Solved Examples: Calculating Charge Remaining During Capacitor Discharge

📚 Quick Study Guide

• ⚡ The voltage across a discharging capacitor at time $t$ is given by: $V(t) = V_0 e^{-\frac{t}{RC}}$, where $V_0$ is the initial voltage, $R$ is the resistance, and $C$ is the capacitance.
• ⏱️ The time constant, denoted by $\tau$, is equal to $RC$. It represents the time it takes for the voltage to drop to approximately 36.8% of its initial value.
• 📉 The charge $Q(t)$ remaining on the capacitor at time $t$ is given by: $Q(t) = Q_0 e^{-\frac{t}{RC}}$, where $Q_0$ is the initial charge.
• 💡 The current $I(t)$ flowing through the circuit during discharge is: $I(t) = I_0 e^{-\frac{t}{RC}}$, where $I_0 = \frac{V_0}{R}$ is the initial current.
• 🧪 The charge $Q_0$ is related to the initial voltage $V_0$ and capacitance $C$ by: $Q_0 = CV_0$.

Practice Quiz

1. What formula calculates the charge remaining on a capacitor during discharge?
   1. $Q(t) = Q_0 e^{t/RC}$
   2. $Q(t) = Q_0 e^{-t/RC}$
   3. $Q(t) = Q_0 (1 - e^{-t/RC})$
   4. $Q(t) = Q_0 \frac{t}{RC}$
2. A 100 $\mu$F capacitor is charged to 10V and then discharged through a 1 k$\Omega$ resistor. What is the time constant of the circuit?
   1. 0.01 s
   2. 0.1 s
   3. 1 s
   4. 10 s
3. A capacitor initially holds a charge of 500 $\mu$C. After a time equal to two time constants, approximately how much charge remains?
   1. 67.7 $\mu$C
   2. 184 $\mu$C
   3. 135 $\mu$C
   4. 250 $\mu$C
4. If a capacitor discharges to half of its initial charge in 5 seconds, what is the time constant of the circuit?
   1. 5 s
   2. 5 / ln(2) s
   3. 5 * ln(2) s
   4. 2.5 s
5. A 200 $\mu$F capacitor is discharged through a 500 $\Omega$ resistor. If the initial voltage is 20V, what is the initial charge on the capacitor?
   1. 0.004 C
   2. 0.04 C
   3. 0.4 C
   4. 4 C
6. During capacitor discharge, the current _________ with time.
   1. increases linearly
   2. decreases linearly
   3. increases exponentially
   4. decreases exponentially
7. What happens to the charge remaining on a capacitor as the resistance in the discharge circuit increases?
   1. The charge decreases more rapidly.
   2. The charge decreases more slowly.
   3. The charge decreases at the same rate.
   4. The charge remains constant.