27.16
RC 电路：电容器放电
RC电路的应用之一是张弛振荡器。 张弛振荡器包括电压源、电容器、电阻器和氖灯。 灯的作用就像开路（无限电阻），直到氖灯两端的电位差达到特定电压。 在该电压下，灯的作用就像短路（零电阻），电容器通过氖灯放电并产生光。 一旦电容器通过灯完全放电，它就会再次开始充电，并重复该过程。
要了解电容器的放电，可…
当电容器充满电时，电容器两端的电势差等于电源电压。
如果电池从电路中断开，电容器将放电。
利用基尔霍夫回路定律可得到一个方程。此处，电流表示电荷离开电容器的速率。由于电容器随时间失去电荷，因此符号为负。
代入电流的定义可得到一个方程。对该方程进一步积分，可得到电容器上的电荷随时间变化的函数关系。
对于放电的电容器，其上的电荷会从初始电荷开始呈指数衰减，该初始电荷是电容器在充电过程中所获得的最大电荷。
电容器上的电荷对时间的导数可得到电流随时间变化的表达式。
表达式中的负号表示电流方向与电容器充电情况下的方向相反。
电流衰减呈指数行为。其幅值随时间指数衰减，并在时间趋于无穷时趋近于零。
View the full transcript and gain access to JoVE Core videos
Q1: What happens to a capacitor when the battery is disconnected from an RC circuit?
When the battery disconnects, the capacitor discharges through the resistor. The charge decreases exponentially from its initial maximum value toward zero. The circuit reduces to a simple series connection of the resistor, capacitor, and switch, with the voltage source completely removed from the circuit.
Q2: How does the current behave during capacitor discharge?
The discharge current decays exponentially with time, starting at its maximum value and approaching zero as time increases. The negative sign in the current expression indicates the current flows opposite to the charging direction. The current magnitude decreases continuously throughout the discharge process.
Q3: Why is there a negative sign in the capacitor discharge current equation?
The negative sign signifies that the current direction during discharge is opposite to the current direction during charging. Since the capacitor loses charge with time, the negative sign reflects this opposite flow direction compared to when the capacitor was acquiring charge from the voltage source.
Q4: What mathematical relationship describes charge decay in a discharging capacitor?
The charge on a discharging capacitor decreases exponentially as a function of time, starting from the maximum charge acquired during charging. Using Kirchhoff's loop rule and integrating the resulting equation yields this exponential decay relationship, where charge approaches zero as time approaches infinity.
Q5: How does a relaxation oscillator use capacitor discharge?
A relaxation oscillator comprises a voltage source, capacitor, resistor, and neon lamp. The lamp acts as an open circuit until voltage reaches a threshold, then acts as a short circuit, allowing the capacitor to discharge through it and produce light. The process repeats as the capacitor recharges, creating flashing controlled by R and C values.
Q6: What is the relationship between voltage and charge during capacitor discharge?
Both voltage and charge magnitudes decrease exponentially during discharge, approaching zero as time increases. Since voltage across the capacitor is proportional to the charge stored, they decay together at the same exponential rate determined by the RC time constant of the circuit.
Q7: How does the discharge process differ from charging in an RC circuit?
During charging, a voltage source drives charge onto the capacitor; during discharge, no external source exists and the capacitor releases stored charge through the resistor. The current direction reverses, indicated by the negative sign in the discharge current equation, though both processes follow exponential behavior.