An uncharged 5.0-µF capacitor and a resistor are connected in series to a 12-V battery and an open switch to form a simple RC circuit. The switch is closed at t = 0 s. The time constant of the circuit is 4.0 s. What is the charge on either plate after one time constant has elapsed?

For a direct current resistor-capacitor circuit where the capacitor is initially uncharged, the charge stored on one of the capacitor's plates is given by:

Q (t) = Cℰ (1-e^{-t / (RC) })

Q (t) is the charge, t is time, ℰ is the battery's terminal voltage, R is the resistor's resistance, and C is the capacitor's capacitance.

The time constant of the circuit τ is the product of the resistance and capacitance:

τ = RC

Q (t) can be rewritten as:

Q (t) = Cℰ (1-e^{-t/τ})

We want to know how much charge is stored when one time constant has elapsed, i. e. what Q (t) is when t = τ. Let us plug in this time value:

Q (τ) = Cℰ (1-e^{-τ/τ})

Q (τ) = Cℰ (1-1/e)

Q (τ) = Cℰ (0.63)

Given values:

C = 5.0*10⁻⁶F

ℰ = 12V

Plug in these values and solve for Q (τ):

Q (τ) = (5.0*10⁻⁶) (12) (0.63)

Q (τ) = 3.8*10⁻⁵C