Direct Current Circuits

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Eler urren (hrges n Moon) Eler urren () The ne moun of hrge h psses hrough onduor per un me ny pon. urren s defned s: Dre urren rus = dq d Eler urren s mesured n oulom s per seond or mperes. ( = /s) n sngle ru, he urren ny nsn s he sme one pon s ny oher pon. (hrge s onsered.) Dre urren rus Dre urren rus 2 Drf Veloy Drf Veloy The free elerons n onduors re n onsn rndom moon, so here s no ne flow of hrge n ny one prulr dreon. f sedy eler feld E s eslshed nsde onduor henhrged prles re sujeed o sedy fore F = E. The ne effe of he eler feld s h n ddon o he rndom moon of he hrged prles whn onduor, here s lso ery smll ne moon or drf of he mong hrged prles s group n he dreon of he eler fore. The hrges mong n onduor he freuen ollsons wh he msse nerly sonry ons of he merl. Ths moon s desred n erms of he drf eloy d of he prles. Dre urren rus 3 Dre urren rus 4 urren Densy Drf Veloy The urren per un ross-seon re s lled he urren densy J. J= Q = Ne d N = onenron of hrges ( m 3 ) Q = = Ne d Dre urren rus Ne d = J = Ne d The eler felde n erms of urren densy J s: E = J The onsn s lled he ressy of he merl. 5 Dre urren rus 6

Beres (emf) Shem Dgrms n order o produe n eler urren n ru, poenl dfferene s needed. Beres re one wy of prodng dfferene n poenl (lled eleromoe fore or emf). The dreon of urren s y onenon he dreon pose hrge moes hrough he ru, whh s owrds he nege ermnl of he ery. Dre urren rus 7 Dre urren rus 8 Ohm s Lw Georg Ohm (787-854) Found h he urren n mel wre s proporonl o he poenl dfferene V ppled o s ends. urren lso depends upon he onduy of he wre. s more ommon o lk ou ressne (nerse of onduy) nd express hs relonshp s: For power soures: Ohm s Lw For resse lods: = V or V = V = V V = V > V = V V = V < The un for ressne s lled he ohm nd s reed $ (omeg) Dre urren rus 9 Dre urren rus Termnl Volge The poenl dfferene n rel ery s no eul o he emf due o nernl ressne whn he ery. Ths lowers he olge lle o he ru. essne () nd essy () n e expermenlly deermned h he ressne of wre s drely proporonl o s lengh l nd nersely proporonl o s ross-seonl re. r V = r V s lled he ermnl olge of he ery. = l The proporonly onsn s lled he ressy nd depends upon he merl used for he wre. [=] $ m Dre urren rus Dre urren rus 2

Eler Power To fnd he power rnsformed y n eler dee rell h energy s Q%V. Power s he re energy s rnsformed n he dee or: P = QV = V The S un for power s J/s or w ( W = J/s). For ressors, omnng he oe wh Ohm s Lw resuls n: P = 2 = V 2 Mesurng Volge Volmeers re pled n prllel wh he pons eween whh he olge mesuremen s mde Volmeers he ery hgh ressne nd do no ffe he ru (hey drw ery smll urren) V V meer = V 3 Dre urren rus 3 Dre urren rus 4 Mesurng Volge Volmeers pled n seres wll lok he urren nd ree n open ru. = V Mesurng urren mmeers re pled n seres wh he dee hrough whh he urren mesuremen s mde mmeers he ery low ressne nd do no ffe he ru (he olge drop s ery low) V meer = E meer = 2 Dre urren rus 5 Dre urren rus 6 Mesurng urren essors n Seres mmeers pled n prllel wll ree shor ru. nyhng prllel o he meer wll he no olge ross hem nd herefore no urren V = V V 2 V 3 = = E = 2 = = V = ( ) V = V s Dre urren rus 7 Dre urren rus 8

essors n Seres urren s he sme hrough eh ressor nd s he sme s he urren n he eulen ressne Volge drop ross eh ressor s dfferen unless he ressne s he sme s = Dre urren rus 9 essors n Seres (Volge Dder) V = V = V 2 = = V = V 2 Dre urren rus 2 essors n Prllel essors n Prllel 2 3 = 2 3 = V = V V V p For ressors n prllel he eulen ressne s p = Dre urren rus 2 Volge drop s he sme ross eh ressor nd he sme s he olge drop ross he eulen ressne urren s dfferen hrough eh ressor, he hgher he ressne he lower he urren p = Dre urren rus 22 essors n Prllel (urren Dder) 2 = = = so nd V = & % ( $ ' = 2 & % ( 2 = V nd = & = V % ( $ ' $ ' Dre urren rus 23 Krhhoff s ules.) Junon ule (onseron of hrge) ny junon pon, he sum of ll urrens enerng he junon mus eul he sum of ll urrens leng he junon. 2 = 2 3 3 Dre urren rus 24

Krhhoff s ules 2.) Loop ule (onseron of energy) The sum of he hnges n poenl round ny losed ph of ru s zero. V V 2 V 3 = Glnomeers glnomeer s smply meer h defles n proporon o he urren runnng hrough. The mxmum defleon s lled he full-sle defleon. The key hrerss of glnomeer re The urren fs reured for full-sle defleon. The ressne of he ol of wre n he meer. V = G Dre urren rus 25 Dre urren rus 26 Volmeers olmeer onsss of ressor pu n seres wh glnomeer. The lue of hs ressne s deermnes he full-sle redng of he meer V. fs s G V = fs ( s ) ( s = V fs ) fs Dre urren rus 28 Dre urren rus 27 mmeers n mmeer onsss of ressor pu n prllel (lled shun ressor or shun) wh glnomeer. The lue of hs ressne sh deermnes he fullsle redng of he meer. fs G sh sh = fs ( fs ) fs = ( fs ) sh rus (hrgng) rus Q o = = - = = Dre urren rus 29 Dre urren rus 3

rus (hrgng) = E d d = E d d = E d d = E d E = d d = d E d = d E Dre urren rus 3 rus (hrgng) d = d E = ln( E ) = ln( E ) ln ( E ) E & = ln% ( $ E ' e = E E Ee = E = E Ee () = Q f e ( ) where Q f = E Dre urren rus 32 rus (hrgng) The hrge on he por res ordng o: rus (hrgng) ( ) = Q f e ( ) = e ( ) () = Q f e ( ) Q f The urren ny me s gen y: = d d = = o e e o s lled he me onsn (& ) nd s he me kes he por o eome 63.2% hrged. Dre urren rus 33 = o e o /e Dre urren rus 34 rus (hrgng) rus (Dshrgng) Q f nresng = Q o -Q o = = = = d d - Dre urren rus 35 Dre urren rus 36

rus (Dshrgng) d d = d = d d d = Q o ln( ) ln( Q ) = ln$ % ' = ( Q & Q = e = Q e rus (Dshrgng) The hrge on he por res ordng o: The urren ny me s gen y: = d d = Q o e = Q o e = o e Dre urren rus 37 Dre urren rus 38 rus (Dshrgng) rus (Dshrgng) = Q o e Q o Q o Q o /e nresng o = o e o /e Dre urren rus 39 Dre urren rus 4 rus Unhrged = shor ru Fully hrged = open ru hrged por hs sme olge s he dee h s elerlly prllel o. Dre urren rus 4

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