Capacitors | 电容器

📚 Capacitors | 电容器

A capacitor is a device that stores electrical energy by holding separated charges. In IB Physics, capacitors appear in both standard and higher level topics, covering their definition, energy storage, charging and discharging behaviour, time constant, and circuit combinations. Understanding capacitors is essential for analysing DC circuits and many real-world applications.

电容器是一种通过保持分离电荷来储存电能的器件。在 IB 物理中，电容器出现在标准水平和高水平的内容中，涵盖其定义、储能、充放电行为、时间常数以及电路组合。理解电容器对于分析直流电路和许多实际应用至关重要。

1. What is a Capacitor? | 什么是电容器？

A capacitor consists of two conducting plates separated by an insulator (dielectric). When connected to a voltage source, one plate gains a net positive charge +Q while the other gains an equal negative charge -Q. The plates hold these charges, creating a uniform electric field in the dielectric and storing potential energy.

电容器由被绝缘体（电介质）隔开的两个导电板组成。当连接到电压源时，一块板获得净正电荷 +Q，而另一块获得等量的负电荷 -Q。两板保持这些电荷，在介质中产生均匀电场并储存势能。

In a circuit symbol, a capacitor is represented by two parallel lines of equal length, perpendicular to the connecting wires. It acts as a temporary store of charge, releasing it when the circuit demands.

在电路符号中，电容器用两条等长且垂直于导线的平行线表示。它充当电荷的临时储存器，在电路需要时释放电荷。

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2. Capacitance – Definition and Units | 电容 – 定义与单位

Capacitance C is defined as the ratio of the charge Q stored on one plate to the potential difference V across the plates. The formal definition is:

电容 C 定义为储存在一块极板上的电荷量 Q 与极板间电势差 V 之比。其正式定义如下：

C = Q / V

mb per volt (1 F = 1 C V⁻¹). In practice, capacitors are usually measured in microfarads (µF), nanofarads (nF), or picofarads (pF).

电容的 SI 单位是法拉（F）。一法拉等于每伏特一库仑（1 F = 1 C V⁻¹）。实际上，电容器通常以微法（µF）、纳法（nF）或皮法（pF）计量。

For a given capacitor, C is constant and depends only on its geometry and the dielectric material, not on Q or V. This linear relationship is the foundation of capacitor behaviour in DC circuits.

对于给定的电容器，C 是常数，仅取决于其几何形状和电介质材料，而不取决于 Q 或 V。这种线性关系是电容器在直流电路中行为的基础。

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3. The Parallel Plate Capacitor | 平行板电容器

The simplest capacitor model is the parallel plate capacitor: two metallic plates of area A separated by a distance d with a vacuum or dielectric between them. Its capacitance is given by:

最简单的电容器模型是平行板电容器：面积为 A 的两块金属板，间距为 d，板间为真空或电介质。其电容由下式给出：

C = ε₀ A / d

lace ε₀ with ε = κ ε₀, where κ (or εᵣ) is the dielectric constant (relative permittivity).

这里 ε₀ 是真空介电常数，ε₀ ≈ 8.85 × 10⁻¹² F m⁻¹。如果间隙充满电介质材料，则用 ε = κ ε₀ 替换 ε₀，其中 κ（或 εᵣ）是介电常数（相对介电常数）。

s with plate area and dielectric constant, and decreases with plate separation. This relationship is key to designing capacitors with specific values.

因此，电容变为 C = κ ε₀ A / d。电容随极板面积和介电常数的增大而增大，随极板间距的增大而减小。这种关系是设计具有特定电容值的电容器的关键。

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4. The Role of Dielectrics | 电介质的作用

A dielectric is an insulating material placed between the plates. It increases capacitance by a factor κ (dielectric constant) because it reduces the effective electric field for a given charge. This happens via polarization of the dielectric molecules, which produces an opposing field and allows more charge to be stored at the same voltage.

电介质是放置在极板之间的绝缘材料。它通过因子 κ（介电常数）增大电容，因为它减少了给定电荷下的有效电场。这是通过电介质分子的极化实现的，极化会产生反向电场，使在相同电压下能储存更多电荷。

Additionally, the dielectric increases the maximum voltage a capacitor can withstand before breakdown. The dielectric strength is the maximum electric field the material can tolerate without conducting. Practical capacitors use materials like ceramic, polyester, or electrolytic solutions with high κ values.

此外，电介质提高了电容器在击穿前能承受的最大电压。介电强度是材料在不导电的情况下能耐受的最大电场。实际电容器使用具有高 κ 值的材料，如陶瓷、聚酯或电解液。

When a dielectric is inserted into a charged capacitor (with disconnected battery), the voltage drops and stored energy decreases; if the battery remains connected, the voltage stays constant and more charge flows onto the plates, increasing stored energy.

当将电介质插入已充电的电容器（电池已断开）时，电压下降，储存的能量减少；如果电池保持连接，电压保持不变，更多电荷流向极板，从而增加储存的能量。

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5. Energy Stored in 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 work done by the battery to charge the capacitor to voltage V with charge Q is equal to the area under the Q-V graph, leading to three equivalent expressions for stored energy E:

已充电的电容器在其极板间的电场中储存电势能。电池将电容器充电至电压 V 并带有电荷 Q 所做的功等于 Q-V 图下方的面积，由此得出储存能量 E 的三个等价表达式：

E = ½ Q V = ½ C V² = ½ Q² / C

These formulas show that energy scales with the square of voltage and inversely with capacitance for a given charge. In an RC circuit, this energy is dissipated as heat in the resistor during discharge.

这些公式表明，能量与电压的平方成正比，对于给定的电荷量则与电容成反比。在 RC 电路中，放电时该能量以热量形式在电阻中耗散。

• E = ½ Q V: useful when both Q and V are known.
• E = ½ Q V：当已知 Q 和 V 时很有用。
• E = ½ C V²: highlights the dependence on voltage, critical for safe voltage ratings.
• E = ½ C V²：突出了对电压的依赖，这对安全额定电压至关重要。
• E = ½ Q² / C: shows that for a fixed charge, a smaller capacitance stores more energy at higher voltage.
• E = ½ Q² / C：表明对于固定电荷，较小的电容在较高电压下储存更多能量。

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

When capacitors are connected in series, the reciprocal of the total capacitance equals the sum of the reciprocals of individual capacitances. This is because the charge on each capacitor is the same, but the total voltage divides.

当电容器串联时，总电容的倒数等于各电容倒数之和。这是因为每个电容器上的电荷相同，但总电压会分配。

1 / Cₜₒₜₐₗ = 1 / C₁ + 1 / C₂ + ...

For two capacitors in series, the combined capacitance is smaller than the smallest individual capacitance. This arrangement is used to obtain a higher voltage rating.

对于两个串联电容器，组合电容小于单个最小电容。这种安排用于获得更高的耐压值。

When connected in parallel, the total capacitance is simply the sum of individual capacitances. All capacitors share the same voltage, and total charge is the sum of individual charges.

当并联连接时，总电容就是各个电容之和。所有电容器电压相同，总电荷为各个电荷之和。

Cₜₒₜₐₗ = C₁ + C₂ + ...

Parallel combination is used to increase capacitance without affecting the voltage limit. A table summarises the rules:

并联组合用于增大电容而不影响电压限制。下表总结了规则：

Connection	Charge	Voltage	Total Capacitance
Series	Same on each capacitor	Vₜₒₜₐₗ = V₁ + V₂ + ...	1/C = Σ 1/Cᵢ
Parallel	Qₜₒₜₐₗ = Q₁ + Q₂ + ...	Same across each	C = Σ Cᵢ

Note: ᵢ here represents the index; in practice use numeric subscripts like C₁, C₂. (The total voltage in series must not exceed the breakdown voltage of any individual capacitor unless proper balancing resistors are used.)

注意：这里 ᵢ 表示索引；实际中使用数字下标如 C₁、C₂。（串联总电压不得超过任何单个电容器的击穿电压，除非使用适当的平衡电阻。）

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7. RC Circuits: Charging a Capacitor | RC 电路：电容器的充电

e capacitor acts like a short circuit (uncharged), and the initial current is I₀ = V₀ / R.

当电容器与电阻 R 和电动势为 V₀ 的电池串联时，电容器两端的电压 V 逐渐升高，电流 I 逐渐减小。在 t=0 时，电容器相当于短路（未充电），初始电流为 I₀ = V₀ / R。

The charge q on the capacitor as a function of time follows an exponential saturation:

电容器上的电荷 q 随时间按指数规律趋于饱和：

q(t) = Q₀ (1 − e^{−t / RC})

where Q₀ = C V₀ is the final charge. The voltage across the capacitor is Vc(t) = V₀ (1 − e^{−t / RC}). The charging current decays exponentially:

其中 Q₀ = C V₀ 是最终电荷。电容器两端电压为 Vc(t) = V₀ (1 − e^{−t / RC})。充电电流按指数衰减：

I(t) = I₀ e^{−t / RC}

These equations are derived from Kirchhoff' + q/C. The term RC, called the time constant τ, determines how fast the capacitor charges.

这些方程由基尔霍夫回路定律 V₀ = IR + q/C 推导而来。乘积 RC 称为时间常数 τ，它决定了电容器充电的快慢。

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8. RC Circuits: Discharging a Capacitor | RC 电路：电容器的放电

Consider a charged capacitor (initial voltage V₀) connected across a resistor R, with the battery removed. The capacitor discharges through the resistor, with both charge and voltage decaying exponentially towards zero.

考虑一个已充电的电容器（初始电压 V₀）连接在电阻 R 两端，并将电池移除。电容器通过电阻放电，电荷和电压均按指数规律衰减至零。

q(t) = Q₀ e^{−t / RC}

V(t) = V₀ e^{−t / RC}

I(t) = −I₀ e^{−t / RC}

is opposite to the charging current. The magnitude of the initial current is I₀ = V₀ / R.

电流中的负号表示流动方向与充电电流相反。初始电流的大小为 I₀ = V₀ / R。

During discharge, the energy initially stored in the capacitor (½ C V₀²) is entirely converted into thermal energy in the resistor by the time the capacitor is fully discharged.

在放电过程中，电容器最初储存的能量（½ C V₀²）在电容器完全放电时全部转换为电阻中的热能。

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9. The Time Constant τ and Its Significance | 时间常数 τ 及其意义

ds (Ω × F = s). It represents the time required for the charge/voltage to rise to about 63% of its final value during charging, or to fall to about 37% of its initial value during discharging.

时间常数 τ（tau）定义为 τ = RC。它的单位是秒（Ω × F = s）。它表示充电时电荷/电压上升至最终值约 63%，或放电时下降至初始值约 37% 所需的时间。

After a time of 5τ, the capacitor is considered practically fully charged (99.3%) or fully discharged (0.7% remaining). The value of τ determines the response speed of RC circuits used in timing, filtering, and smoothing applications.

经过 5τ 的时间后，电容器被认为几乎完全充电（99.3%）或完全放电（剩余 0.7%）。τ 值决定了 RC 电路在定时、滤波和平滑应用中的响应速度。

• For charging: at t = τ, Vc = 0.63 V₀; at t = 2τ, Vc = 0.86 V₀; at t = 5τ, Vc ≈ V₀.
• 充电时：在 t = τ 时，Vc = 0.63 V₀；在 t = 2τ 时，Vc = 0.86 V₀；在 t = 5τ 时，Vc ≈ V₀。
• For discharging: at t = τ, Vc = 0.37 V₀; at t = 5τ, Vc ≈ 0.
• 放电时：在 t = τ 时，Vc = 0.37 V₀；在 t = 5τ 时，Vc ≈ 0。

The exponential nature of RC transients is independent of the shape of the circuit; only R and C matter. In IB exam questions, you may be asked to find τ from a graph or to calculate unknown R or C values.

RC 瞬态的指数性质与电路形状无关；只有 R 和 C 决定。在 IB 考试题中，可能会要求你从图形中找出 τ，或计算未知的 R 或 C 值。

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10. Capacitor Applications and Safety | 电容器的应用与安全

Capacitors are widely used in electronic circuits: for smoothing rectified AC in power supplies, in timing circuits (e.g., flashing LEDs, pulse generation), in camera flashes (rapid discharge), and in touch screens (sensing capacitance changes).

电容器广泛用于电子电路中：在电源中平滑整流后的交流电、用于定时电路（例如闪烁 LED、脉冲生成）、相机闪光灯（快速放电）以及触摸屏（检测电容变化）。

In IB Physics, a common context is the use of a capacitor to store energy for a flash tube. The rapid release of stored energy provides a high current pulse that ignites the flash.

在 IB 物理中，常见的情境是使用电容器为闪光灯管储存能量。储存能量的快速释放提供高电流脉冲来点亮闪光。

Safety: Large capacitors can retain dangerous voltages long after the power is removed. They must be discharged before handling. The maximum voltage rating and polarity (for electrolytic capacitors) must be observed to prevent explosion or circuit failure.

安全注意事项：大型电容器在断电后仍可能长时间保留危险电压。在操作前必须对其进行放电。必须注意最大额定电压和极性（针对电解电容器），以防止爆炸或电路故障。

Dielectric breakdown occurs if the electric field exceeds the dielectric strength, causing a short circuit and often permanent damage. Understanding energy storage and time constants also helps in designing safe discharge paths.

如果电场超过介电强度，就会发生介质击穿，导致短路并通常造成永久性损坏。了解能量储存和时间常数也有助于设计安全的放电路径。

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11. Summary of Key Equations and Concepts | 关键公式与概念总结

For IB Physics revision, the essential capacitor concepts and formulas are:

对于 IB 物理复习，基本的电容器概念和公式如下：

• Definition: C = Q / V
• 定义： C = Q / V
• Parallel plate capacitance: C = κ ε₀ A / d
• 平行板电容： C = κ ε₀ A / d
• Stored energy: E = ½ Q V = ½ C V² = ½ Q² / C
• 储存能量： E = ½ Q V = ½ C V² = ½ Q² / C
• Series: 1/Cₜₒₜₐₗ = Σ 1/Cᵢ (smaller total)
• 串联： 1/Cₜₒₜₐₗ = Σ 1/Cᵢ（总电容变小）
• Parallel: Cₜₒₜₐₗ = Σ Cᵢ (larger total)
• 并联： Cₜₒₜₐₗ = Σ Cᵢ（总电容变大）
• Charging: q = Q₀ (1 − e^{−t / RC}), Vc = V₀ (1 − e^{−t / RC})
• 充电： q = Q₀ (1 − e^{−t / RC}), Vc = V₀ (1 − e^{−t / RC})
• Discharging: q = Q₀ e^{−t / RC}, V = V₀ e^{−t / RC}
• 放电： q = Q₀ e^{−t / RC}, V = V₀ e^{−t / RC}
• Time constant: τ = RC; at t = τ, charge reaches ~63% (charging) or drops to ~37% (discharging).
• 时间常数： τ = RC；在 t = τ 时，充电量达到约 63%（充电）或降至约 37%（放电）。

Mastering these relationships allows you to tackle both calculation and explanation questions on capacitors with confidence.

掌握这些关系，你就能自信地回答有关电容器的计算和解释题。

Published by TutorHao | Physics Revision Series | aleveler.com

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