CIE A-Level Physics: Energy Stored in a Capacitor — Calculation and Applications | CIE A-Level 物理：电容器储存能量的计算与应用

📚 CIE A-Level Physics: Energy Stored in a Capacitor — Calculation and Applications | CIE A-Level 物理：电容器储存能量的计算与应用

When a capacitor is charged, an external source drives charge onto its plates. The charge separation creates an electric field, and the work done by the source is stored as electric potential energy. This energy can be released later to produce a rapid pulse of current. In CIE A-Level Physics, you need to be able to derive the energy equations, choose the correct form, and apply them to practical devices such as flash lamps and defibrillators.

当电容器充电时，外部电源把电荷推动到极板上。电荷分离会建立电场，电源做功转化为电势能储存起来。这些能量以后可以释放，形成瞬时电流脉冲。在 CIE A-Level 物理中，你需要会推导能量公式、选择合适的表达式，并将其应用于闪光灯和心脏除颤器等实际设备。

1. Why a Capacitor Stores Energy | 为什么电容器能储存能量

For a capacitor with capacitance C, the charge Q and potential difference V are proportional: Q = CV. Because the voltage across a capacitor is zero when it is empty and rises as charge accumulates, small amounts of charge ΔQ are transferred against a gradually increasing voltage.

对于电容为 C 的电容器，电荷 Q 与电压 V 成正比：Q = CV。当电容器不带电时，两端电压为零；随着电荷积累，电压逐渐升高，因此每一小份电荷 ΔQ 都是在不断增大的电压下被转移的。

On a graph of charge Q against voltage V, the line is straight through the origin with gradient C. The area under this graph between V = 0 and the final voltage is exactly equal to the total energy stored, because energy is the sum of VΔQ over every small step.

在 Q-V 图像上，这是一条过原点的直线，斜率为 C。在 V = 0 到最终电压之间，曲线下方的面积正好等于储存的总能量，因为能量就是所有微小步骤 VΔQ 的累加。

2. Deriving the Energy Formula | 推导能量公式

To derive the energy formula, consider a capacitor that already holds charge q. The potential difference across it is V = q/C. If a further small charge dq is transferred, the work done is dW = V dq = (q/C)dq. Integrating from q = 0 to q = Q gives the total energy:

推导能量公式时，考虑电容器已经带有电荷 q，此时两端电压 V = q/C。若再转移微小电荷 dq，做功 dW = V dq = (q/C)dq。把 q 从 0 积分到 Q，就得到总能量：

U = ∫ (q/C) dq = Q² / (2C) = ½QV = ½CV²

Notice that the factor ½ arises because the voltage is not constant during charging; the average voltage is ½V. If we used the final voltage V throughout, we would overestimate the energy by a factor of two.

注意系数 ½ 来自充电过程中电压并非恒定，平均电压只有最终电压的一半。如果全程都用最终电压 V 计算，能量会被高估一倍。

3. Three Equivalent Energy Equations | 三个等价的能量公式

The three equivalent forms are:

三个等价形式分别为：

U = ½QV — Use when both the final charge and voltage are known.

U = ½QV —— 当电荷和电压都知道时使用。
U = ½CV² — Most useful when a capacitor is connected to a known potential difference.

U = ½CV² —— 最常用于已知电压和电容的情况。
U = Q²/(2C) — Useful when the charge is common, for example with capacitors in series.

U = Q²/(2C) —— 当电荷保持不变时很有用，例如串联电容器的情况。

All three formulas are algebraically identical because Q = CV. Choose the one that matches the quantities given in the question.

由于 Q = CV，这三个公式在代数上完全等价。解题时选择与已知量最匹配的形式即可。

4. Energy Density in the Electric Field | 电场中的能量密度

For a parallel-plate capacitor, C = ε₀A/d and V = Ed. Substituting into U = ½CV² gives U = ½(ε₀A/d)(Ed)² = ½ε₀E²(Ad). Since Ad is the volume between the plates, the energy density is:

对于平行板电容器，C = ε₀A/d，V = Ed。代入 U = ½CV² 可得 U = ½(ε₀A/d)(Ed)² = ½ε₀E² Ad。由于 Ad 是两极板间的体积，能量密度为：

u = U / (Ad) = ½ε₀E²

This result shows that energy is stored in the electric field itself, not simply on the surfaces of the plates. The stronger the electric field, the more energy is stored in each cubic metre of space.

这个结果说明能量储存在电场本身，而不仅仅储存在极板表面。电场越强，每立方米空间内储存的能量就越多。

5. Energy Losses When Charging Through a Resistor | 通过电阻充电时的能量损耗

In a simple charging circuit, a battery of e.m.f. V transfers total charge Q to the capacitor, so the energy supplied by the battery is QV. The capacitor stores only ½QV. The remaining ½QV is dissipated as thermal energy in the resistance of the circuit, whether in a fixed resistor, the battery internal resistance, or the wires.

在简单充电电路中，电动势为 V 的电池把总电荷 Q 送到电容器，因此电池提供的能量是 QV。电容器只储存 ½QV，剩余 ½QV 以热能形式耗散在电路电阻上——无论这个电阻是外接电阻、电池内阻还是导线电阻。

This result is independent of the size of the resistance. A larger resistance makes the capacitor charge more slowly, but the energy split remains exactly half stored and half dissipated.

这一结果与电阻大小无关。电阻越大，电容器充电越慢，但能量分配始终是储存一半、耗散一半。

6. Practical Applications | 实际应用

In a camera flash, a capacitor is charged slowly from a battery over a few seconds, then discharged through the flash tube in about a millisecond. The capacitor releases its stored energy as a bright pulse of light. This requires a high instantaneous current, which the battery may not be able to provide directly.

在相机闪光灯中，电容器在几秒内被电池缓慢充电，然后在约 1 毫秒内通过闪光灯管放电。电容器把储存的能量释放为明亮的闪光。这需要很大的瞬时电流，而电池本身难以直接提供。

A defibrillator charges a capacitor to a high voltage, typically around 5000 V, and then delivers a controlled energy pulse through the chest. The capacitor allows the device to store energy safely before rapid delivery at the required moment.

心脏除颤器把电容器充电到约 5000 V 的高电压，然后通过胸部释放受控的能量脉冲。电容器可以先安全地储存能量，等到需要的时刻再快速释放。

Supercapacitors can provide short-term backup power when a battery is briefly disconnected, for example to keep memory alive in devices or to smooth power supply in electric vehicles. Their high power density makes them useful for rapid bursts, even though their energy density is lower than that of batteries.

超级电容器可以在电池短暂断开时提供短期备用电源，例如维持设备内存供电，或在电动车中平抑电源波动。虽然其能量密度低于电池，但高功率密度使其适合快速突发的能量释放。

7. Energy Stored in Series and Parallel Combinations | 串联与并联电容器组的能量

When capacitors are connected in parallel, the total capacitance is C_total = C₁ + C₂. If both have the same voltage V, the stored energy is U = ½C_total V² = ½(C₁ + C₂)V², which is simply the sum of the energies of the individual capacitors.

当电容器并联时，总电容 C_total = C₁ + C₂。若两端电压同为 V，储存能量 U = ½C_total V² = ½(C₁ + C₂)V²，等于各电容器储存能量之和。

When capacitors are connected in series, the total capacitance is found from 1/C_total = 1/C₁ + 1/C₂. Since series capacitors share the same charge Q, the energy is U = Q²/(2C_total). The total energy is the sum of the individual stored energies, but the voltage across each capacitor is different.

串联时，总电容由 1/C_total = 1/C₁ + 1/C₂ 求出。串联电容器带有相同电荷 Q，因此能量为 U = Q²/(2C_total)。总能量仍是各个电容器能量之和，但每个电容器上的电压不同。

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CIE A-Level Physics: Energy Stored in a Capacitor — Calculation and Applications | CIE A-Level 物理：电容器储存能量的计算与应用