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www.GOALias.blogspot.comAlternating CurrentExample 7.11 Show that in the free oscillations of an LC circuit, thesum of energies stored in the capacitor and the inductor is constantin time.Solution Let q 0be the initial charge on a capacitor. Let the chargedcapacitor be connected to an inductor of inductance L. As you havestudied in Section 7.8, this LC circuit will sustain an oscillation withfrquency⎛1 ⎞ω ⎜= 2πν=⎝⎟LC ⎠At an instant t, charge q on the capacitor and the current i are givenby:q (t) = q 0cos ωti (t) = – q 0ω sin ωtEnergy stored in the capacitor at time t is2 21 2 1 q q02UE= C V = = cos ( ωt)2 2 C 2CEnergy stored in the inductor at time t isUM1= L i221sin ( )=( )2 2 202 Lq ω ωtq= =2C20 2sin ( ωt) 2∵ω1/Sum of energiesLCq2C⎣20 2 2UE+ UM= ⎡cosωt + sin ωt⎤⎦2q0=2CThis sum is constant in time as q oand C, both are time-independent.Note that it is equal to the initial energy of the capacitor. Why it isso? Think!EXAMPLE 7.117.9 TRANSFORMERSFor many purposes, it is necessary to change (or transform) an alternatingvoltage from one to another of greater or smaller value. This is done witha device called transformer using the principle of mutual induction.A transformer consists of two sets of coils, insulated from each other.They are wound on a soft-iron core, either one on top of the other as inFig. 7.20(a) or on separate limbs of the core as in Fig. 7.20(b). One of thecoils called the primary coil has N pturns. The other coil is called thesecondary coil; it has N sturns. Often the primary coil is the input coiland the secondary coil is the output coil of the transformer.259