📚 A-Level OCR Physics: Capacitance – Key Points & Revision | 电容 考点精讲
Capacitance is a core topic in OCR A-Level Physics, linking electric fields, circuits and energy storage. This article breaks down the key formulas, graphs and experiments you need to master for your exams, with clear English‑Chinese explanations for every concept.
电容是 OCR A-Level 物理的核心考点，它将电场、电路与能量储存联系起来。本文逐一拆解你必须掌握的关键公式、图线和实验，每个概念都配有清晰的中英双语讲解。
1. Definition of Capacitance | 电容的定义
Capacitance (C) is defined as the charge (Q) stored per unit potential difference (V) across a capacitor: a component stores 1 coulomb of charge when 1 volt is applied across it has a capacitance of 1 farad (F).
电容（C）定义为电容器每单位电势差（V）所储存的电荷（Q）：在 1 伏特电压下储存 1 库仑电荷的电容为 1 法拉（F）。
C = Q / V
The unit farad is often very large, so practical capacitors are labelled in microfarads (μF), nanofarads (nF) or picofarads (pF). The relationship is linear for a fixed capacitor; a Q‑V graph yields a straight line through the origin, with the gradient giving the capacitance.
法拉这个单位通常很大，因此实际电容器常用微法（μF）、纳法（nF）或皮法（pF）标注。对固定电容器，电荷与电压成线性关系；Q‑V 图线是一条通过原点的直线，斜率即为电容。
2. Parallel Plate Capacitor | 平行板电容器
For a parallel plate capacitor, the capacitance depends on the area A of the overlapping plates, the separation d between them and the permittivity ε of the dielectric material filling the gap:
对于平行板电容器，其电容取决于极板正对面积 A、极板间距 d 以及填充在极板间介质的介电常数 ε：
C = ε A / d
Here ε = ε0 εr, where ε0 is the permittivity of free space (8.85×10⁻¹² F m⁻¹) and εr is the relative permittivity (dielectric constant) of the insulator. Increasing the plate area, reducing the separation, or using a dielectric with a higher εr all increase capacitance.
式中 ε = ε0 εr，ε0 是真空介电常数（8.85×10⁻¹² F m⁻¹），εr 是绝缘介质的相对介电常数。增大极板面积、减小间距或使用 εr 更高的介质都能增大电容。
tions, you may be asked to combine this formula with C = Q / V to find unknown quantities.
电介质不仅能增大电容，还能通过提高最大工作电压来防止击穿。考题中常要求将本式与 C = Q / V 结合，求解未知量。
3. Capacitors in Series and Parallel | 电容器的串联与并联
When capacitors are connected in series, the total capacitance is smaller than the smallest individual capacitance. The formula is:
电容器串联时，总电容小于各电容中最小的那个。公式为：
1 / Ctotal = 1 / C₁ + 1 / C₂ + …
In a series arrangement, the charge Q on each capacitor is the same, and the potential differences add up to the supply voltage. This is analogous to resistors in parallel.
串联时，每个电容器上的电荷 Q 相等，各自电势差之和等于电源电压。这类似于电阻的并联。
When capacitors are connected in parallel, the total capacitance is simply the sum:
电容器并联时，总电容为各电容之和：
Ctotal = C₁ + C₂ + …
Here the potential difference across each branch is the same, while the charges add up. The combined effect increases the plate area available for storing charge.
此时各支路两端电势差相等，而电荷量相加。并联的总效果相当于增大了可供储电荷的极板面积。
You should be able to derive these rules from conservation of charge and energy. Typical OCR questions will ask you to calculate combined capacitance and deduce how voltage or charge divides.
你应能从电荷与能量守恒出发推导这些规律。典型的 OCR 试题会要求计算组合电容，并推导电压或电荷的分配。
4. Energy Stored by a Capacitor | 电容器储存的能量
A charged capacitor stores electrical potential energy in the electric field between its plates. The energy E can be expressed in three equivalent forms:
已充电的电容器将电势能储存在极板间的电场中。能量 E 有三种等价的表达式：
E = ½ Q V = ½ C V² = ½ Q² / C
The factor ½ appears because the average potential difference during charging is half the final value. When a capacitor discharges through a resistor, this stored energy is dissipated as heat in the resistor.
½ 因子源于充电过程中平均电势差为最终值的一半。当电容器通过电阻放电时，储存的能量以热的形式在电阻上耗散。
You can confirm the relationship by finding the area under a Q‑V graph, which is a triangle for a linear capacitor. Exam problems often involve calculating energy changes when a capacitor discharges or when two capacitors are connected.
可通过求 Q‑V 图线下面积来验证，线性电容的图线构成三角形。考题常涉及放电过程或两个电容器连接时能量变化的计算。
5. Charging a Capacitor through a Resistor | RC 充电过程
When an uncharged capacitor is connected in series with a resistor to a d.c. supply of voltage V₀, the charge, p.d. and current change exponentially with time. The governing equations are:
将一个未充电的电容器与电阻串联后接到电压为 V₀ 的直流电源上，电荷、电压和电流均随时间作指数规律变化。基本方程如下：
Q = Q₀ (1 – e–t/RC)
V = V₀ (1 – e–t/RC)
I = I₀ e–t/RC
r charges. After a long time, the capacitor behaves like an open circuit: current falls to zero and the p.d. equals the supply voltage.
初始时刻电流最大（I₀ = V₀ / R），并随充电过程的进行而减小。长时间后，电容器相当于开路：电流降至零，两端电势差等于电源电压。
The product RC governs the rate of charging; it has units of seconds and is called the time constant (τ). The charging curve is an inverted exponential that rises rapidly at first and then gradually levels off.
乘积 RC 决定了充电的快慢，其单位为秒，称为时间常数（τ）。充电曲线是一条倒置的指数曲线，起初上升很快，随后逐渐趋于平缓。
6. Discharging a Capacitor | RC 放电过程
If a charged capacitor is disconnected from the supply and connected across a resistor, it discharges. The charge, p.d. and current all decay exponentially:
若将已充电的电容器脱离电源并接在电阻两端，电容器便开始放电。电荷、电势差和电流均按指数规律衰减：
Q = Q₀ e–t/RC
V = V₀ e–t/RC
I = I₀ e–t/RC
es at t = 0. The discharge current direction is opposite to the charging current, so the I‑t graph falls below the time axis if signed conventions are used.
式中 Q₀、V₀ 和 I₀ 是 t = 0 时的初始值。放电电流的方向与充电电流相反，因此若考虑符号规定，I‑t 图线会落在时间轴下方。
The exponential nature means that the quantity halves in equal time intervals. The ‘half‑life’ t½ = RC ln 2, which is often used in experimental analysis to determine RC.
指数规律意味着在相等的时间间隔内，物量每次减半。“半衰期” t½ = RC ln 2，常被用于实验分析中确定 RC。
7. Time Constant τ | 时间常数 τ
defined as τ = RC. Its significance lies in how quickly a capacitor charges or discharges:
RC 电路的时间常数定义为 τ = RC。它的重要意义在于衡量电容器充电或放电的快慢：
After a time t = τ during charging, the p.d. reaches 63% of its final value.
充电过程中，经过 t = τ 时间后，电势差达到其最终值的 63%。
After t = τ during discharging, the p.d. falls to 37% of its initial value.
放电过程中，经过 t = τ 时间后，电势差降至初始值的 37%。
After about 5τ, the capacitor is considered fully charged (99.3%) or fully discharged (0.7%).
约 5τ 后，电容器被认为已充满（99.3%）或已放空（0.7%）。
ts the time axis or the final value line after a time τ. This geometric property is frequently examined in OCR papers.
时间常数还可从充电或放电曲线在 t = 0 处的切线求得：该切线与时间轴或最终值线的交点对应的时间即 τ。这一几何性质在 OCR 试卷中频繁出现。
8. Exponential Graphs and Their Features | 指数曲线特征
Both charging and discharging produce characteristic exponential graphs that you must be able to sketch and interpret. For discharging a capacitor:
充电和放电过程都会产生典型的指数曲线，你必须能够绘制和解释它们。对于电容器的放电：
The Q‑t and V‑t graphs start at initial value and decay asymptotically towards zero.
Q‑t 和 V‑t 图线从初始值开始，渐近衰减至零。
The I‑t graph magnitude also decays, but the direction may be shown as negative.
I‑t 图线的幅度同样衰减，但方向可能显示为负值。
For charging a capacitor:
The Q‑t and V‑t graphs start at zero and rise asymptotically towards the final value.
Q‑t 和 V‑t 图线从零开始，渐近上升至最终值。
The I‑t graph starts at a maximum and decays to zero.
I‑t 图线从最大值开始衰减至零。
You should be able to use the curves to determine the time constant, either by reading 63% values or by drawing tangents. The natural logarithmic form ln V = ln V₀ – t/RC is also useful for linearising data to find C.
你应能利用曲线求时间常数，可通过读取 63% 的值或作切线实现。自然对数形式 ln V = ln V₀ – t/RC 也常用于将数据线性化，以求出 C。
9. Experimental Determination of Capacitance | 测定电容的实验方法
OCR often asks about practical investigations. A common method is to charge/discharge a capacitor through a known resistor, using a voltmeter and a stopwatch to record V‑t data. Plotting ln V against t gives a straight line of gradient –1/RC, from which C can be found if R is known.
OCR 常考查实验探究。常用的方法是：通过已知电阻对电容器充电或放电，用电压表和秒表记录 V‑t 数据。绘制 ln V‑t 图得到一条斜率为 –1/RC 的直线，若已知 R 便能求出 C。
ises linearly (V = (I/C) t + constant), and C can be calculated from the gradient of the V‑t graph. For an electrolytic capacitor, careful attention to polarity is essential.
另一种方法是使用恒流源：若用恒定电流 I 对电容器充电，电势差将线性上升（V = (I/C) t + 常数），由 V‑t 图线的斜率即可计算 C。对于电解电容器，必须格外注意极性。
In all experiments, you should discuss sources of error such as meter resistance, leakage currents and heating effects. Repeating readings and using sensors/data loggers improve accuracy.
在所有实验中，应讨论误差来源，如电表内阻、漏电流和热效应等。重复读数并使用传感器/数据记录器可提高精度。
10. Practical Applications of Capacitors | 电容器的实际应用
Capacitors appear in many real‑ duce a brief, intense flash. The stored energy E = ½ C V² determines the flash brightness.
电容器出现在许多实际电路中。在相机闪光灯中，电容器由电池缓慢充电，然后通过氙灯快速放电，产生短暂而强烈的闪光。储存的能量 E = ½ C V² 决定了闪光亮度。
In power supplies, large capacitors are used for smoothing: they charge when the rectified voltage rises and discharge when it falls, reducing the ripple. The required capacitance depends on the load resistance and the acceptable ripple voltage.
在电源中，大电容用于平滑滤波：当整流后的电压升高时充电，电压下降时放电，从而减小纹波。所需电容值取决于负载电阻和可接受的纹波电压。
Timing circuits, such as those found in intermittent wipers or pacemakers, rely on the predictable RC charging/discharging time. Touch‑sensitive screens and capacitive sensors also exploit the principles of capacitance change due to a nearby conductor.
定时电路（如间歇式雨刷器或心脏起搏器中的电路）依赖于可预知的 RC 充放电时间。触摸屏和电容式传感器也利用了邻近导体会改变电容的原理。
11. Key Revision Points | 考点小结
E = ½ QV, series and parallel rules, exponential charging and discharging equations with e–t/RC, and the significance of the time constant τ = RC.
想在 OCR 电容部分拿分，请牢记这些要点：C = Q/V、平行板电容器 C = εA/d、能量储存 E = ½ QV、串并联规律、含 e–t/RC 的指数充电放电方程，以及时间常数 τ = RC 的意义。
Practice sketching Q‑t, V‑t and I‑t graphs for both charging and discharging, and be ready to interpret straight‑line graphs derived from exponential data. Strong understanding of the underlying physics will help you handle unfamiliar contexts confidently.
请多练习绘制充电和放电的 Q‑t、V‑t 和 I‑t 图线，并准备好解读由指数数据得到的直线图。扎实理解物理本质将帮助你从容应对陌生情境。
Published by TutorHao | Physics Revision Series | aleveler.com
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