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:

.

.

Apply the equation :

.

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

.

Take the natural logarithm of both sides:

.

.

.

.

Apply the equation :

.

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

.

Take the natural logarithm of both sides:

.

.

.

it takes about 30 days for it to break down

answer; equivalent to a one(1) horsepower motor;