Simple Nature - Light and Matter

of the charges on the plates from q to q + dq:V C = (U q+dq − U q )/ dq= dU Cdq= d ( ) 1dq 2C q2= q CMany books use this as the definition of capacitance. This equation,by the way, probably explains the historical reason why C was definedso that the energy was inversely proportional to C for a givenvalue of C: the people who invented the definition were thinking of acapacitor as a device for storing charge rather than energy, and theamount of charge stored for a fixed voltage (the charge “capacity”)is proportional to C.In the case of an inductor, we know that if there is a steady, constantcurrent flowing through it, then the magnetic field is constant,and so is the amount of energy stored; no energy is being exchangedbetween the inductor and any other circuit element. But what ifthe current is changing? The magnetic field is proportional to thecurrent, so a change in one implies a change in the other. For concreteness,let’s imagine that the magnetic field and the current areboth decreasing. The energy stored in the magnetic field is thereforedecreasing, and by conservation of energy, this energy can’t justgo away — some other circuit element must be taking energy fromthe inductor. The simplest example, shown in figure l, is a seriescircuit consisting of the inductor plus one other circuit element. Itdoesn’t matter what this other circuit element is, so we just call it ablack box, but if you like, we can think of it as a resistor, in whichcase the energy lost by the inductor is being turned into heat bythe resistor. The junction rule tells us that both circuit elementshave the same current through them, so I could refer to either one,and likewise the loop rule tells us V inductor + V black box = 0, so thetwo voltage drops have the same absolute value, which we can referto as V . Whatever the black box is, the rate at which it is takingenergy from the inductor is given by |P | = |IV |, sol / The inductor releases energyand gives it to the black box.|IV | =dU L∣ dt ∣( )∣ =d 1 ∣∣∣∣dt 2 LI2 =dI∣LI dt ∣ ,Section 10.5 LRC Circuits 597