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## Homework Statement

In an oscillating series RLC circuit, with resistance R and inductance L, find the time required for the maximum energy in the capacitor during an oscillation to fall to 1/5 its initial value. Assume

*q*=

*Q*at

*t*= 0

## Homework Equations

## U(t) = \frac{Q^2}{2C}e^{\frac{-Rt}{L}}\cos^2(\omega't+\phi) ##

## U_0 = \frac{Q^2}{2C} ##

## \omega' = \sqrt(\frac{1}{LC} - \frac{R}{2L}^2) ##

## The Attempt at a Solution

## 0.5 \frac{Q^2}{2C} = \frac{Q^2}{2C}e^{\frac{-Rt}{L}}\cos^2(\omega't+\phi) ##

## 0.5 = e^{\frac{-Rt}{L}}\cos^2(\omega't+\phi) ##

Since q = Q @ t=0, I dropped the phase angle:

## 0.5 = e^{\frac{-Rt}{L}}\cos^2(\omega't) ##

I'm not even sure on how to begin solving this without getting that t out of the cosine squared. To do this I would either need to add some other ## \sin(t)^2 ## to get ## \cos(t)^2 + \sin(t)^2 = 1 ## on the resulting function or I would need to find some expression ## \cos(t) = \frac{adj}{hyp} ##

Any hints, tips? Thanks.