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Introductory Physics Homework Help
After how many time constants is capacitor energy 1/4?
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[QUOTE="David Day, post: 6037031, member: 647840"] [h2]Homework Statement[/h2] [/B] After how many time constants is the stored energy of a discharged capacitor one-fourth its initial value? [h2]Homework Equations[/h2] [/B] (1) U = Q[SUP]2[/SUP]/2C (2) U = Q[SUB]max[/SUB]ε/2 (3) q(t) = Q[SUB]i[/SUB]e[SUP]-t/RC[/SUP] [h2]The Attempt at a Solution[/h2] The solution can be correctly attained using the first equation: (1/4) Q[SUP]2[/SUP]/2C = (Q[SUB]i[/SUB]e[SUP]-t/RC[/SUP])[SUP]2[/SUP]/2C (1/4) = e[SUP]-2t/RC[/SUP] 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[SUB]max[/SUB]ε/2 = Q[SUB]max[/SUB]e[SUP]-t/RC[/SUP]ε/2 (1/4) = e[SUP]-t/RC[/SUP] As can be seen here, this answer is off from the first by an exponent of two. Q[SUP]2[/SUP]/2C and Q[SUB]max[/SUB]ε/2 are mathematically equivalent, so I'm not sure where I'm making the error. Is it not correct to assume Q[SUB]max[/SUB] = Q[SUB]i[/SUB]? 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... [/QUOTE]
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Introductory Physics Homework Help
After how many time constants is capacitor energy 1/4?
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