Capacitance | 电容 - aleveler.com

📚 Capacitance | 电容

Capacitance is a core topic in CIE A-Level Physics. It describes how much electric charge a capacitor can store per unit potential difference. This article explains the definition, capacitor combinations, energy storage, exponential charging and discharging, and the role of the time constant.

电容是 CIE A-Level 物理中的一个核心主题。它描述电容器每单位电势差能够储存多少电荷。本文讲解电容的定义、电容器的组合、能量储存、指数式充放电以及时间常数的作用。

1. What is Capacitance? | 什么是电容？

Capacitance C is defined as the charge Q stored on one plate per unit potential difference V across the plates.

电容 C 定义为电容器一块极板上储存的电荷 Q 与两极板间电势差 V 之比。

C = Q / V

The SI unit of capacitance is the farad, F. One farad is one coulomb per volt, so 1 F = 1 C V⁻¹.

电容的国际单位是法拉（F）。1 法拉等于每伏特 1 库仑，因此 1 F = 1 C V⁻¹。

2. The Farad and Charge Separation | 法拉与电荷分离

A capacitor is an electrical component that stores charge on two conducting plates separated by an insulator, called the dielectric.

电容器是一种电气元件，它在两块被绝缘体（称为电介质）隔开的导电板之间储存电荷。

When a p.d. is applied, electrons move onto one plate and away from the other. The plates therefore carry equal but opposite charges, +Q and −Q.

当施加电势差时，电子移动到一块极板上并从另一块极板移走。因此两极板带等量异种电荷 +Q 和 −Q。

The farad is a very large unit. Practical capacitors are usually in microfarads (μF), nanofarads (nF) or picofarads (pF).

法拉是一个非常大的单位。实际电容器的电容通常以微法（μF）、纳法（nF）或皮法（pF）为单位。

3. Parallel-plate Capacitors | 平行板电容器

For a parallel-plate capacitor, the capacitance is proportional to the area A of the plates and inversely proportional to their separation d.

对于平行板电容器，电容与极板面积 A 成正比，与极板间距 d 成反比。

C = ε₀ εᵣ A / d

Here ε₀ is the permittivity of free space and εᵣ is the relative permittivity, also called the dielectric constant, of the material between the plates.

其中 ε₀ 是真空介电常数，εᵣ 是两极板间材料的相对介电常数，也称为介电常数。

Increasing A or inserting a dielectric increases C. Increasing d decreases C.

增大面积 A 或插入电介质会增大电容 C。增大间距 d 会减小电容 C。

4. Dielectric Materials | 介电材料

A dielectric is an insulating material placed between the plates. It increases capacitance because it reduces the effective electric field for the same charge.

电介质是置于两极板之间的绝缘材料。它能增大电容，因为在相同电荷下它削弱了有效电场。

The molecules in a dielectric become polarised. This creates an opposing field, so the p.d. across the capacitor is smaller for the same charge.

电介质中的分子会被极化。这会产生一个相反电场，因此在相同电荷下电容器两端的电势差变小。

C = εᵣ C₀

Thus a dielectric increases stored charge and energy capacity for a given applied voltage.

因此，在给定外加电压下，电介质提高了储存电荷和能量的能力。

5. Capacitors in Series and Parallel | 电容器的串联与并联

For capacitors in parallel, the total capacitance is the sum of individual capacitances.

对于并联电容器，总电容等于各个电容之和。

C_total = C₁ + C₂ + C₃ + …

This is because the p.d. across each capacitor is the same, but the total stored charge is shared.

这是因为每个电容器两端的电势差相同，但总储存电荷由各电容器共同承担。

For capacitors in series, the reciprocal total capacitance is the sum of reciprocals.

对于串联电容器，总电容的倒数等于各个电容倒数之和。

1 / C_total = 1 / C₁ + 1 / C₂ + 1 / C₃ + …

In series, the charge on each capacitor is the same, and the total p.d. is divided across them.

串联时，每个电容器上的电荷相同，总电势差按各电容器分配。

6. Energy Stored in a Charged Capacitor | 电容器储存的能量

Work must be done to move charge onto a capacitor. The stored electrical potential energy is given by:

必须做功才能将电荷移到电容器上。储存的电势能为：

E = ½ Q V = ½ C V² = ½ Q² / C

Use E = ½ Q V when charge and p.d. are known, E = ½ C V² for a fixed capacitor, and E = ½ Q² / C when charge is fixed.

当已知电荷和电势差时使用 E = ½ Q V；对于固定电容器使用 E = ½ C V²；当电荷固定时使用 E = ½ Q² / C。

The factor ½ appears because the average p.d. during charging is V / 2, not V, if the capacitor starts uncharged.

公式中的 ½ 出现是因为如果电容器从零开始充电，充电过程中的平均电势差为 V / 2，而不是 V。

7. Charging a Capacitor | 电容器的充电

When a capacitor is charged through a resistor from a battery of e.m.f. E, the charge increases exponentially to a final value Q₀ = C E.

当电容器通过电阻由电动势为 E 的电池充电时，电荷按指数规律增长到最终值 Q₀ = C E。

q = Q₀ ( 1 − e^(−t/RC) )

Similarly, the p.d. across the capacitor increases as V = V₀ ( 1 − e^(−t/RC) ), and the charging current decreases as I = I₀ e^(−t/RC).

同理，电容器两端电势差按 V = V₀ ( 1 − e^(−t/RC) ) 增大，而充电电流按 I = I₀ e^(−t/RC) 减小。

Initially the current is large because the p.d. across the resistor is large. As the capacitor charges, the current falls toward zero.

开始时电流很大，因为电阻两端的电势差很大。随着电容器充电，电流逐渐减小并趋近于零。

8. Discharging a Capacitor | 电容器的放电

When a charged capacitor is connected across a resistor, the charge, p.d. and current all decay exponentially from their initial values.

当已充电的电容器与电阻连接时，电荷、电势差和电流都从各自的初始值按指数规律衰减。

Q = Q₀ e^(−t/RC), V = V₀ e^(−t/RC), I = I₀ e^(−t/RC)

The product RC appears in the exponent. It controls how quickly the capacitor discharges.

乘积 RC 出现在指数中。它控制电容器放电的快慢。

During discharge, the current direction is opposite to the charging current. Energy stored in the capacitor is dissipated as heat in the resistor.

放电时，电流方向与充电电流相反。电容器中储存的能量以热量形式在电阻中耗散。

9. The Time Constant and Exponential Graphs | 时间常数与指数图像

The time constant τ is defined as τ = RC. It has units of seconds when R is in ohms and C is in farads.

时间常数 τ 定义为 τ = RC。当 R 以欧姆、C 以法拉为单位时，它的单位是秒。

τ = R C

After one time constant, a discharging capacitor retains about 37% of its initial charge, because e⁻¹ ≈ 0.37.

经过一个时间常数后，放电中的电容器大约保留初始电荷的 37%，因为 e⁻¹ ≈ 0.37。

For charging, after one time constant the charge reaches about 63% of its final value, since 1 − e⁻¹ ≈ 0.63.

对于充电过程，经过一个时间常数后，电荷达到最终值的约 63%，因为 1 − e⁻¹ ≈ 0.63。

Exponential graphs have a constant half-life. The half-life can be calculated using t½ = R C ln 2 ≈ 0.693 R C.

指数图像具有恒定的半衰期。半衰期可用 t½ = R C ln 2 ≈ 0.693 R C 计算。

10. Applications of Capacitors | 电容器的应用

Capacitors are widely used in smoothing circuits, where they reduce ripple in rectified power supplies by storing and releasing charge.

电容器广泛用于平滑电路，在整流电源中通过储存和释放电荷来减小纹波。

They are also used in timing circuits, flash photography, camera flashes, and touch screens. In a camera flash, a capacitor discharges rapidly to deliver a brief high-power burst.

它们还用于定时电路、闪光摄影、相机闪光灯和触摸屏。在相机闪光灯中，电容器快速放电，以提供短暂的高功率脉冲。

In A-Level problems, recognising the exponential charge and discharge equation and calculating RC is essential for both descriptive and quantitative questions.

在 A-Level 题目中，识别指数式充放电方程并计算 RC 对描述性和定量问题都至关重要。

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Capacitance | 电容