PROBLEMS FROM FISHBANE and other ADDITIONAL PROBLEMS
                               
                             JULY 26 - AUG 1
        
         Problems from Fishbane, Gasiorowicz, and Thornton (2nd edition) 

         CHAPTER 28
         57.  Find a differential equation which can be solved for Q(t), the 
              charge as a function of time, on a discharging capacitor.  The 
              figure for this problem is Fig. 28-13, with an initial charge 
              of Q_0 on capacitor C when the switch is moved to position b at 
              t = 0.  Let positive current represent a charging capacitor, 
              i.e. i = (dQ/dt).  Show by direct substitution that Q(t) = 
              (Q_0)e^(-t/RC) is a solution of this differential equation.

         CHAPTER 33
         46.  The figure for this problem is Fig. 31-17.  After being at 
              position a for a long time, switch S is thrown to b.  Find the 
              differential equation which can be solved for the resulting 
              I(t), and show that I(t) = (EMF/R)e^(-tR/L) is a solution of 
              that differential equation.

         EXTRA PROBLEMS FOR CHAPTER 33

         G1 and G2:
         The differential equation for an undriven LRC circuit is

           (q/C) + R(dq/dt) + L(d^2q/dt^2) = 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.

         G1.  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.

         G2.  Show that the cos((omega)t) terms sum to zero only if

                 omega^2 = omega0^2 - alpha^2 where omega0^2 = 1/LC.

         G3 and G4:
         The differential equation for a driven LRC circuit is

           (q/C) + R(dq/dt) + L(d^2q/dt^2) = (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.

         G3.  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.

         G4.  (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 sin^2(phi) + cos^2(phi) = 1.

              (b) From (a), show that Z = sqrt((X_L-X_C)^2+R^2).
                  HINT: sec^2(phi) = 1 + tan^2(phi).