📚 IB & CCEA Physics: Capacitance Key Points | IB 和 CCEA 物理：电容考点精讲
Capacitance is a core topic in both IB and CCEA Physics, bridging electrostatics and circuit analysis. A clear understanding of how capacitors store charge and energy, how they behave in DC circuits, and the role of dielectrics is essential for high marks in exams. This guide walks you through every key concept, equation, and graph you need to master, with paired bilingual explanations to reinforce your learning.
电容是 IB 和 CCEA 物理共同的核心专题，连接着静电学和电路分析。清晰理解电容器如何储存电荷与能量、它们在直流电路中的行为以及电介质的作用，是考试取得高分的关键。这篇考点精讲将带你逐一攻克所有重要概念、公式和图像，中英双语对照讲解帮助你牢固掌握。
1. Definition of Capacitance | 电容的定义
Capacitance C is defined as the amount of charge Q stored per unit potential difference V across a conductor. The SI unit is the farad (F), where 1 F = 1 C V⁻¹. For any isolated conductor or capacitor, the relationship is linear for a given geometry, so C = Q / V.
电容 C 定义为导体上储存的电荷量 Q 与其两端电势差 V 之比。国际单位是法拉（F），1 F = 1 C V⁻¹。对于任何孤立导体或电容器，给定几何形状下该关系是线性的，因此 C = Q / V。
C = Q / V
A capacitor is a practical device designed to store charge, typically made of two conducting plates separated by an insulator or dielectric. The capacitance depends only on the physical dimensions and the properties of the dielectric, not on the applied voltage or stored charge.
电容器是一种设计用来储存电荷的实际装置，通常由两片被绝缘体或电介质隔开的导电板构成。电容值仅取决于物理尺寸和电介质的性质，与施加的电压或储存的电荷量无关。
2. The Parallel Plate Capacitor | 平行板电容器
For an ideal parallel plate capacitor, the capacitance is given by C = ε₀ εᵣ A / d. Here A is the area of one plate, d is the separation between plates, ε₀ = 8.85 × 10⁻¹² F m⁻¹ is the permittivity of free space, and εᵣ is the relative permittivity (dielectric constant) of the material between the plates.
对于理想的平行板电容器，电容由 C = ε₀ εᵣ A / d 给出。其中 A 是一片板的面积，d 是板间距离，ε₀ = 8.85 × 10⁻¹² F m⁻¹ 为真空介电常数，εᵣ 是板间材料的相对介电常数（介电常数）。
C = ε₀ εᵣ A / d
This equation shows that capacitance increases with larger plate area and higher permittivity, but decreases with greater separation. In a vacuum, εᵣ = 1. Inserting a dielectric with εᵣ > 1 increases the capacitance by a factor of εᵣ.
该公式表明，电容随板面积增大和介电常数升高而增大，随间距增大而减小。在真空中 εᵣ = 1。插入 εᵣ > 1 的电介质会使电容增大 εᵣ 倍。
3. 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: Ctotal = C₁ + C₂ + C₃ + …. All capacitors share the same voltage, and the total stored charge is the sum of the charges on each.
当电容器并联时，总电容为各电容之和：C总 = C₁ + C₂ + C₃ + …。所有电容器承受相同的电压，总储存电荷为各电容器电荷之和。
For a series connection, the reciprocal of the total capacitance equals the sum of the reciprocals: 1/Ctotal = 1/C₁ + 1/C₂ + 1/C₃ + …. Here, each capacitor carries the same charge, and the total voltage is the sum of the individual voltages.
对于串联连接，总电容的倒数等于各电容倒数之和：1/C总 = 1/C₁ + 1/C₂ + 1/C₃ + …。此时每个电容器带的电荷量相同，总电压为各电压之和。
- Parallel: C increases, same V.
- Series: C decreases, same Q.
并联：电容增大，电压相同。串联：电容减小，电荷相同。
Parallel: Ctotal = C₁ + C₂ | Series: 1/Ctotal = 1/C₁ + 1/C₂
4. Energy Stored in a Capacitor | 电容器储存的能量
The energy U stored in a charged capacitor can be derived from the work done to move charge against the growing potential difference. The total energy is given by U = ½ Q V = ½ C V² = ½ Q² / C.
充了电的电容器中储存的能量 U 可以从反抗逐渐升高的电势差移动电荷所做的功推导出来。总能量由 U = ½ Q V = ½ C V² = ½ Q² / C 给出。
E = ½ C V² = ½ Q V = ½ Q² / C
This energy is stored in the electric field between the plates. For a parallel plate capacitor, the energy density (energy per unit volume) is u = ½ ε₀ εᵣ E², where E is the electric field strength.
这份能量储存在板间的电场中。对于平行板电容器，能量密度（单位体积的能量）为 u = ½ ε₀ εᵣ E²，其中 E 是电场强度。
5. Charging a Capacitor through a Resistor | 通过电阻对电容器充电
When a capacitor is charged through a resistor from a constant voltage source V₀, the charge and voltage grow exponentially. The voltage across the capacitor follows V(t) = V₀ (1 − e^(−t/RC)), where R is the resistance and C the capacitance. The current decays as I(t) = (V₀/R) e^(−t/RC).
当一个电容器通过电阻从恒定电压源 V₀ 充电时，电荷和电压呈指数增长。电容器两端电压遵循 V(t) = V₀ (1 − e^(−t/RC))，其中 R 是电阻，C 是电容。电流按 I(t) = (V₀/R) e^(−t/RC) 衰减。
V(t) = V₀ (1 − e^(−t/RC))
Initially, at t = 0, the uncharged capacitor behaves like a short circuit (zero voltage), and the initial current is V₀/R. As time passes, the voltage approaches V₀ and the current drops to zero.
初始时刻 t = 0，未充电的电容器相当于短路（电压为零），初始电流为 V₀/R。随着时间推移，电压趋近于 V₀，电流降至零。
6. Discharging a Capacitor | 电容器放电
When a charged capacitor is discharged through a resistor, both charge and voltage fall exponentially. The discharge equations are Q(t) = Q₀ e^(−t/RC), V(t) = V₀ e^(−t/RC), and the current I(t) = I₀ e^(−t/RC), where Q₀, V₀, and I₀ are the initial values.
当充了电的电容器通过电阻放电时，电荷和电压均按指数规律下降。放电方程为 Q(t) = Q₀ e^(−t/RC)，V(t) = V₀ e^(−t/RC)，以及电流 I(t) = I₀ e^(−t/RC)，其中 Q₀、V₀ 和 I₀ 为初始值。
V(t) = V₀ e^(−t/RC)
The product RC appears in the exponent and governs how quickly the discharge occurs. The larger the RC, the slower the decay.
乘积 RC 出现在指数中，决定了放电的快慢。RC 越大，衰减越慢。
7. Time Constant and Its Significance | 时间常数及其意义
The time constant τ (tau) of an RC circuit is defined as τ = RC. It has units of seconds (Ω × F = s). After one time constant during charging, the capacitor voltage reaches about 63% of the supply voltage; during discharging, the voltage falls to about 37% of its initial value.
RC 电路的时间常数 τ（tau）定义为 τ = RC。它的单位是秒（Ω × F = s）。充电时经过一个时间常数，电容电压达到电源电压的约 63%；放电时电压降至其初始值的约 37%。
After a time of 5τ, the capacitor is considered fully charged or fully discharged (over 99%). The time constant can be found experimentally from graphs by locating the intercept of the initial tangent with the time axis, or by reading the time taken for the voltage to halve and using t₁/₂ = τ ln2.
经过 5τ 的时间后，可以认为电容器已完全充电或完全放电（超过 99%）。时间常数可通过实验图像求得：寻找初始切线在时间轴上的截距，或测出电压减半所需时间并利用 t₁/₂ = τ ln2。
8. Exponential Decay Equations and Logarithmic Analysis | 指数衰减方程与对数分析
Because the discharge follows V = V₀ e^(−t/τ), taking natural logs gives ln V = ln V₀ − t/τ. A plot of ln V against time yields a straight line with gradient −1/τ. This linearisation is a common exam requirement for determining the time constant from data.
由于放电遵循 V = V₀ e^(−t/τ)，取自然对数得 ln V = ln V₀ − t/τ。绘制 ln V 随时间变化的图像会得到一条斜率为 −1/τ 的直线。这种线性化方法是考试中根据数据求时间常数的常见要求。
ln V = ln V₀ − t / RC
Similarly, for charging, the equation for the voltage across the resistor is also exponential, and you may be asked to sketch or interpret I–t, V–t, and Q–t graphs, identifying key features such as initial gradients and asymptotic behaviour.
类似地，对于充电过程，电阻两端的电压也是指数形式，你可能需要画出或分析 I–t、V–t 和 Q–t 图像，并识别初始梯度、渐近行为等关键特征。
9. Dielectric Materials and Their Effect | 电介质材料及其影响
A dielectric is an insulating material that, when placed between the plates of a capacitor, increases its capacitance by a factor εᵣ. This happens because the electric field polarises the dielectric molecules, producing an opposing field that reduces the net electric field and hence the potential difference for the same stored charge.
电介质是一种绝缘材料，当置于电容器板间时，会使电容增大 εᵣ 倍。这是因为电场使电介质分子极化，产生一个反向电场，削弱了净电场，从而在储存相同电荷的情况下降低了电势差。
In practical capacitors, dielectrics also allow the plates to be placed very close without shorting, further increasing capacitance. Common dielectrics include paper, ceramic, and electrolytic films. Electrolytic capacitors are polarised and must be connected the correct way round in a DC circuit.
在实际电容器中，电介质还能让板间距非常小而不发生短路，从而进一步增大电容。常见的电介质有纸、陶瓷和电解膜。电解电容器有极性，在直流电路中必须正确连接。
10. Capacitor Charge and Discharge Graphs | 电容器充放电图像
You must be able to sketch and interpret curves for charge, voltage, and current during charging and discharging. Key characteristics:
– Charging: V and Q rise asymptotically towards maximum; I starts at a maximum and decays to zero.
– Discharging: All quantities decay exponentially to zero.
– The initial gradient of a V–t graph for charging equals V₀/RC; for discharging, it equals −V₀/RC.
– The area under an I–t graph gives the total charge transferred.
你必须能够画出并分析充电和放电过程中电荷、电压、电流的曲线。关键特征：
– 充电：V 和 Q 渐近上升至最大值；I 从最大值衰减至零。
– 放电：所有量均以指数方式衰减至零。
– 充电时 V–t 图像的初始梯度等于 V₀/RC；放电时初始梯度为 −V₀/RC。
– I–t 图像下的面积表示转移的总电荷量。
Time constant can be read directly from such graphs: on a discharge curve, it is the time for V to drop to 37% of its initial value.
时间常数可以直接从这类图像上读出：在放电曲线上，它等于电压降至初始值 37% 所需的时间。
11. Exam Tips and Common Mistakes | 考试技巧与常见错误
Unit conversions: Capacitance values in formulas must be in farads; often you need to convert from µF, nF, or pF. Remember 1 µF = 10⁻⁶ F, 1 nF = 10⁻⁹
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