After how many time constants is capacitor energy 1/4?

[David Day]

Homework Statement

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After how many time constants
is the stored energy of a discharged capacitor one-fourth its initial value?

Homework Equations

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(1) U = Q²/2C

(2) U = Q_maxε/2

(3) q(t) = Q_ie^-t/RC

The Attempt at a Solution

The solution can be correctly attained using the first equation:

(1/4) Q²/2C = (Q_ie^-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) Q_maxε/2 = Q_maxe^-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 Q_maxε/2 are mathematically equivalent, so I'm not sure where I'm making the error. Is it not correct to assume Q_max = Q_i?

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...

 

[cnh1995]

After how many time constants
is the stored energy of a discharged capacitor

Do you mean a "discharging" capacitor?

(1/4) Qmaxε/2 = Qmaxe-t/RCε/2

If the capacitor is discharging, can you use the same ε on the RHS of this equation as in the LHS?

Edit: I see you found the error in your equation by yourself while I was typing my response. Well done!