27.15:
RC Circuits: Charging A Capacitor
A camera flashlight works on the capacitor's charging and discharging principle. On pressing the shutter button, a short burst of current provided by the charged capacitor creates a flash.
To understand the charging of a capacitor, consider an RC circuit connected in series to an ideal battery.
When the switch is closed, the capacitor begins to charge. The battery's emf is distributed between the resistor and capacitor.
The potential difference across the capacitor increases until it is equal to the battery's emf.
Applying Kirchoff's rule, a differential equation is obtained, which can be integrated to obtain an expression for the charge of the capacitor as a function of time.
By differentiating with respect to time, an expression for the charging current is obtained.
The current has its maximum value when time equals zero and decays exponentially to zero when time approaches infinity.
Conversely, the charge of the capacitor increases exponentially from zero to maximum as time approaches infinity.
Here, R–C is the time constant. It is the time interval when current and charge attain the given values.
27.15:
RC Circuits: Charging A Capacitor
A circuit containing resistance and capacitance is called an RC circuit. A capacitor is an electrical component that stores electric charge by storing energy in an electric field. Consider a simple RC circuit having a DC (direct current) voltage source ε, a resistor R, a capacitor C, and a two-way position switch. In the circuit, the capacitor can be charged or discharged depending on the position of the switch.
When the switch is moved to connect the battery, the circuit reduces to a simple series connection of the voltage source, the resistor, the capacitor, and the switch. In this circuit, if we apply Kirchhoff's law, we obtain a differential equation that is further integrated to obtain an expression for the charge on the charging capacitor as a function of time.
The charge on the capacitor and the current through the resistor has an inverse relationship: as the charge increases, the current decreases. The current through the resistor can be obtained by taking the time derivative of the charge.
At time "t " equal to RC, the current in the RC circuit decreases to 0.368 of its initial value. At the same time, the capacitor charge has reached 0.632 of its maximum value. Therefore, the product RC is a measure of how quickly the capacitor charges. This term RC is called the circuit's time constant or the relaxation time.
When the time constant is small, the capacitor charges quickly; when the time constant is larger, the charging takes more time. If the resistance is small, it's easier for the current to flow, and the capacitor charges more quickly.
Suggested Reading
- Young, H.D. and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson. Section 20.4; pages 864–866.
- OpenStax. (2019). University Physics Vol. 2. [Web version]. Retrieved from https://openstax.org/details/books/university-physics-volume-2; section 10.5; pages 469–471.