Chapter 30 nductance 30. Self-nductance Cnsider a lp f wire at rest. f we establish a current arund the lp, it will prduce a magnetic field. Sme f the magnetic field lines pass thrugh the lp. et! be the flux f! thrugh the lp. "! and! thus "! The cnstant f prprtinality is called the self-inductance f the lp. That is,! ( lp) N! (N lps) (a) The S unit f inductance is the Henry. H Tesla m /Amp (b) nductance is a purely gemetric quantity! Fr example, let s cnsider the self-inductance f a
slenid f length!, crss-sectinal area A, and n turns per unit length. Thus, N! N [ A] N [µ ο n A] N µ ο n A nte that N n! s that the self-inductance f the slenid becmes µ n slenid which is a purely gemetric quantity! We saw a similar result fr the capacitance f a parallel-plate capacitr f area A, separated by a distance d, and with vacuum between the plates. There we saw that C! A/ d, als a purely gemetric quantity! An induced emf! appears in any cil in which the current is changing ver time. Faraday s law f electrmagnetic inductin says that d # " ( N! ) A!
"! d ( ) "! Resistance is a measure f the ppsitin t current. nductance is a measure f the ppsitin t a change in the current. d 30.3 Magnetic Field Energy du et U energy stred in an inductr. Then is the rate at which energy is stred (pwer). Thus, du! 3
du & d $ #!" % du d T find the ttal energy stred in an inductr, integrate frm a current f zer t a current f. That is, U! du 0! d U The energy is stred in the magnetic field in the inductr! Fr the specific case f the slenid wrked ut abve, we had that the self-inductance f the slenid is µ n A!. Thus the energy stred in the slenid is slenid U U [ µ n A! ] & # $! % µ n" 4
U µ ( A! ) nte that A! is equal t the vlume ccupied by the slenid. Define the energy density u as the energy stred per unit vlume, u U! Vlume U A! Hence, u µ The energy is stred in the magnetic field! We saw that fr a parallel-plate capacitr the stred energy density u E was given by u E! E this says that in that case the energy was stred in the electric field! 5
30. Mutual nductance Cnsider lps f wire at rest. f we establish a current arund lp, it will prduce a magnetic field!. Sme f these magnetic field lines pass thrugh lp. et " be the magnetic flux f! thrugh lp. Since! is prprtinal t by the it-savart law {! µ d! " # r ˆ 4" $ } r and since " is prprtinal t! accrding t!! { " #! da}, then " is prprtinal t. The cnstant f prprtinality M is called the mutual inductance f the lps. That is, " # M The mutual inductance M is a purely gemetrical quantity, just like self-inductance and capacitance C. nteresting result: Frm " M and frm " M we see that M M " " which means that if, then " " 6
Nte als that frm " M it s clear that a changing current in induces an emf in lp equal t " # d$ " # d ( M ) " #M d 30.4 The R- Circuit Current Grwth in an R- Circuit An inductr in a circuit makes it difficult fr rapid changes in the current t ccur, thanks t the effect f self-induced emf. Cnsider the circuit belw. Suppse that at sme initial time t 0 we clse switch S. The current cannt change suddenly frm zer t sme final value, since di/ and the induced emf in the inductr wuld be infinite. nstead, the current begins t grw at a rate that depends n the value f in the circuit. 7
et i be the current at sme time t after switch S is clsed. Applying Kirchhff s lp rule yields!! ir! di 0 Nte that the rate f increase in the current di/ is di!! R i At the instant that switch S is clsed, i 0 and the vltage acrss R is zer. The initial rate f increase in the current is! di $ # &! " % initial and as we wuld expect, the greater the inductance, the mre slwly the current increases t its final value. When the current reaches its final steady-state value f, then its rate f increase is zer s that di/ 0. Thus! # " di $ & % final 0! ' R and! R 8
The equatin fr the current i at time t is di + R i! and as shwn in class, the slutin t this first-rder differential with cnstant cefficients is i! ( R! e!(r/)t ) The time cnstant fr the circuit is τ /R and equals the time it takes the current t build up t -/e 63% f its final value. Energy cnsideratins tell us that if we multiply the equatin! ir! di 0 by i, we btain 9
!i i R + i di εi is the instantaneus pwer supplied by the surce, the instantaneus pwer dissipated by the resistr is i R, and (d/){(/) i } i di/ is the instantaneus pwer stred in the inductr. S f the pwer supplied by the surce, part f the pwer is dissipated by the resistr, and the rest ges t stred pwer in the inductr. Current Decay in an R- Circuit Nw cnsider an R- circuit that has a current that has reached the steady-state value. And then we cnsider the circuit belw s that at t 0 the current is. 0
Kirchhff s lp rule yields The slutin f which is!ir! di 0 i e!(r/)t The time cnstant τ /R here is the time fr the current t decrease t /e 37% f its initial value. Energy cnsideratins here lead t 0 i R + i di which shws that in this case, i di is negative, indicating that the energy stred in the inductr decreases at a rate equal t the rate f energy dissipatin f energy i R in the resistr.