Extra Problems for Ch 33. G1: The differential equation for an undriven LRC circuit is (q/C) + R(dq/dt) + L(d²q/dt²) = 0 Eq (1) Your task is to show that q(t) = (Q_0)e^(-(alpha)t)cos((omega)t) is a solution of this differential equation, where Q_0, alpha, and omega are constants. (A) Substitution of q(t) in Eq (1) gives terms in sin((omega)t) and terms in cos((omega)t). Show that the sin((omega)t) terms sum to zero only if alpha = R/2L. (B) Show that the cos((omega)t) terms sum to zero only if omega² = omega0² - alpha² where omega0² = 1/LC. G2 and G3: The differential equation for a driven LRC circuit is (q/C) + R(dq/dt) + L(d²q/dt²) = (EMFm)sin((omegaD)t) Eq (2) where (EMFm)sin((omegaD)t) is the driving EMF as a function of time, EMFm is the amplitude of that driving EMF, and omegaD is the driving angular frequency. Your task is to show that q(t) = -(Q_0)cos((omegaD)t - phi) is a solution of this differential equation, where Q_0 and phi are constants. G2. Substitution of q(t) in Eq (2) gives terms in sin((omegaD)t) and terms in cos((omegaD)t). HINT: Use the trig identities for sin(alpha+-beta) and cos(alpha+-beta) as found on page A9 of HRW. Show that the sum of the cos((omegaD)t) terms = 0 only if tan(phi) = (1/R)(X_L-X_C) where X_L and X_C are the inductive and capacitive reactances. G3. (a) Show that the sum of the sin((omegaD)t) terms = (EMFm)sin((omegaD)t) only if EMFm/(omegaD)Q_0 = R/cos(phi) and therefore that Z = R/cos(phi), where Z is the inductance. HINT: Use sin²(phi) + cos²(phi) = 1. (b) From (a), show that Z = sqrt((X_L-X_C)²+R²). HINT: sec²(phi) = 1 + tan²(phi).