Notes for the Physics GRE - Harvard University Department of Physics

When the battery is turned off:The energy stored in the magnetic field of an inductor:I(t) = E R e−t/τ . (108)U = 1 2 LI2 . (109)The energy per unit volume in the B-field region of the inductor:u B = B22µ 0. (110)The current and charge of an LC-circuit oscillate in time:Q(t) = Qmax cos(ωt + φ) (111)I(t) = −ωQmax sin(ωt + φ). (112)Here, ω = √ 1LCand the phase φ depends upon the initial state of the circuit.The energy U stored in an LC-circuit:U = U C + U L = Q2 max2Ccos2 (ωt + φ) + LI2 max2sin 2 (ωt + φ). (113)Note that U is constant in time. In an RLC-circuit, we get damped harmonicmotion for I, Q.In AC-circuits,E = Vmax sin(ωt). (114)If only a resistor, I is in phase with V , andIrms = Imax/ √ 2, (115)Vrms = Vmax/ √ 2. (116)For RLC in series, inductive reactance χ L = ωL, capacitive reactance χ C = 1ωC ,impedenceZ = √ R 2 + (χ L − χ C ) 2 . (117)Voltage and current are out of phase by φ, wheretan φ = χ L − χ C. (118)RThe average power produced by the generator of RLC AC-circuit isPavg = IrmsVrms cos φ = I 2 rmsR, (119)and the energy is dissipated as heat in the resistor. For this general RLC circuit,Irms = V rmsZ . (120)15