RC Circuit

RC Circuits

• Overview

  • An RC (Resistor-Capacitor) circuit is a circuit that contains both resistors and capacitors.
  • These circuits are fundamental in understanding how electrical charge is stored and dissipated in various components.

• Charging a Capacitor

  • When a capacitor is charged through a resistor, the voltage across the capacitor and the current through the circuit change over time according to the following relationships:
    • $V_C(t)=V_0\left(1-e^{-\frac{t}{RC}}\right)$
    • $I(t)=\frac{V_0}{R}{e}^{-\frac{t}{RC}}$
    • $V_0$ is the initial voltage, $R$ is the resistance, $C$ is the capacitance, and $t$ is time.
  • The time constant ($\tau$) of an RC circuit is defined as $\tau =RC$. It indicates how quickly the capacitor charges or discharges.

• Example of Charging RC Circuit

  • Given:

    • Capacitance: $C=0.30\mu \text{F}$
    • Resistance: $R=20\text{k}\Omega$
    • Battery emf: $E=12\text{V}$

  • Calculate:

    • Time constant ($\tau$): $\tau =RC=(0.30\times 10^{-6}\text{F})(20\times 10^3\Omega )=6\text{ms}$
    • Maximum charge ($Q_{\text{max}}$): $Q_{\text{max}}=CE=(0.30\times 10^{-6}\text{F})(12\text{V})=3.6\mu \text{C}$
    • Time to reach 99% of $Q_{\text{max}}$: Solve for $t$ in $Q(t)=Q_{\text{max}}\left(1-e^{-\frac{t}{\tau }}\right)$

• Discharging a Capacitor

  • When a fully charged capacitor discharges through a resistor, the charge and the current through the circuit decrease over time as given by:
    • $Q(t)=Q_0{e}^{-\frac{t}{RC}}$
    • $I(t)=-\frac{Q_0}{RC}{e}^{-\frac{t}{RC}}$
    • $Q_0$ is the initial charge on the capacitor.

• Example of Discharging RC Circuit

  • Given:

    • Capacitance: $C=1.02\mu \text{F}$
    • Battery emf: $E=20\text{V}$
    • Current decreases to 0.50 of its initial value in $40\mu \text{s}$

  • Calculate:

    • Initial charge on the capacitor ($Q_0$): $Q_0=CE=(1.02\times 10^{-6}\text{F})(20\text{V})=20.4\mu \text{C}$
    • Value of $R$: Derived from given info that current reduces by half in $40\mu \text{s}$[1]``[2]``[3].