RC Circuit: Charging and Discharging Notes

RC Capacitor Overview

Components of RC Circuit

  • RC Capacitor: A capacitor connected in a circuit with a resistor.
  • Battery: Provides voltage (emf) to charge the capacitor.
  • Switch: Controls the flow of current in the circuit.
  • Mini LabQuest: Provides resistance for the circuit and allows measurements.

Charging Phase

  • Kirchhoff’s Equation for Charging:
    • The total voltage in any closed loop must add up to zero.
    • Voltage from the battery is used up by the capacitor and resistor.

Behaviour of Charge

  • Charge Equation:
    • Starts at 0 (uncharged).
    • Asymptotically approaches maximum charge ( Q{max} = C \times V{battery} )
    • The charge approaches maximum but never truly reaches it (takes infinite time).

Current Dynamics

  • Current Equation:
    • Begins at maximum value ( I{max} = \frac{V{emf}}{R} ) when capacitor is uncharged.
    • Decreases over time as charge builds up (increased voltage across capacitor).
    • Current never fully reaches zero.

Capacitor Voltage

  • Voltage Across Capacitor (VC):
    • Starts at 0 (initially uncharged).
    • Increases over time as charge accumulates, approaching the emf.
    • After time ( t = RC ) (time constant), VC reaches approximately 63% of maximum voltage.
    • After about 5RC, VC is around 99% of emf.
    • Time constant (RC) governs the charging speed: larger RC results in slower charging.

Discharging Phase

  • Kirchhoff’s Equation for Discharging:
    • The total voltage in a closed loop still equals zero.
    • When the battery is disconnected, the capacitor releases its stored charge.

Charge during Discharge

  • Charge Equation:
    • Starts at maximum charge ( Q{max} = C \times V{emf} ) and decreases to 0 asymptotically.

Current During Discharge

  • Current Dynamics:
    • Starts at maximum value ( I{max} = \frac{V{emf}}{R} ) (fully charged).
    • Decreases over time corresponding to decreasing voltage across the capacitor.

Voltage Across Capacitor in Discharging

  • VC Changes Over Time:
    • Starts at maximum voltage (emf) and decreases as charges leave the capacitor.
    • Voltage and charge are directly related (as charge decreases, voltage decreases).

Energy Considerations

  • Energy Stored in Capacitor:
    • Equation: ( E = \frac{1}{2} C V^2 )
    • Energy is derived from work done to move charges against the electric field.
    • As charge leaves the capacitor, the electric field weakens, leading to decreased voltage and energy stored.

Key Equations:

  • Charge during charging: ( Q(t) = C \times V_{battery} \times (1 - e^{-t/RC}) )

  • Current during charging: ( I(t) = \frac{V_{emf}}{R} e^{-t/RC} )

  • Voltage across capacitor during charging: ( VC(t) = V{battery} (1 - e^{-t/RC}) )

  • Charge during discharging: ( Q(t) = Q_{max} e^{-t/RC} )

  • Current during discharging: ( I(t) = -\frac{V_{emf}}{R} e^{-t/RC} )

  • Voltage during discharging: ( VC(t) = V{emf} e^{-t/RC} )

  • Energy stored: ( E = \frac{1}{2} C V^2 )