DUCTCE 3. Sef nductance Cnsider the circuit shwn in the Figure. S R When the switch is csed the current, and s the magnetic fied, thrugh the circuit increases frm zer t a specific vaue. The increasing magnetic fux induces an emf. By enz's aw, ε this induced emf ppses the change in fux. The effect f this induced emf is t retard the change f the rigina current, that is, retard its increasing. The same phenmena ccurred when the switch is pened where the current in this case decreases frm a specific vaue t zer. The emf induced due t the decreasing f the magnetic fux nw tends t ppse the decreasing f the rigina current. This phenmena is caed the sef inductin since the changing fux thrugh the circuit arises frm the circuit itsef. The emf induced due t this phenmena is caed the sef-induced emf. f the emf induced in a circuit is due t the changing f the magnetic fux set up by anther circuit we have the mutua inductin phenmena. T btain a quantitative descriptin f the sef inductin, we knw frm Faraday's aw that the induced emf is prprtina t the time rate f the magnetic fux, i.e., dφ ε m = 3. But Φ m B and B ε d = 3.
The prprtinaity cnstant is caed the sef-inductance, r simpy the inductance f the ci. The S unit f inductance is Henry (H), which, frm Equatin 3., is equivaent t V.s H = w cmparing Eqs. 3. & 3. Φm = 3.3 s it is cear frm Eq. 3.3 depends n the gemetric features f the ci. t shud be nted that a eements in a circuit have sme inductance but it is t sma t be significant except that f a ci. ci that has significant inductance is caed inductr, and is represented in the circuits by the symb Exampe 3. Find the inductance f an idea senid f turns and ength. Sutin Knwing that, inside the senid B is unifrm and given by B = µ n = µ Φ = Bcs 0 = m µ w using Eq. 3.3 = µ µ = 3. R Circuits T study expicity the effect f sef inductin in a circuit we refer t the circuit shwn. Suppse that the switch is thrwn tm pint at t = 0. ppying Kirchhff's p rue t the circuit at time t ε S R
we get d ε R =0 3.4 t is nt difficut t verify that the sutin f the differentia equatin given in Equatin 3.4 is t = e max 3.5 with the maximum current is ε max = 3. R and the time cnstant f the R circuit is = 3.7 R Frm Equatin 3.5 we cncude that at t = 0, = 0, whie = max as t. This means that: the inductr acts as an pen circuit at t = 0 and acts as an rdinary wire after a ng time. f the battery is suddeny remved, by thrwing the switch t pint in the circuit and appying Kirchhff's rue again we get d R + =0 t = max e 3.8 The reatins f Equatins 3.5 and 3.8 are ptted in as a functin f time. s it is cear frm the graph (a), the current takes sme time t reach its maximum vaue. The graph f Figure (b) tes that the current takes sme time t reach it zer
m m t t (a) (b) (a) The current versus time in an R circuit when cnnected t a battery. (b) The current versus time in an R circuit when the battery is discnnected. vaue. n anther wrd, the inductr has the effect t hinder the current frm reaching its fina vaue fr sme time. Exampe 3.3 Cnsider the circuit shwn, find a) the time cnstant f the circuit, b) the current in the circuit at t =. 0 ms, and c) cmpare the P.D acrss the resistr with that acrss the inductr. S Ω V 30 mh Sutin a) The time cnstant is given by the Equatin 3.0 0 = = R.0 b) The current is 3 = 5.0 ms 5 t e = e = max = 0.
c) The P.D. acrss the resistr is given by V t = R = R e R Whie the P.D. acrss the inductr is given by V = d = e t = Re t VR+ V= R= ε = V 3.3 Energy in Magnetic Fied Mutipying Eq. 3.4 by ε R d = 0 The st term represents the pwer f the battery, whie the nd term represents the pwer deivered t the resistr the 3 rd term represents the pwer deivered t the inductr, i.e., du d P = = du = d 0 U m = 3.9
3.5 Osciatins in an C Circuit Cnsider the circuit shwn with the capacitr is charged with Q max. fter csing S the charge wi fw thrugh the inductr. t sme time et the charge in the capacitr t be q and the current in the inductr t be. The tta energy in the circuit at this time is C S U tta = U C + U q = + C Deriving the abve Eq. with respect t time But du tta = + dq C d du tta = 0 q dq C + d =0 dq d d q Knwing that = and = q C d q + = 0 d q + q = 0 C q = Q cs t max ( ω +γ ) T find the cnstant γ we knw that q = Q at t = 0 γ 0 max = q = Q cs( ωt) 3.9 max
With w With ω = 3.0 C dq = = Qmaxω sin ωt = max sin ωt 3. = ω 3. max Q max Exampe 3.8 Cnsider the Circuit shw. First S is pen and S is csed such that the capacitr is charged. w if S is pened t remve the battery and then S is csed t cnnect the capacitr with the inductr. a) Find ω f the circuit. b) Find Q max and max. c) Find (t) and Q(t). Sutin a) The frequency is given by V 9 pf.8 mh S S ω = = C 3 (.8 0 )( 9 0 ) =.3 0 b) The maximum charge n the capacitr is the initia charge befre pening S, i.e., w Q max Hz ( 9 0 ) =.08 0 0 C = Cε = max = ω Q 0 max =.3 0.08 0 =.79 0 4 c) q( t) = Q cs( t) 0 max ω =.08 0 cs(.3 0 t) q( t) = sin( t).79 4 max ω = 0 sin(.3 0 t)