Energy Sources: Key Formula Derivations | 能源：关键公式推导

📚 Energy Sources: Key Formula Derivations | 能源：关键公式推导

In A-Level Physics, a deep understanding of energy sources involves not only knowing the facts but also being able to derive and apply the mathematical relationships that govern energy conversion. This article focuses on deriving key formulas for wind power, hydropower, efficiency, energy density, solar cells, fuels, nuclear energy, and pumped storage. Each derivation is explained step by step to help you excel in your Oxford AQA International A-Level Physics topic test.

在A-Level物理中，对能源的深入理解不仅需要掌握事实，还要能够推导和应用支配能量转换的数学关系。本文将重点推导风能、水力、效率、能量密度、太阳能电池、燃料、核能以及抽水蓄能的关键公式。每一步推导都将详细解释，帮助你在牛津AQA国际A-Level物理专题测试中取得优异成绩。

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1. Derivation of Wind Power Equation | 风能功率公式推导

Imagine a cylinder of air of length vΔt moving towards a wind turbine with sweep area A. The volume of air in this cylinder is A × vΔt, hence the mass is Δm = ρ A vΔt, where ρ is air density.

想象一个长度为vΔt、向扫掠面积为A的风力涡轮机移动的空气柱。该空气柱的体积为A × vΔt，因此其质量为Δm = ρ A vΔt，其中ρ为空气密度。

The mass flow rate, i.e. mass per unit time passing through the turbine, is therefore dm/dt = ρ A v.

因此，质量流量，即单位时间通过涡轮机的空气质量，为 dm/dt = ρ A v。

The kinetic energy of the moving air is E_k = ½ Δm v². Substituting Δm gives the energy delivered in time Δt: ΔE = ½ ρ A v³ Δt. The power in the wind, P = ΔE/Δt, is thus:

运动空气的动能为 E_k = ½ Δm v²。代入Δm可得在Δt时间内传递的能量：ΔE = ½ ρ A v³ Δt。因此，风中的功率 P = ΔE/Δt 为：

P = ½ ρ A v³

This is the maximum theoretical power available in the wind. In practice, a turbine can only extract a fraction of this power due to the Betz limit, with the actual power given by P_actual = ½ C_p ρ A v³, where C_p is the power coefficient (maximum about 0.59).

这是风中可用的最大理论功率。实际上，由于贝茨极限，涡轮机只能提取其中的一部分，实际功率为 P_actual = ½ C_p ρ A v³，其中C_p为功率系数（最大值约为0.59）。

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2. Hydroelectric Power Formula | 水力发电功率公式

In a typical hydroelectric plant, water from a reservoir falls through a height h, converting gravitational potential energy into kinetic energy and then electrical energy. The volume flow rate of water is Q (m³ s⁻¹), so the mass flow rate is dm/dt = ρ Q.

在典型的水力发电站中，水库中的水通过高度h下落，将重力势能转化为动能，再转化为电能。水的体积流量为Q (m³ s⁻¹)，因此质量流量为dm/dt = ρ Q。

The gravitational potential energy lost per unit mass is g h. Hence the power generated (ignoring losses) is the product of mass flow rate and g h:

单位质量损失的重力势能为 gh。因此，产生的功率（忽略损耗）为质量流量与gh的乘积：

P = ρ Q g h

If the system has an overall efficiency η, the useful electrical output power becomes P_out = η ρ Q g h. This formula is essential for calculating the electrical output from a given flow rate and head, and it shows clearly that doubling the head or flow doubles the power.

如果系统的总效率为η，则有用电输出功率为 P_out = η ρ Q g h。该公式对于计算给定流量和水头下的电输出至关重要，并且清楚地表明水头或流量加倍会使功率加倍。

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3. Efficiency of Energy Conversion | 能量转换效率

Efficiency is defined as the ratio of useful output energy (or power) to total input energy (or power). The basic definition can be written as:

效率定义为有用输出能量（或功率）与总输入能量（或功率）之比。基本定义可写为：

η = E_out / E_in or η = P_out / P_in

Because energy is always conserved, the ‘lost’ energy is dissipated as thermal energy or other non-useful forms. Starting from E_in = E_out + E_waste, we can derive η = 1 − (E_waste/E_in). This makes it clear that reducing waste increases efficiency.

由于能量总是守恒的，“损失”的能量会以热能或其他无用的形式耗散。从 E_in = E_out + E_waste 出发，可推导出 η = 1 − (E_waste/E_in)。这清楚地表明减少浪费可以提高效率。

When multiple energy conversion stages are chained, the overall efficiency is the product of individual efficiencies: η_total = η₁ × η₂ × η₃ … This multiplicative rule is derived from the fact that the output of one stage becomes the input for the next, so E_out_final = η₁ η₂ η₃ … E_in_initial.

当多个能量转换阶段串联时，总效率是各个效率的乘积：η_total = η₁ × η₂ × η₃ … 这一乘法规则源于前一级的输出成为后一级的输入，因此 E_out_final = η₁ η₂ η₃ … E_in_initial。

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4. Energy Density and Specific Energy | 能量密度与比能

Energy density (symbol u) is the amount of energy stored per unit volume: u = E / V. Specific energy (or gravimetric energy density) is the energy stored per unit mass: e = E / m. These relationships are definitions but are derived directly from the concepts of energy storage.

能量密度（符号u）是单位体积储存的能量：u = E / V。比能（或重量能量密度）是单位质量储存的能量：e = E / m。这些关系是定义性的，但直接从能量储存概念导出。

For a fuel with a known calorific value Q (energy released per kilogram), its specific energy is simply e = Q. The energy density can then be found by multiplying by the fuel’s density ρ: u = ρ Q. This derivation links the two key metrics used to compare energy carriers.

对于已知热值Q（每千克释放的能量）的燃料，其比能就是 e = Q。然后，能量密度可以通过乘以燃料密度ρ得到：u = ρ Q。这一推导将用于比较能量载体的两个关键指标联系起来。

For example, a lithium-ion battery with a specific energy of 0.8 MJ kg⁻¹ and density 2000 kg m⁻³ has an energy density u = 0.8×10⁶ × 2000 = 1.6×10⁹ J m⁻³. This number helps engineers decide whether a battery is suitable for a particular application.

例如，一个比能为0.8 MJ kg⁻¹、密度为2000 kg m⁻³的锂离子电池，其能量密度为 u = 0.8×10⁶ × 2000 = 1.6×10⁹ J m⁻³。这个数字有助于工程师判断电池是否适合特定应用。

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5. Solar Cell Efficiency | 太阳能电池效率

Solar cells convert light into electricity. The input power is the solar irradiance G (W m⁻²) multiplied by the cell area A. The useful output electrical power is the maximum power point P_max, which can be expressed as:

太阳能电池将光转化为电。输入功率为太阳辐照度G (W m⁻²) 乘以电池面积A。有用的输出电功率是最大功率点 P_max，可表示为：

P_max = I_sc × V_oc × FF

where I_sc is the short-circuit current, V_oc is the open-circuit voltage, and FF is the fill factor. The fill factor arises from the shape of the I–V curve and is theoretically derived from diode equations, but at A-Level you simply use it.

其中I_sc为短路电流，V_oc为开路电压，FF为填充因子。填充因子源于I–V曲线的形状，理论上由二极管方程导出，但在A-Level阶段你只需直接使用。

The efficiency of the solar cell is therefore:

因此，太阳能电池的效率为：

η = P_max / (G × A) = (I_sc × V_oc × FF) / (G × A)

This derivation shows that to increase efficiency, manufacturers aim to maximise all three factors. Typical silicon cells have efficiencies around 15–22%.

这一推导表明，为了提高效率，制造商会力求最大化所有三个因子。典型的硅电池效率约为15–22%。

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