27.16:
RC Circuits: Discharging A Capacitor
When a capacitor is fully charged, the potential difference across the capacitor is equal to the voltage source.
If the battery gets disconnected from the circuit, the capacitor discharges.
Using Kirchhoff's loop rule yields an equation. Here, the current is the rate at which the charge is leaving the capacitor. Since the capacitor is losing charge with time, the sign is negative.
Substituting the definition of current yields an equation. This equation is further integrated to get the charge on the capacitor as a function of time.
For a discharging capacitor, the charge on the capacitor decreases exponentially from the initial charge, which is the maximum charge acquired by the capacitor while charging.
The time derivative of the— charge on a capacitor— equation results in the current expression as a function of time.
The negative sign in the expression signifies the opposite direction of the current compared to the capacitor's charging case.
The current decay has an exponential behavior. Its magnitude decreases exponentially and tends to zero as time tends to infinity.
27.16:
RC Circuits: Discharging A Capacitor
One of the applications of an RC circuit is the relaxation oscillator. The relaxation oscillator comprises a voltage source, a capacitor, a resistor, and a neon lamp. The lamp acts like an open circuit (infinite resistance) until the potential difference across the neon lamp reaches a specific voltage. At that voltage, the lamp acts like a short circuit (zero resistance), and the capacitor discharges through the neon lamp and produces light. Once the capacitor is fully discharged through the lamp, it again begins to charge, and the process repeats.
To understand the discharging of a capacitor, consider a simple RC circuit with a two-way position switch connected to a voltage source. Once the capacitor is fully charged, the position of the switch is moved to disconnect the battery from the circuit. Now the circuit reduces to a simple series connection of the resistor, the capacitor, and the switch. The voltage source is completely removed from the circuit. Instantly, the capacitor then discharges through the resistor, and its charge decreases to zero. The relaxation oscillator controls indicator lights that flash at a frequency determined by the values for R and C.
Kirchhoff's loop rule is used to analyze the circuit. Using the definition of the current and integrating the loop equation gives the charge on the capacitor as a function of time:
The expression for current can be found by taking the time derivative of the charge:
The negative sign in the expression indicates that the direction of current flow is opposite compared to the direction of the current in the case of charging. The charge, current, and voltage magnitudes decrease exponentially, approaching zero as time increases.
Suggested Reading
- Young, H.D. and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco,CA: Pearson. Section 20.4; pages 867-868.
- OpenStax. (2019). University Physics Vol. 2. [Web version]. Retrieved from https://openstax.org/details/books/university-physics-volume-2; section 10.5; pages 471–472.