📚 Electrical Power | 电功率
Electrical power is the rate at which electrical energy is transferred or converted in a circuit. In A-Level Physics, understanding power helps explain how components such as resistors, lamps, and motors convert electrical energy into heat, light, or mechanical work.
电功率是电路中电能被转移或转换的速率。在 A-Level 物理中，理解功率有助于解释电阻、灯泡和电动机等元件如何将电能转换为热、光或机械能。
1. Definition of Electrical Power | 电功率的定义
through a potential difference. The SI unit of power is the watt (W), where 1 W = 1 J s⁻¹.
功率定义为单位时间内所做的功。在电学中，它是电荷载流子通过电势差时转移的能量。功率的国际单位是瓦特 (W)，其中 1 W = 1 J s⁻¹。
The word 'power' is often confused with energy. Energy is the total amount transferred, while power tells us how quickly that transfer happens.
'功率' 一词常与能量混淆。能量是转移的总量，而功率告诉我们转移发生的快慢。
2. The Fundamental Equation P = VI | 基本公式 P = VI
For any circuit component, electrical power is given by the product of the potential difference across it and the current through it. The defining equation is:
对于任何电路元件，电功率等于其两端电势差与通过其电流的乘积。定义公式为：
P = V × I
This equation arises because potential difference V is energy per unit charge, and current I is charge per unit time. Multiplying them gives energy per unit time, which is power.
该公式来源于：电势差 V 是单位电荷的能量，电流 I 是单位时间的电荷量。两者相乘得到单位时间的能量，即功率。
3. Energy Transferred and the Kilowatt-hour | 能量传递与千瓦时
= P × t. If power is measured in watts and time in seconds, energy is measured in joules.
当元件以恒定功率工作时，转移的总能量用 E = P × t 计算。若功率以瓦特为单位、时间以秒为单位，则能量以焦耳为单位。
att-hour is the energy used by a 1 kW device operating for one hour, equal to 3.6 × 10⁶ J.
电能的一个较大实用单位是千瓦时 (kWh)。1 千瓦时是功率为 1 kW 的电器工作 1 小时所用的能量，等于 3.6 × 10⁶ J。
Electricity meters in homes measure energy in kWh, not power, because households are charged for total energy consumed over time.
家庭中的电表以 kWh 为单位测量电能而非功率，因为家庭按一段时间内消耗的总能量计费。
4. Power Dissipation in Resistors: P = I²R and P = V²/R | 电阻中的功率耗散：P = I²R 与 P = V²/R
For a resistor or any ohmic conductor, Ohm' substituted into P = V × I to obtain two alternative forms. Replacing V with I × R gives:
对于电阻或任何欧姆导体，将欧姆定律 V = I × R 代入 P = V × I 可得到两种等效形式。用 I × R 替换 V 得到：
P = I² × R
Replacing I with V ÷ R gives:
用 V ÷ R 替换 I 得到：
P = V² ÷ R
R when voltage is constant or known, as in parallel circuits or mains supply calculations.
这些公式适用于不同情况。当电流恒定或已知时（如串联电路）使用 P = I²R；当电压恒定或已知时（如并联电路或市电计算）使用 P = V²/R。
5. Deriving Power Equations from Ohm's Law | 由欧姆定律推导功率公式
ightforward. Starting from P = V × I and substituting V = I × R gives P = I × R × I = I²R.
P = I²R 的推导很直接。从 P = V × I 出发，代入 V = I × R，得到 P = I × R × I = I²R。
ing I = V ÷ R gives P = V × V ÷ R = V²/R. These derivations assume the component obeys Ohm's law and has a constant resistance.
类似地，从 P = V × I 出发，代入 I = V ÷ R，得到 P = V × V ÷ R = V²/R。这些推导假设元件遵循欧姆定律且电阻恒定。
that P = VI is always true, but P = I²R and P = V²/R are only valid for ohmic conductors.
需要记住的是 P = VI 始终成立，但 P = I²R 和 P = V²/R 仅对欧姆导体有效。
6. Heating Effect and Joule's Law | 热效应与焦耳定律
by P = I²R, so doubling the current quadruples the heating effect for the same resistance.
当电流通过电阻时，电能转化为热能，这称为焦耳热效应。每秒产生的热量由 P = I²R 给出，因此在相同电阻下，电流加倍会使热效应增加为原来的四倍。
This heating can be useful, for example in filament lamps, kettles, and electric heaters. However, it is also a source of energy loss in power transmission lines, which is why high voltages are used to reduce current and therefore reduce I²R losses.
这种热效应可以是有用的，例如在白炽灯、水壶和电加热器中。但它也是输电线路中能量损耗的来源，这就是为什么采用高电压来降低电流，从而减少 I²R 损耗。
7. Rated Power and Appliances | 额定功率与电器
Every electrical appliance has a rated power and a rated voltage. For example, a lamp marked '230 V, 60 W' is designed to operate at 230 V and will convert 60 J of electrical energy per second into light and heat under normal conditions.
每个电器都有额定功率和额定电压。例如，标有 '230 V, 60 W' 的灯泡设计在 230 V 下工作，正常条件下每秒将 60 J 电能转化为光和热。
= 60 W ÷ 230 V ≈ 0.26 A. The resistance can be found using R = V² ÷ P = 230² ÷ 60 ≈ 882 Ω.
根据额定值，可用 I = P ÷ V 求出工作电流。对于上述灯泡，I = 60 W ÷ 230 V ≈ 0.26 A。电阻可用 R = V² ÷ P = 230² ÷ 60 ≈ 882 Ω 求出。
8. Series and Parallel Power Calculations | 串联与并联功率计算
easiest with P = I²R. The component with the largest resistance dissipates the most power.
在串联电路中，通过所有元件的电流相同，因此用 P = I²R 比较功率最方便。电阻最大的元件耗散的功率最多。
easiest with P = V²/R. The branch with the smallest resistance dissipates the most power.
在并联电路中，所有支路两端的电压相同，因此用 P = V²/R 比较功率最方便。电阻最小的支路耗散的功率最多。
first and then apply P = I²R, or find the voltage across each resistor and apply P = V²/R.
对于混联电路，先计算每个电阻中的电流再应用 P = I²R，或先求出每个电阻两端的电压再应用 P = V²/R。
9. Internal Resistance and Power Output | 内阻与输出功率
A real cell or battery has internal resistance r. When it delivers current I to an external load R, the terminal potential difference V is less than the electromotive force E. The power delivered to the external load is Pout = I²R.
实际电池具有内阻 r。当它向外部负载 R 提供电流 I 时，端电压 V 小于电动势 E。传递给外部负载的功率为 Pout = I²R。
The current in the circuit is I = E ÷ (R + r). Substituting this into Pout gives:
电路中的电流为 I = E ÷ (R + r)。将其代入 Pout 得到：
Pout = E² × R ÷ (R + r)²
Some power is also wasted inside the cell as heat, given by Pint = I²r. The total power supplied by the cell is Ptotal = E × I.
部分功率也以热量形式在电池内部损耗，由 Pint = I²r 给出。电池提供的总功率为 Ptotal = E × I。
10. Maximum Power Transfer Theorem | 最大功率传输定理
To find the load resistance that receives maximum power from a source, we analyse Pout is maximised when the load resistance equals the internal resistance of the source, R = r.
为了找到从电源获得最大功率的负载电阻，我们将 Pout 作为 R 的函数进行分析。当负载电阻等于电源内阻，即 R = r 时，外部功率最大。
Under this condition, the maximum power delivered to the load is:
在此条件下，传递给负载的最大功率为：
Pmax = E² ÷ (4r)
This theorem is important in electronics and communications, where matching source and load resistance is needed for maximum power transfer. However, at maximum power transfer the efficiency is only 50%.
该定理在电子学和通信中很重要，在这些领域中需要匹配源电阻和负载电阻以实现最大功率传输。但在最大功率传输时效率仅为 50%。
11. Efficiency of Electrical Systems | 电气系统的效率
Efficiency is the ratio of useful output power to total input power. It can be expressed as a percentage using:
效率是有用输出功率与总输入功率之比。可用以下公式以百分比表示：
η = (Pout ÷ Pin) × 100%
wer is I²R and the input power is I²(R + r). The efficiency is therefore R ÷ (R + r) × 100%. The larger R is compared with r, the higher the efficiency.
对于有内阻的电路，输出功率为 I²R，输入功率为 I²(R + r)。因此效率为 R ÷ (R + r) × 100%。R 与 r 相比越大，效率越高。
In power transmission systems, high efficiency is more important than maximum power transfer, so R is kept much larger than r.
在电力传输系统中，高效率比最大功率传输更重要，因此 R 保持远大于 r。
12. Safety, Fuses and Power Rating | 安全、保险丝与额定功率
A fuse is a safety device containing a thin wire that melts when current exceeds a rated value. The power rating of an appliance determines the normal operating current, which is used to select a suitable fuse.
保险丝是一种安全装置，内含一根细金属丝，当电流超过额定值时会熔断。电器的额定功率决定了正常工作电流，以此选择合适的保险丝。
ing a fuse with too high a rating fails to protect the appliance, while too low a rating causes unnecessary melting.
例如，一台 2.3 kW 的加热器接在 230 V 电源上，电流为 I = 2300 W ÷ 230 V = 10 A，因此适合使用 13 A 保险丝。额定值过高的保险丝无法保护电器，而过低则会导致不必要的熔断。
Understanding power calculations is therefore essential for both circuit analysis and practical electrical safety.
因此，理解功率计算对于电路分析和实际用电安全都至关重要。
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