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## 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 = Q

^{2}/2C

(2) U = Q

_{max}ε/2

(3) q(t) = Q

_{i}e

^{-t/RC}

## The Attempt at a Solution

The solution can be correctly attained using the first equation:

(1/4) Q

^{2}/2C = (Q

_{i}e

^{-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

_{max}ε/2 = Q

_{max}e

^{-t/RC}ε/2

(1/4) = e

^{-t/RC}

As can be seen here, this answer is off from the first by an exponent of two. Q

^{2}/2C and Q

_{max}ε/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...

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