Problem 5.10. Due to the random motion and discrete nature of electrons, and LRC series circuit experiences a random electromotive from (EMF), (t). This, in turn, induces a random varying charge, Q(t), on the capacitor plates and a random current, I(t)=(dQ(t)/dt), through the resistor and inductor. The random charge, Q(t), satisfies the Langevin equation

Assume that the EMF is delta-correlated, ((12)(11)) = 88(121), and ((t)) = 0. Assume that the circuit is at temperature T and that the average magnetic energy in the inductor and average electric energy in the capacitor satisfy the equipartition theorem, LTBT and (Q), =kBT, where 2(0) = 20 and 1(0) = Io. Assume that (20)=(10) = (2010) = 0.

(a) Compute the current correlation function, ((I(2)I(t)))

(b) Compute the variance of the charge distribution, (((Q(1) - Q0)))T

Transcribed Image Text:

L dt dt d0(1) do(1) 2(1), R + =(1). C