A capacitor with initial charge q0 is discharged through a resistor. a) In terms of the time constant τ, how long is required for the capacitor to lose the first one-ninth of its charge? b) How long is required for the capacitor to lose the first two-ninths of its charge?

It would take time for the capacitor to discharge from to .

It would take time for the capacitor to discharge from to .

Note that , and that.

Explanation:

In an RC circuit, a capacitor is connected directly to a resistor. Let the time constant of this circuit is , and the initial charge of the capacitor be . Then at time , the charge stored in the capacitor would be:

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Apply the equation :

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The goal is to solve for in terms of . Rearrange the equation:

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Take the natural logarithm of both sides:

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Apply the equation :

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The goal is to solve for in terms of . Rearrange the equation:

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Take the natural logarithm of both sides:

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