An Equation for Current | 电流方程

📚 An Equation for Current | 电流方程

In CIE A-Level Physics, electric current is not just a reading on an ammeter; it can be expressed quantitatively by the defining equation I = ΔQ / Δt and, at the microscopic level, by the transport equation I = nAvq. Understanding this equation links the measurable current in a circuit to the motion of charge carriers inside a conductor.

在 CIE A-Level 物理中，电流不仅是电流表上的读数；它可以通过定义式 I = ΔQ / Δt 以及微观输运方程 I = nAvq 定量表达。理解这个方程将电路中可测量的电流与导体内电荷载流子的运动联系起来。

1. Defining Electric Current | 电流的定义

Electric current I is the rate at which electric charge passes through a given cross-section of a conductor. If a net charge ΔQ passes through a cross-section in time Δt, the average current is given by:

电流 I 是电荷通过导体某一横截面的速率。如果在时间 Δt 内有净电荷 ΔQ 通过横截面，则平均电流为：

I = ΔQ / Δt

The SI unit of current is the ampere (A), where 1 A = 1 C s⁻¹. Although current is usually shown with a direction, it is a scalar quantity because it does not combine like a vector in three-dimensional space.

电流的国际单位是安培（A），1 A = 1 C s⁻¹。虽然电流通常带有方向，但它是标量，因为它在三维空间中不像矢量那样合成。

2. Deriving the Transport Equation | 推导输运方程

To derive I = nAvq, consider a cylindrical conductor with cross-sectional area A. In a small time interval Δt, the charge carriers move an average distance vΔt, where v is their average drift velocity along the conductor.

为了推导 I = nAvq，考虑一根横截面积为 A 的圆柱形导体。在一小段时间 Δt 内，电荷载流子沿导体平均移动距离 vΔt，其中 v 是它们的平均漂移速度。

The volume of charge that passes through the cross-section in this time is A × vΔt. If the conductor contains n charge carriers per unit volume, and each carrier carries charge q, the total mobile charge in that volume is:

在这段时间内通过横截面的电荷体积为 A × vΔt。如果导体每单位体积含有 n 个电荷载流子，且每个载流子所带电荷为 q，则该体积内可移动的总电荷为：

ΔQ = nAvΔt × q

Dividing both sides by Δt gives the current equation:

两边同时除以 Δt，得到电流方程：

I = nAvq

This equation is especially useful because it connects a macroscopic measurement, current, with microscopic properties of the material.

这个方程特别有用，因为它将宏观测量量电流与材料的微观性质联系起来。

3. Meanings of the Symbols | 各符号的物理意义

The equation I = nAvq contains four physical quantities that determine the current in a conductor:

方程 I = nAvq 包含决定导体中电流的四个物理量：

Symbol / 符号	Meaning / 意义	SI unit / 国际单位
n	number density of charge carriers / 载流子数密度	m⁻³
A	cross-sectional area / 横截面积	m²
v	average drift velocity / 平均漂移速度	m s⁻¹
q	charge per carrier / 每个载流子的电荷量	C

For a metal, q is the elementary charge e = 1.60 × 10⁻¹⁹ C carried by each free electron. For other conductors, q may be a multiple of e, as in ions in an electrolyte.

对于金属，q 是每个自由电子所带的元电荷 e = 1.60 × 10⁻¹⁹ C。对于其他导体，q 可能是 e 的整数倍，例如电解质中的离子。

4. Drift Velocity and Carrier Motion | 漂移速度与载流子运动

The drift velocity v in the equation is the average velocity that charge carriers acquire due to an electric field. It is surprisingly small in a typical metal wire, often of the order of 10⁻⁴ m s⁻¹ even for currents of several amperes.

方程中的漂移速度 v 是电荷载流子在电场作用下获得的平均速度。在典型金属导线中，这个速度小得惊人，即使电流达到几安培，漂移速度通常也只有 10⁻⁴ m s⁻¹ 量级。

This low drift velocity does not mean the electrical signal travels slowly. When a switch is closed, the electric field is established almost instantly along the circuit at a speed close to the speed of light, but the individual electrons themselves drift only slowly.

漂移速度低并不意味着电信号传播慢。闭合开关时，电场几乎瞬间以接近光速沿电路建立，但单个电子本身只是缓慢漂移。

The random thermal velocities of free electrons are much larger, around 10⁶ m s⁻¹, but they are random in direction and therefore do not produce a net current. Only the small average drift velocity contributes to I = nAvq.

自由电子的随机热运动速度要大得多，约为 10⁶ m s⁻¹，但方向随机，因此不产生净电流。只有很小的平均漂移速度对 I = nAvq 有贡献。

5. Conductors, Semiconductors and Electrolytes | 导体、半导体与电解质

The transport equation applies to any material that contains mobile charge carriers, but the values of n and q depend on the material. In metals, n is very large, roughly 10²⁸ to 10²⁹ m⁻³, and the carriers are free electrons with q = e.

输运方程适用于任何含有可移动电荷载流子的材料，但 n 和 q 的取值取决于材料。在金属中，n 非常大，约为 10²⁸ 至 10²⁹ m⁻³，载流子是 q = e 的自由电子。

In semiconductors, n is much smaller and both negative electrons and positive holes contribute to the current. The total current is the sum of the electron current and the hole current, so it may be written as I = nₑAvₑe + nₕAvₕe, where the subscripts refer to electrons and holes.

在半导体中，n 要小得多，负电子和正空穴都对电流有贡献。总电流是电子电流与空穴电流之和，因此可以写为 I = nₑAvₑe + nₕAvₕe，其中下标分别表示电子和空穴。

In an electrolyte, positive and negative ions move in opposite directions under the electric field. Both motions contribute to conventional current in the same direction, so each type of ion adds a term of the form nAvq to the total current.

在电解质中，正离子和负离子在电场作用下朝相反方向运动。两种运动对常规电流的贡献方向相同，因此每种离子都会对总电流增加一个 nAvq 形式的项。

6. Current Density | 电流密度

Dividing both sides of I = nAvq by the cross-sectional area A gives the current density J:

将 I = nAvq 两边同时除以横截面积 A，得到电流密度 J：

J = I / A = nvq

Current density is the current per unit cross-sectional area and has the SI unit A m⁻². It is useful when comparing how concentrated the current is in wires of different thicknesses.

电流密度是单位横截面积上的电流，SI 单位为 A m⁻²。在比较不同粗细导线中电流的集中程度时，电流密度非常有用。

For the same current, a thinner wire has a larger current density and therefore a larger drift velocity if n and q are fixed by the material.

对于相同电流，较细的导线具有更大的电流密度，因此如果材料的 n 和 q 固定，其漂移速度也更大。

7. Kirchhoff’s First Law from the Equation | 基尔霍夫第一定律与电流方程

Kirchhoff’s first law states that at any junction in a circuit, the total current entering the junction equals the total current leaving it. This is a direct consequence of conservation of electric charge.

基尔霍夫第一定律指出，在电路中的任一节点，流入该节点的总电流等于流出该节点的总电流。这是电荷守恒的直接结果。

In equation form, for steady currents at a junction:

对于节点处的恒定电流，方程形式为：

Σ Iᵢₙ = Σ Iₒᵤₜ

Each branch current can be understood using I = nAvq. If a wire splits into two branches, the sum of the drift-current terms in the branches must equal the drift-current term in the main wire, provided no charge is stored at the junction.

每一支路电流都可以用 I = nAvq 来理解。如果一根导线分成两条支路，支路中的漂移电流项之和必须等于主路中的漂移电流项，前提是节点处没有电荷积累。

8. Instantaneous Current and Alternating Current | 瞬时电流与交流电

The equation I = ΔQ / Δt gives an average current over time Δt. For a current that changes with time, the instantaneous current is the charge passing per unit time in an infinitesimally small interval.

方程 I = ΔQ / Δt 给出的是时间 Δt 内的平均电流。对于随时间变化的电流，瞬时电流是在无穷小时间间隔内单位时间通过的电荷量。

In alternating current, the current varies sinusoidally with time. A typical form is:

在交流电中，电流随时间呈正弦变化。典型形式为：

I = I₀ sin(2πft)

Here I₀ is the peak current and f is the frequency. The transport equation I = nAvq still applies at each instant, but the drift velocity of the carriers reverses direction periodically.

其中 I₀ 是峰值电流，f 是频率。输运方程 I = nAvq 在每个瞬时仍然成立，但载流子的漂移速度会周期性地改变方向。

9. Worked Example: Copper Wire | 计算示例：铜导线

A copper wire has a cross-sectional area of 1.0 × 10⁻⁶ m² and carries a current of 2.0 A. The number density of free electrons in copper is 8.5 × 10²⁸ m⁻³. Calculate the average drift velocity of the electrons.

一根铜导线的横截面积为 1.0 × 10⁻⁶ m²，通过的电流为 2.0 A。铜中自由电子的数密度为 8.5 × 10²⁸ m⁻³。计算电子的平均漂移速度。

Using the equation I = nAvq and rearranging for v:

使用方程 I = nAvq，并对 v 进行整理：

v = I / (nAe)

Substituting the values:

代入数值：

v = 2.0 / (8.5 × 10²⁸ × 1.0 × 10⁻⁶ × 1.60 × 10⁻¹⁹)

v = 1.47 × 10⁻⁴ m s⁻¹

This confirms that the average electron drift velocity in a metal wire is very small even for a current of 2 A.

这证实了即使在 2 A 电流下，金属导线中电子的平均漂移速度也非常小。

10. Worked Example: Electron Beam | 计算示例：电子束

A beam of electrons carries a current of 1.0 A. Calculate the number of electrons passing a point in the beam per second.

一束电子携带 1.0 A 的电流。计算每秒通过束流中某一点的电子数量。

Since current is charge per unit time, the number of electrons per second is the total charge per second divided by the charge of one electron:

由于电流是单位时间的电荷量，每秒通过的电子数等于每秒总电荷量除以单个电子的电荷量：

N/t = I / e = 1.0 / (1.60 × 10⁻¹⁹)

N/t = 6.25 × 10¹⁸ s⁻¹

This large number shows how small the fundamental unit of charge is compared with everyday currents.

这个巨大的数字表明，与日常电流相比，基本电荷单位是多么小。

11. Experimental Measurement of Current | 电流的实验测量

Current is measured with an ammeter connected in series with the component or circuit branch. An ideal ammeter has zero resistance so that it does not reduce the current it is measuring.

电流用电流表测量，电流表与被测元件或电路支路串联。理想电流表的电阻为零，因此不会减小它所测量的电流。

In school experiments, a digital milliammeter or a moving-coil ammeter is often used. The reading can be checked against I = Q/t by using a coulombmeter, which measures the total charge that flows in a known time.

在学校实验中，常用数字毫安表或动圈式电流表。读数可以通过库仑计测量已知时间内流过的总电荷，用 I = Q/t 进行校验。

For rapidly changing currents, an oscilloscope can be used with a known shunt resistor to display the current waveform, making it possible to observe how I changes with time.

对于快速变化的电流，可以使用示波器和已知分流电阻来显示电流波形，从而观察 I 如何随时间变化。

12. Common Misconceptions and Exam Tips | 常见误区与考试技巧

A common misconception is that electrons travel at the speed of light in a circuit. In fact, the signal travels close to the speed of light, but the electron drift speed is very small.

一个常见误区是认为电子在电路中以光速运动

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An Equation for Current | 电流方程