Problem 7 For the example above what is the RC timeconstant

Problem 7: For the example above, what is the RC time-constant (in seconds)? How long must we wait before the charge on the capacitor is only 0.5 C? What is the current in the resistor at a time t? The rate of dissipation of energy in the resistor is the product IV; and both I and V decrease as the the capacitor discharges. When the capacitor is fully discharged, how much energy will have been dissipated in the resistor? How does this energy loss compare with Q^2/2C where Q is the initial charge on the capacitor?

Solution

a)

Time Constant

T=RC =(10*10^-6)*20

T=2*10^-4 s

b)

For a RC circuit discharging

Q=Q_maxe^-t/RC

0.5 =1 e^{-t/2*10^-4}

-t/2*10^-4 =Ln(0.5)=-0.693

t=1.386*10^-4 s

c)

Maximum current

I_max=Q_max/T =10^-6/(2*10^-4) =5*10^-3A

In a RC circuit current as a function of time is given by

I=I_maxe^-t/T=(5*10^-3A)e^{-t/(2*10^-4)}

I=(5*10^-3A)e^-5000t

d)

Energy dissipated in resistor while complete discharging is

U=Q²/2C =(1*10^-6)²/2*(10*10^-6)

U=5*10^-8 J

e)

Both are equal