Capacitance

考纲要求

• 19.1 Capacitors and capacitance
  • Define capacitance (including isolated spherical conductors and parallel plate capacitors)
  • Use $C = Q / V$
  • Derive and use formulae for capacitors in series and parallel
• 19.2 Energy stored in a capacitor
  • Determine energy from area under $V$-$Q$ graph
  • Use $W = \frac{1}{2} QV = \frac{1}{2} CV^2$
• 19.3 Discharging a capacitor
  • Analyse graphs of $V$, $Q$, $I$ against $t$ for RC discharge
  • Use time constant $\tau = RC$
  • Use $x = x_0 e^{-t/RC}$ for $I$, $Q$ or $V$ during discharge

常见题型

• 定义题：Define capacitance
• 计算题：Combined capacitance / RC discharge / Energy stored
• 作图题：Sketch $Q$-$t$, $I$-$t$, $V$-$t$ 放电曲线
• 解释题：Explain the function of capacitors in circuits

核心公式

$C = \frac{Q}{V}$

$\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} \quad \text{(series)}$

$C = C_1 + C_2 \quad \text{(parallel)}$

$W = \frac{1}{2} QV = \frac{1}{2} CV^2 = \frac{1}{2} \frac{Q^2}{C}$

$\tau = RC$

$x = x_0 e^{-t/RC}$

常见错误

• 电容串并联公式与电阻搞反
• 放电用 $x = x_0(1 - e^{-t/RC})$（这是充电公式）
• 时间常数 $\tau = RC$ 的单位是 s（不是 $\Omega$F 等奇怪单位）