📚 IB OCR Physics: Capacitance – Exam Essentials | IB OCR 物理：电容考点精讲
Capacitance is a core concept in both IB Physics (Topic 11.3) and OCR A Level Physics (Module 6.1). A thorough understanding of capacitors, their behaviour in circuits, energy storage, and the mathematics of RC charge/discharge is essential for high marks. This article covers all key points, linking theory to typical exam questions to help you master capacitance.
电容是 IB 物理（Topic 11.3）和 OCR A Level 物理（Module 6.1）的核心概念。透彻理解电容器、它们在电路中的行为、能量储存以及 RC 充放电的数学计算，是获得高分的关键。本文将涵盖所有重点，将理论与典型考题联系起来，帮助你掌握电容知识。
1. Definition of Capacitance | 电容的定义
Capacitance (C) is defined as the charge stored per unit potential difference across a capacitor. It is a measure of a capacitor’s ability to store charge.
电容（C）定义为电容器每单位电势差所储存的电荷量。它衡量了电容器储存电荷的能力。
C = Q / V
The unit of capacitance is the farad (F), where 1 F = 1 C V⁻¹. In practice, capacitors often have values in microfarads (µF), nanofarads (nF) or picofarads (pF).
电容的单位是法拉（F），1 F = 1 C V⁻¹。实际应用中，电容器的数值常用微法（µF）、纳法（nF）或皮法（pF）表示。
2. Capacitance of a Parallel-Plate Capacitor | 平行板电容器的电容
For a parallel-plate capacitor, the capacitance is proportional to the plate area (A) and inversely proportional to the plate separation (d). The permittivity of free space (ε₀) is the constant of proportionality for a vacuum.
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。在真空中，比例常数为真空介电常数（ε₀）。
C = ε₀ A / d
Where ε₀ ≈ 8.85 × 10⁻¹² F m⁻¹. If a dielectric material is inserted, the capacitance increases by a factor εᵣ (relative permittivity).
其中 ε₀ ≈ 8.85 × 10⁻¹² F m⁻¹。如果插入电介质材料，电容会增大 εᵣ（相对介电常数）倍。
3. Dielectrics and Relative Permittivity | 电介质与相对介电常数
A dielectric is an insulating material that becomes polarised in an electric field, reducing the effective field strength and allowing more charge to be stored for the same p.d. The relative permittivity (εᵣ), also called dielectric constant, is the ratio of the capacitance with the dielectric to the capacitance in a vacuum.
电介质是一种绝缘材料，在电场中会发生极化，削弱有效电场强度，从而在相同电势差下允许储存更多电荷。相对介电常数（εᵣ），也称介电常数，是有电介质时的电容与真空电容的比值。
C = ε₀ εᵣ A / d
Exam questions often ask you to explain the effect of inserting a dielectric on capacitance, charge, potential difference, and energy stored. Remember that if the capacitor is connected to a battery, V remains constant; if isolated, Q remains constant.
考题常常要求解释插入电介质对电容、电荷、电势差和储存能量的影响。请记住：如果电容器连接着电池，V 保持不变；如果已隔离，则 Q 保持不变。
4. Energy Stored in a Capacitor | 电容器储存的能量
A charged capacitor stores electrical potential energy. The energy stored can be derived from the work done to move charge against the potential difference, and it is given by the three equivalent expressions.
充电的电容器储存电势能。储存的能量可由反抗电势差移动电荷所做的功推导得出，共有三种等价表达式。
E = ½ QV = ½ CV² = ½ Q² / C
In exam problems, choose the most convenient formula based on the given quantities. The area under a Q–V graph represents the energy stored, which is important for multiple-choice and data-analysis questions.
在解题时，根据已知量选择最方便的公式。Q–V 图下的面积代表储存的能量，这在选择题和数据分析题中很重要。
5. Capacitors in Series and Parallel | 电容器的串联与并联
When capacitors are connected in parallel, the total capacitance is the sum of the individual capacitances. This is because each capacitor experiences the same p.d. and total charge is the sum of charges.
电容器并联时，总电容等于各电容之和。这是因为每个电容器承受相同电势差，总电荷为各电荷之和。
C_total = C₁ + C₂ + C₃ + …
For series combinations, the reciprocal of the total capacitance is the sum of the reciprocals of the individual capacitances. All capacitors in series store the same charge, and the p.d. across each is inversely proportional to its capacitance.
串联时，总电容的倒数等于各电容倒数之和。串联的所有电容器储存相同电荷，各电容器上的电势差与其电容成反比。
1 / C_total = 1 / C₁ + 1 / C₂ + …
Common exam traps involve mixing series and parallel circuits or applying these rules to energy and charge calculations. Always identify the configuration first.
常见考题陷阱包括混合串并联电路，或将这些规则应用于能量和电荷计算。务必先识别连接方式。
6. RC Circuits: Charging a Capacitor | RC 电路：电容器充电
When a capacitor is charged through a resistor from a steady p.d. source (V₀), the charge, voltage across the capacitor, and current follow exponential functions. The capacitor voltage grows as it approaches the supply voltage asymptotically.
当电容器通过电阻从恒压源（V₀）充电时，电荷、电容两端电压和电流均服从指数函数。电容器电压渐近地趋近电源电压。
V = V₀ (1 − e^(-t/RC))
The current decreases from its initial maximum (I₀ = V₀/R) according to I = I₀ e^(-t/RC). Similarly, the charge follows q = Q₀ (1 − e^(-t/RC)) where Q₀ = CV₀.
电流从初始最大值（I₀ = V₀/R）按 I = I₀ e^(-t/RC) 减小。类似地，电荷遵循 q = Q₀ (1 − e^(-t/RC))，其中 Q₀ = CV₀。
7. RC Circuits: Discharging a Capacitor | RC 电路：电容器放电
When a charged capacitor discharges through a resistor, the charge, voltage, and current all decay exponentially from their initial values (Q₀, V₀, I₀).
当充电的电容器通过电阻放电时，电荷、电压和电流均从其初始值（Q₀, V₀, I₀）开始指数衰减。
q = Q₀ e^(-t/RC)
V = V₀ e^(-t/RC)
The discharge current flows in the opposite direction to the charging current, and its magnitude decreases as |I| = I₀ e^(-t/RC).
放电电流方向与充电电流相反，其大小按 |I| = I₀ e^(-t/RC) 减小。
8. Time Constant and Graphical Analysis | 时间常数与图像分析
The time constant τ (tau) is defined as τ = RC. It has the unit of seconds. After a time equal to τ, the charge on a discharging capacitor falls to about 37% of its initial value; during charging, the charge reaches about 63% of its final value.
时间常数 τ（tau）定义为 τ = RC，单位为秒。经过时间 τ 后，放电电容器的电荷降至初始值的约 37%；充电过程中，电荷会达到最终值的约 63%。
τ = RC
Graphs of Q, V, or I against time show exponential decay or growth. You must be able to determine τ from a graph by finding the time for the quantity to fall to 37% of its initial value, or by using the initial gradient method. In log-linear plots, the gradient equals –1/τ for discharge.
Q、V 或 I 随时间变化的图像呈现指数衰减或增长。你必须能从图像中求出 τ：找到电量降至初始值 37% 所需的时间，或使用初始切线法。在对数-线性图中，放电曲线的梯度等于 –1/τ。
9. Exponential Decay and Half-Life | 指数衰减与半衰期
For an RC discharging circuit, the half-life t₁/₂ is the time taken for the charge, voltage, or current to reduce by half. It is related to the time constant by t₁/₂ = τ ln 2 ≈ 0.693 RC. This relationship is analogous to radioactive decay and is a favourite topic for linking physics concepts.
对于 RC 放电电路，半衰期 t₁/₂ 是电荷、电压或电流减半所需的时间。它与时间常数的关系为 t₁/₂ = τ ln 2 ≈ 0.693 RC。这个关系与放射性衰变类似，是跨概念关联的热门考点。
Exam questions may ask you to calculate R or C from half-life measurements, or to sketch graphs showing the effect of doubling R or C on the discharge curve.
考题可能要求你根据半衰期测量值计算 R 或 C，或者画出 R 或 C 加倍后对放电曲线影响的示意图。
10. Practical Applications and Exam Tips | 实际应用与考试技巧
Capacitors are used in smoothing circuits for rectified power supplies, in timing circuits, camera flashes, and touch screens. Understanding the energy discharge rate helps explain why a flash can deliver a brief, intense burst of light.
电容器用于整流电源的平滑滤波、定时电路、相机闪光灯和触摸屏等。理解能量释放速率可以解释闪光灯为何能发出短暂而强烈的光线。
In both IB and OCR exams, common mistakes include confusing series and parallel rules, misusing average power, or misinterpreting V–t graphs. Always check whether the capacitor is charging or discharging, and note if the circuit includes multiple resistors that affect τ.
在 IB 和 OCR 考试中，常见错误包括混淆串并联规则、误用平均功率或误解 V–t 图像。务必先确认电容器在充电还是放电，并注意电路中是否包含影响 τ 的多个电阻。
For data-analysis questions, practice finding τ from graphs, applying C = ε₀εᵣA/d to real electrode dimensions, and rearranging logarithmic forms of exponential equations, such as ln(V) = ln(V₀) – t/RC.
对于数据分析题，要练习从图像求 τ、对实际电极尺寸应用 C = ε₀εᵣA/d，以及将指数方程转换为对数形式，如 ln(V) = ln(V₀) – t/RC。
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