After how many time constants
is the stored energy of a discharged capacitor one-fourth its initial value?
(1) U = Q2/2C
(2) U = Qmaxε/2
(3) q(t) = Qie-t/RC
The Attempt at a Solution
The solution can be correctly attained using the first equation:
(1/4) Q2/2C = (Qie-t/RC)2/2C
(1/4) = e-2t/RC
t = 0.693RC
However, I first attempted to use equation (2) but could not get the correct answer. I can't figure out what mistake or incorrect assumption I'm making. My attempt follows:
(1/4) Qmaxε/2 = Qmaxe-t/RCε/2
(1/4) = e-t/RC
As can be seen here, this answer is off from the first by an exponent of two. Q2/2C and Qmaxε/2 are mathematically equivalent, so I'm not sure where I'm making the error. Is it not correct to assume Qmax = Qi?
I'd really appreciate some help in clearing this up!
EDIT: I may see my problem. The ε in equation (2) is proportional to Q and will thus be changing, i.e, it's not equal to the ε on the other side of the equation except when t = 0, so I need to change it to Q/C, which will be the same form as equation (1). I'm not sure if this is the problem or not...