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).