Energy Storage in Electric and Magnetic Fields: Capacitors and Inductors, Study notes of Physics

Download Energy Storage in Electric and Magnetic Fields: Capacitors and Inductors and more Study notes Physics in PDF only on Docsity! CPHY 122 Class Notes 15 Instructor: H. L. Neal 1 Energy of Electric and Magnetic Fields In this section we calculate the energy stored by a capacitor and an inductor. It is most pro…table to think of the energy in these cases as being stored in the electric and magnetic …elds produced respectively in the capacitor and the inductor. From these calculations we compute the energy per unit volume in electric and magnetic …elds. These results turn out to be valid for any electric and magnetic …elds –not just those inside parallel plate capacitors and inductors! Let us …rst consider a capacitor. Recall that the energy stored is UE = q2 2C : Assuming that we have a parallel plate capacitor, let’s insert the formula for the capacitance of such a device C = 0A d : Let us further recall that the electric …eld in a parallel plate capacitor is E = =0 = q=(0A); so that q = 0EA and UE = (E0A) 2 2(0A=d) = 0E 2Ad 2 : The combination Ad is just the volume between the capacitor plates. The energy density in the capacitor is therefore uE = UE Ad = 0E 2 2 : This formula for the energy density in the electric …eld is speci…c to a parallel plate capacitor. However, it turns out to be valid for any electric …eld. 1 A similar analysis of a current increasing from zero in an inductor yields the energy density in a magnetic …eld. The work done by the generator in time dt is dW = Edq = EIdt so that the power is P = dW dt = EI = LI dI dt = d dt  1 2 LI2  : This implies that W = 1 2 LI2: But
U = W = 1 2 LI2: Or U = 1 2 LI2 + constant. The constant term is usually ignored. Now recall that for a solenoid B = 0 N ` I L = 0 N2 ` A Putting this int0 to the equation for U gives UB = 1 2  0 N2 ` A  ` 0N B 2 2 For underdamping I (t) = Imax exp  R 2L t  cos (!dt+ ) !d = s 1 LC  R 2L 2 : For overdamping I = A1e 1t + A2e 2t 1; 1 = R 2L  s R 2L 2 1 LC : 5 For critical damping 1p LC = R 2L and I (t) = (A+Bt) exp  R 2L t  : 6