chapte Outline n Use when chage distibutin is knwn n Gauss s aw Ψ k F n Use when chage distibutin is smmetical n Electic Ptential (uses scala, nt ects) n Use when ptential V is knwn d V ( ) Electic Flu ensit is independent f the medium in which the chage is placed. ε E d 4π [C / m ] Then the electic flu is: Ψ d [C] Electical Engineeing, UPM (please pint n OTH sides f pape) 4

5 .. Cu-Pl, INE 45- Electmagnetics I Theefe: Ψ Gauss s aw d d d d d This is the st f the Mawell s equatins deied hee." n The ttal electic flu Ψ, thugh an clsed suface is equal t the ttal chage lsed b that suface. Gauss s aw d ε E 4π Ψ d d The ke is t chse the Gauss suface t simplif the pblem." Fllw the smmet f the paticula case. " Pick suface s that is me eamples: Finding at pint P fm the chages: chage P ˆ a 4π Pint Chage is at the igin. d n Chse a spheical d n Nte whee is pependicula t this suface. d 4π me eamples: Finding at pint P fm the chages: ine chage P a ˆ π Infinite ine Chage dl d l n Chse a clindical d n Nte that integal at tp and bttm sufaces f clinde d πl sheet f chage me eamples: Find at pint P fm the chages: ea n Infinite heet f chage d d s n Chse a clindical b cutting the sheet s d d tp bttm Nte that is paallel t the sides f the b. [ ] s P.E. 4.7 pint chage f nc is lcated at the igin, while plane caies chage nc/m. Find at (, 4, ) 4π 9 s 4π n ( ) [ (,4,) (,,) ] 4 4π 9 ( 5) (,4,) 5n nc/m n Electical Engineeing, UPM (please pint n OTH sides f pape) 5

6 .. Cu-Pl, INE 45- Electmagnetics I P.E. 4.8 ( ) 4 If C/m. Find : n lume chage densit at (-,,) (,,) 4 4C/m n Flu thu the cube defined b,, Ψ d n Ttal chage lsed b the cube Ψ C 4ddd Pint chage lume Chage distibutin ine chage distibutin heet chage distibutin eiew a ˆ π a 4π n ˆ We will stud Electic chages: n Culmb's aw (geneal cases) n Gauss s aw (smmetical cases) n Electic Ptential (uses scala, nt ects) Electic Ptential, V n The wk dne t me a chage fm t is dw F dl E dl n The (-) means the wk is dne b an etenal fce. n The ttal wk ptential eneg equied in ming : W E dl n The eneg pe unit chage ptential diffee between the pints: W J V [ V] C V is independent f the path taken. The Ptential at an pint is the ptential diffee between that pint and a chsen efee pint at which the ptential is e. (chsing infinit): V ( ) d'ˆ a ' F man Pint chages at k : (appl supepsitin) F ine Chages: F uface chages: ' V ( ) V (ˆ) n k ˆ ˆ k k ( ˆ' ) πε ˆ ˆ' 4 [ V] dl' s( ˆ' ) d' V (ˆ) ˆ ˆ' F Vlume chages: ( ˆ' ) d' V (ˆ) ˆ ˆ' [ V] P.E. 4. V ( ) k 4 pint chage f -4µC is lcated at (,-,) pint chage f 5µC is lcated at (,4,-) pint chage f µc is lcated at the igin ssume V() and Find the ptential at (-, 5, ) πε k 6 V (,5,) / 9 k C (,5,) (,,) (,5,) (,4, ) (,5,) (,,) kv Electical Engineeing, UPM (please pint n OTH sides f pape) 6

7 .. Cu-Pl, INE 45- Electmagnetics I Eample line chage f 5nC/m is lcated n line, ssume V(,,) and Find the ptential at (,, 5) V ( ˆ) πε V ( ˆ) ln C πε V igin V V 4.8 πε [ ln ln ] d (,,)-(,,).6 and (,,5)-(,,). V 4.8V P.E. 4. UIZ #: pint chage f 7nC is lcated at the igin V(,,-5)V and Find C V C P.E. 4. pint chage f 5nC is lcated at the igin V(,6,-8)V and Find the ptential at (-,, 6) Find the ptential at (,5,7), the ptential diffee V (,,) V C 5n 4 πε ( ) (,6, 8) C C.5 5n V C. 9 V 4 πε (,,6) (,,) 5n V V 4 πε (,5,7) (,,) V V V. V V V elatin between E and V V is independent f the path taken. V V *Est aplica sól a camps estátics. ignifica que n ha tabaj NETO en me una caga en un pas cead dnde haa un camp estátic E. ( E) d tatic E satisfies: E Cnditin f Cnseatie field independent f path f integatin dv E dl E d E d E d V V V dv d d d E V Eample Gien the ptential V csφ, π, Find at. ε E ε V In spheical cdinates: ( ) V V V E θ φ θ φ V E csφ csθ csφ θ φ φ ε E ε θ (, π /,) 8.ˆ a C / m φ Electical Engineeing, UPM (please pint n OTH sides f pape) 7

8 .. Cu-Pl, INE 45- Electmagnetics I Gien that E( )a a kv/m, find the P.E. 4. wk dne in ming a -µc chage fm (,5,) t (,-,) b taking the staight-line path. a) (,5,) (,5,) (,-,) W ( ) d d b) 5- W [ ] W ( ) d d 5 W ( )(8 ) W 6 ( µ ) mj [( 5 ) ( )] d d ( 6 5) W d d d W 8 6 W mj d - Electic iple n Is fmed when pint chages f equal but ppsite sign ae sepaated b a small distance. P V F fa awa bseatin pints (>>d): d csθ V Eneg ensit in Electstatic fields n It can be shwn that the ttal electic wk dne is: W E ε Ed E d Electical Engineeing, UPM (please pint n OTH sides f pape) 8