Ampère's Circuital Law and Maxwell's Equations | 安培环路定理与麦克斯韦方程

📚 Ampère’s Circuital Law and Maxwell’s Equations | 安培环路定理与麦克斯韦方程

Ampère’s circuital law is a fundamental statement in classical electromagnetism that links the circulation of the magnetic field around a closed loop to the electric current passing through the surface bounded by that loop. Maxwell’s addition of the displacement current transformed this law into a full electromagnetic field equation and paved the way for the prediction of electromagnetic waves.

安培环路定理是经典电磁学中的基本定律，它将闭合回路周围磁场的环流与穿过该回路所围曲面的电流联系起来。麦克斯韦引入位移电流后，把该定律推广为完整的电磁场方程，并由此预言了电磁波的存在。

1. The Original Ampère’s Circuital Law | 原始安培环路定理

For a steady current, Ampère’s circuital law states that the line integral of the magnetic field B around any closed loop equals μ₀ times the net current enclosed by the loop:

对恒定电流而言，安培环路定理表明：磁感应强度 B 沿任意闭合回路的线积分等于 μ₀ 乘以该回路所包围的净电流。

∮ B · dl = μ₀ I_enc

Here, μ₀ = 4π × 10⁻⁷ T·m/A is the permeability of free space, I_enc is the algebraic sum of currents threading the surface bounded by the loop, and dl is an infinitesimal element of the path.

其中，μ₀ = 4π × 10⁻⁷ T·m/A 为真空磁导率，I_enc 为穿过回路所围曲面的电流的代数和，dl 为路径上的无穷小线元。

2. Understanding the Line Integral | 理解线积分

The symbol ∮ B · dl represents the circulation of the magnetic field around a closed path. The dot product includes cos θ, where θ is the angle between B and dl; only the component of B parallel to the path contributes.

符号 ∮ B · dl 表示磁感应强度沿闭合路径的环流。点积中包含了 cos θ，θ 为 B 与 dl 之间的夹角；只有 B 沿路径方向的分量才产生贡献。

For a long straight wire carrying current I, symmetry tells us that the magnitude of B is constant on a circular path of radius r. Choosing this path, the integral simplifies:

对于载流 I 的长直导线，对称性告诉我们：在半径为 r 的圆形路径上，B 的大小处处相等。取该路径后积分简化为：

∮ B · dl = B × 2πr = μ₀ I

B = μ₀ I / (2πr)

This result matches the Biot-Savart law and illustrates the power of symmetry in applying Ampère’s law.

这一结果与毕奥-萨伐尔定律完全一致，也体现了对称性在应用安培环路定理时的强大作用。

3. The Capacitor Paradox | 电容器充电中的悖论

Consider a parallel-plate capacitor being charged by a steady current I. Apply the original Ampère’s law to a loop surrounding the connecting wire. The surface bounded by the loop can be chosen in two different ways.

考虑一个由恒定电流 I 充电的平行板电容器。对环绕导线的回路应用原始的安培环路定理，可以发现回路所张的曲面有两种取法。

If the surface is a flat disk cut by the wire, the enclosed current is I, so ∮ B · dl = μ₀ I. But if the surface is a bulging balloon that passes between the capacitor plates without crossing the wire, no conduction current flows through it, and I_enc = 0.

若取被导线穿过的平圆盘为曲面，则所围电流为 I，故 ∮ B · dl = μ₀ I。但若取一个鼓起的球面，使其穿过电容器极板之间的区域而不与导线相交，则该曲面没有传导电流穿过，I_enc = 0。

This contradiction shows that the original Ampère’s law cannot be correct for a changing electric field. Something else must “flow” between the plates to sustain the magnetic field there.

这一矛盾表明，原始的安培环路定理在电场变化时并不成立。极板之间一定存在某种“流动”的东西，来维持该区域中的磁场。

4. Maxwell’s Displacement Current | 麦克斯韦的位移电流

Maxwell resolved the paradox by proposing that a changing electric field itself acts like a current. This idea is called the displacement current I_d.

麦克斯韦提出，变化的电场本身就相当于一种电流，即位移电流 I_d，从而解决了这一矛盾。

I_d = ε₀ × dΦ_E / dt

Here, Φ_E = ∫ E · dA is the electric flux through the surface, and ε₀ = 8.85 × 10⁻¹² F/m is the permittivity of free space. For a parallel-plate capacitor, the changing charge on the plates produces a changing electric flux.

其中，Φ_E = ∫ E · dA 是通过曲面的电通量，ε₀ = 8.85 × 10⁻¹² F/m 是真空介电常数。对平行板电容器而言，极板上电荷的变化产生了变化的电通量。

Between the plates, the electric field is approximately uniform and related to the charge by E = Q / (ε₀ A). Therefore:

在极板之间，电场近似均匀，且满足 E = Q / (ε₀ A)。因此：

I_d = ε₀ × d(Q / ε₀) / dt = dQ / dt = I

The displacement current between the plates is numerically equal to the conduction current in the wire. Thus, the magnetic field between the plates is real and measurable.

极板间的位移电流在数值上等于导线中的传导电流。因此，极板之间的磁场是真实存在且可测量的。

5. The Ampère-Maxwell Law | 安培-麦克斯韦定律

With the displacement current included, the modified law is:

纳入位移电流后，修正后的定律为：

∮ B · dl = μ₀ (I_enc + I_d)

∮ B · dl = μ₀ I_enc + μ₀ ε₀ × d/dt ∫ E · dA

The total current includes both conduction current and displacement current. This unified form is valid for both steady and time-varying fields.

总电流包含传导电流和位移电流两种。这种统一形式对恒定场和时变场均适用。

Examination tip: In problems, first ask whether the chosen surface cuts any conducting wires. If not, I_enc = 0, but a changing electric field may still produce a magnetic field through the displacement-current term.

考试提示：解题时先判断所选曲面是否穿过导线。若没有，则 I_enc = 0，但变化的电场仍可能通过位移电流项产生磁场。

6. Applications to Common Geometries | 常见几何形状的应用

Three standard applications are especially important for A-level and international exams:

以下三个标准应用在 A-level 及国际课程考试中尤为重要：

Long straight wire: B = μ₀ I / (2πr)

长直导线：B = μ₀ I / (2πr)
Solenoid (long, tightly wound): B = μ₀ n I, where n is the number of turns per unit length.

长直螺线管（密绕）：B = μ₀ n I，其中 n 为单位长度的匝数。
Toroid: B = μ₀ N I / (2πr), where N is the total number of turns and r is the radius inside the toroid.

环形螺线管：B = μ₀ N I / (2πr)，其中 N 为总匝数，r 为环内的半径。

For the solenoid, choose a rectangular Amperian loop with one side inside the solenoid and one side outside; outside the ideal solenoid, B = 0, giving the result directly.

对螺线管，取一个矩形安培回路，其中一条边在螺线管内部、另一条在外部；理想螺线管外部 B = 0，直接得到上述结果。

7. The Differential Form | 微分形式

Using vector calculus, the Ampère-Maxwell law can be written in local form. The curl of the magnetic field equals the current density plus the time derivative of the electric field:

利用矢量微积分，安培-麦克斯韦定律可以写成局域形式。磁场的旋度等于电流密度加上电场的时间导数：

∇ × B = μ₀ J + μ₀ ε₀ ∂E / ∂t

Here, J is the conduction current density (A/m²), and the term μ₀ ε₀ ∂E/∂t is the displacement-current density. In regions with no conduction current, J = 0, so:

式中 J 为传导电流密度（A/m²），μ₀ ε₀ ∂E/∂t 为位移电流密度。在没有传导电流的区域中 J = 0，因此：

∇ × B = μ₀ ε₀ ∂E / ∂t

This shows that a time-varying electric field generates a solenoidal (curling) magnetic field, even in free space.

这表明，即使是在自由空间，时变电场也能产生有旋的磁场。

8. The Complete Maxwell’s Equations | 完整的麦克斯韦方程组

With Maxwell’s addition, the four fundamental equations of classical electromagnetism take their final form:

加入麦克斯韦修正后，经典电磁学的四个基本方程具有如下最终形式：

Law	Integral Form	名称
Gauss’s law for electricity	∮ E · dA = Q_enc / ε₀	电场高斯定律
Gauss’s law for magnetism	∮ B · dA = 0	磁场高斯定律
Faraday’s law	∮ E · dl = − dΦ_B / dt	法拉第定律
Ampère-Maxwell law	∮ B · dl = μ₀ I_enc + μ₀ ε₀ dΦ_E / dt	安培-麦克斯韦定律

Faraday’s law couples a changing magnetic field to an induced electric field, while the Ampère-Maxwell law couples a changing electric field to a magnetic field. This symmetric coupling is the basis of electromagnetic wave propagation.

法拉第定律将变化磁场与感应电场联系起来，而安培-麦克斯韦定律将变化电场与磁场联系起来。这种对称的耦合正是电磁波传播的基础。

9. Electromagnetic Waves and the Speed of Light | 电磁波与光速

When the Ampère-Maxwell law is combined with Faraday’s law in free space (no charges or currents), the equations can be manipulated to yield the wave equation for E and B. The wave speed is:

在自由空间中（无电荷、无电流），将安培-麦克斯韦定律与法拉第定律结合，可以推导出 E 和 B 的波动方程。其波速为：

c = 1 / √(μ₀ ε₀)

Substituting the known values gives c ≈ 3.00 × 10⁸ m/s, exactly the speed of light. Maxwell therefore concluded that light is an electromagnetic wave.

代入已知数值可得 c ≈ 3.00 × 10⁸ m/s，与光速完全一致。因此麦克斯韦断定：光就是一种电磁波。

This unification of electricity, magnetism, and optics is one of the greatest achievements in physics. In exams, you may be asked only to state that c = 1/√(μ₀ε₀) and explain its origin qualitatively.

电、磁和光学由此得到统一，这是物理学最伟大的成就之一。考试中，通常只需写出 c = 1/√(μ₀ε₀) 并定性说明其来源。

10. Worked Example: Magnetic Field Near a Charging Capacitor | 例题：充电电容器附近的磁场

A parallel-plate capacitor has circular plates of radius R = 0.05 m. A constant charging current I = 2.0 A flows. Find the magnitude of the magnetic field at a distance r = 0.02 m from the central axis between the plates.

一个平行板电容器的圆形极板半径为 R = 0.05 m，以恒定电流 I = 2.0 A 充电。求极板间离中心轴线 r = 0.02 m 处的磁感应强度大小。

Step 1: For r < R, the electric field is uniform and perpendicular to the plates. Choose a circular Amperian loop of radius r between the plates.

第一步：当 r < R 时，极板间电场均匀且垂直于极板。在极板间取半径为 r 的圆形安培回路。

Step 2: By symmetry, ∮ B · dl = B × 2πr. The electric flux through the loop is Φ_E = E × πr², and E = Q / (ε₀ πR²), so:

第二步：由对称性，∮ B · dl = B × 2πr。穿过回路的电通量 Φ_E = E × πr²，且 E = Q / (ε₀ πR²)，因此：

Φ_E = Q r² / (ε₀ R²)

Step 3: Apply the Ampère-Maxwell law with no conduction current through the surface:

第三步：应用安培-麦克斯韦定律，此时曲面没有传导电流穿过：

2πr B = μ₀ ε₀ × d(Q r² / (ε₀ R²)) / dt = μ₀ r² / R² × dQ / dt = μ₀ I r² / R²

B = μ₀ I r / (2π R²)

Step 4: Substitute values:

第四步：代入数值：

B = (4π × 10⁻⁷ × 2.0 × 0.02) / (2π × 0.05²) = 3.2 × 10⁻⁵ T

The magnetic field is extremely small but nonzero, confirming that a charging capacitor produces a magnetic field in the surrounding region.

该磁场非常弱小但确实存在，证实了充电电容器周围会产生磁场。

11. Common Mistakes and Exam Tips | 常见错误与考试提示

Forgetting to include the displacement-current term when the electric field is changing between capacitor plates.

在电容器极板间电场变化时，忘记计入位移电流项。
Using the wrong area in the flux derivative: For r < R, the flux through only the Amperian loop area is needed, not the entire plate area.

在通量求导时使用错误的面积：当 r < R 时，应取安培回路所围的部分面积，而不是整个极板面积。
Sign errors: Use the right-hand rule to assign positive directions consistently for the path integral and the current.

符号错误：应用右手定则，使线积分与电流的方向保持协调一致。
Confusing conduction current I and current density J: I = ∫ J · dA only when the current is uniform or when the integral is explicitly evaluated.

混淆传导电流 I 与电流密度 J：只有当电流均匀或明确计算积分时，才有 I = ∫ J · dA。
Stating that displacement current is a “real” moving charge: it is a changing electric flux, not a physical flow of charge.

误称位移电流是真实移动的电荷：位移电流是变化的电通量，并非电荷的物理流动。

12. Conclusion | 结论

Ampère’s circuital law, corrected by Maxwell’s displacement current, completes the family of Maxwell’s equations. The extended law restores consistency for time-varying fields, explains the magnetic field of a charging capacitor, and predicts electromagnetic waves whose speed is exactly that of light.

安培环路定理经麦克斯韦位移电流修正后，补全了麦克斯韦方程组的最后一块拼图。拓展后的定律使时变场情形下理论保持自洽，解释了充电电容器周围的磁场，并预言了速度恰好等于光速的电磁波。

For examinations, master the integral form, the displacement current concept, and the symmetric application to wires, solenoids, and capacitors. Write every formula with clear symbols, state the direction of B using the right-hand rule, and always check which currents—conduction or displacement—thread your chosen surface.

考试复习时，务必掌握积分形式、位移电流概念，以及针对直导线、螺线管和电容器的对称性应用。书写公式时保持符号清晰，用右手定则判断磁场方向，并且始终确认所选曲面中穿过的是传导电流还是位移电流。

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Ampère’s Circuital Law and Maxwell’s Equations | 安培环路定理与麦克斯韦方程