Overview
This lecture covers how capacitors charge and discharge in RC circuits, including key equations, time constants, and how to calculate charging time to a specific voltage.
RC Circuit Basics
- An RC circuit consists of a resistor (R), capacitor (C), and typically a battery and switch.
- When the switch closes, current flows, charging the capacitor.
- Initially (t = 0), the capacitor voltage is 0 and the resistor voltage equals the battery voltage.
- As the capacitor charges, its voltage increases and the resistor's voltage decreases.
- Once fully charged, the capacitor voltage equals the battery voltage, and current stops flowing.
Charging a Capacitor
- The voltage across the capacitor during charging:
( V_C = \epsilon (1 - e^{-t/RC}) )
- The time constant ( \tau ) is ( RC ), determining charging speed.
- After 1 time constant (( t = \tau )): capacitor reaches 63.2% of max voltage.
- 2 time constants: 86.5% charged; 3: 95%; 4: 98.2%; 5: 99.3%.
Discharging a Capacitor
- When disconnected from the battery and connected to a resistor, the capacitor discharges.
- Discharge equation:
( V_C = V_{initial} \cdot e^{-t/RC} )
- After 1 time constant: 36.8% charge remains.
- 2 time constants: 13.5% remains; 3: 5%; 4: 1.8%; 5: 0.7%._
Example Problem: Charging to 90%
- Given: ( R = 1k\Omega ), ( C = 500\mu F ), battery ( = 20V ).
- Time constant ( \tau = RC = 1,000 \times 500 \times 10^{-6} = 0.5s ).
- To reach 90% (( V_C = 18V )), use:
( t = -RC \ln(1 - V/\epsilon) )
- Substitute values: ( t = -0.5 \ln(1 - 18/20) = 1.15s ).
- Number of time constants: ( n = t/\tau = 1.15/0.5 = 2.3 ) (matches between table values for 2 and 3 time constants).
Key Terms & Definitions
- Capacitor (C) — Device that stores electrical energy in an electric field.
- Resistor (R) — Component that resists current, producing a voltage drop.
- RC Circuit — Circuit containing both a resistor and a capacitor.
- Time Constant (( \tau )) — Product of resistance and capacitance (( \tau = RC )); characteristic time for charging/discharging.
- EMF (( \epsilon )) — Electromotive force; the battery or source voltage.
Action Items / Next Steps
- Review the charging and discharging equations for capacitors in RC circuits.
- Practice using the formula ( t = -RC \ln(1 - V/\epsilon) ) for charge problems.
- Know how to calculate and interpret the time constant ( \tau ).