V(t)=Vo sin@t Figure 12.2.3 A purely inductive circuit As we shall see below,a purely inductive circuit corresponds to infinite capacitance C=oo and zero resistance R=0.Applying the modified Kirchhoff's rule for inductors, the circuit equation reads P)-y20=p0-L业=0 (12.2.11) dt which implies dlL-V④_'sinot (12.2.12) dt LL where Vo=Vo.Integrating over the above equation,we find (12.2.13) where we have used the trigonometric identity -cosot=sin @1- (12.2.14) for rewriting the last expression.Comparing Eq.(12.2.14)with Eq.(12.1.2),we see that the amplitude of the current through the inductor is L0= OL XL (12.2.15) where XL=@L (12.2.16) is called the inductive reactance.It has SI units of ohms ()just like resistance. However,unlike resistance,X,depends linearly on the angular frequency @Thus,the resistance to current flow increases with frequency.This is due to the fact that at higher 12-6Figure 12.2.3 A purely inductive circuit As we shall see below, a purely inductive circuit corresponds to infinite capacitance and zero resistance . Applying the modified Kirchhoff’s rule for inductors, the circuit equation reads C = ∞ R = 0 ( ) ( ) ( ) 0 L L dI V t V t V t L dt − = − = (12.2.11) which implies 0 ( ) sin dIL V t VL t dt L L = = ω (12.2.12) where . Integrating over the above equation, we find VL0 =V0 0 0 0 ( ) sin cos sin 2 L L L L L V V V I t dI t dt t t L L L π ω ω ω ω ⎛ ⎞ ⎛ ⎞ ⎛ = = = − = ⎜ − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ∫ ∫ ω ⎞ ⎟ (12.2.13) where we have used the trigonometric identity cos sin 2 t t π ω ω ⎛ − = ⎜ − ⎝ ⎠ ⎞ ⎟ (12.2.14) for rewriting the last expression. Comparing Eq. (12.2.14) with Eq. (12.1.2), we see that the amplitude of the current through the inductor is 0 0 L L L L V V I ωL X = = 0 (12.2.15) where XL =ωL (12.2.16) is called the inductive reactance. It has SI units of ohms (Ω), just like resistance. However, unlike resistance, XL depends linearly on the angular frequency ω. Thus, the resistance to current flow increases with frequency. This is due to the fact that at higher 12-6