Electrical Power | 电功率

📚 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

Power is defined as the work done per unit time. In electrical terms, it is the energy transferred by charge carriers as they move 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

When a component operates at constant power, the total energy transferred is calculated using E = 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。

A larger practical unit for electrical energy is the kilowatt-hour (kWh). One kilowatt-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’s law V = I × R can be 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

These forms are useful in different situations. Use P = I²R when current is constant or known, as in series circuits. Use P = V²/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

The derivation of P = I²R is straightforward. 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。

Similarly, starting from P = V × I and substituting 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。

It is important to remember 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

When current flows through a resistor, electrical energy is converted into thermal energy. This is known as Joule heating. The heat produced per second is given 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。

From the rated values, the operating current can be found using I = P ÷ V. For the lamp above, I = 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

In a series circuit, the current is the same through all components, so power comparisons are 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。

In a parallel circuit, the voltage is the same across all branches, so power comparisons are 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。

For a combination circuit, calculate the current in each resistor 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 P_out = 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 P_out = I²R。

The current in the circuit is I = E ÷ (R + r). Substituting this into P_out gives:

电路中的电流为 I = E ÷ (R + r)。将其代入 P_out 得到：

P_out = E² × R ÷ (R + r)²

Some power is also wasted inside the cell as heat, given by P_int = I²r. The total power supplied by the cell is P_total = E × I.

部分功率也以热量形式在电池内部损耗，由 P_int = I²r 给出。电池提供的总功率为 P_total = 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 P_out as a function of R. The external power 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 P_out 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：

P_max = 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 = (P_out ÷ P_in) × 100%

For a circuit with internal resistance, the output power 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。

For example, a 2.3 kW heater connected to a 230 V supply draws I = 2300 W ÷ 230 V = 10 A, so a 13 A fuse is appropriate. Using 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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Electrical Power | 电功率