📚 Gravitational, Electric and Magnetic Fields | A-Level物理：重力场、电场与磁场知识点解析
In A-Level Physics, understanding field concepts is essential for explaining forces that act at a distance. Gravitational fields govern planetary motion, electric fields drive charges, and magnetic fields influence moving charges. This article unpacks these three fundamental fields, their similarities and differences, key formulas and applications.
在A-Level物理中，理解场的概念对解释超距作用力至关重要。重力场支配行星运动，电场驱动电荷，而磁场影响运动电荷。本文将剖析这三种基本场，它们的相似性与差异、关键公式及应用。
1. The Concept of a Field | 场的概念
A field is a region of space where a particle with a particular property (mass, charge, or moving charge) experiences a non-contact force. Fields are represented by field lines or vectors that indicate both direction and magnitude.
场是空间中的一个区域，其中具有特定属性（质量、电荷或运动电荷）的粒子会受到非接触力。场通过场线或矢量表示，显示方向和大小。
There are two broad categories: scalar fields (potential) and vector fields (field strength). Gravitational and electric fields are conservative, meaning work done around a closed path is zero. Magnetic fields are non-conservative because the force on a moving charge depends on direction.
场分为两大类：标量场（势）和矢量场（场强）。重力场和电场是保守场，意味着沿闭合路径做功为零。磁场是非保守场，因为对运动电荷的力依赖于方向。
2. Gravitational Fields | 重力场
A gravitational field is set up by any object with mass. Newton’s law of universal gravitation states that the force F between two point masses m₁ and m₂ separated by distance r is
F = Gm₁m₂/r²
where G = 6.67 × 10⁻¹¹ N m² kg⁻² is the universal gravitational constant.
任何有质量的物体会产生重力场。牛顿万有引力定律指出，两个相距 r 的质点 m₁ 和 m₂ 之间的力 F 为
F = Gm₁m₂/r²
其中 G = 6.67 × 10⁻¹¹ N m² kg⁻² 是万有引力常量。
Gravitational field strength g at a point is defined as the gravitational force per unit test mass: g = F/m. For a point mass M (or outside a uniform sphere), the radial field is given by
g = GM/r²
directed towards the centre of the mass.
重力场强度 g 定义为每单位检验质量所受的重力：g = F/m。对于点质量 M（或匀质球外），径向场为
g = GM/r²
方向指向质量中心。
3. Gravitational Potential and Energy | 重力势与重力势能
Gravitational potential V_g at a point is the work done per unit mass to bring a small test mass from infinity to that point. For a point mass M,
V_g = −GM/r
The negative sign indicates work is done by the field when attracting a mass. The gravitational potential energy of a mass m placed in this potential is U = mV_g = −GMm/r.
重力势 V_g 是把单位检验质量从无穷远移到该点外力所做的功。对于点质量 M，
V_g = −GM/r
负号表示质量被吸引时场做正功。处于该势中的质量 m 的重力势能为 U = mV_g = −GMm/r。
Escape velocity from a planet of mass M and radius R is the minimum speed that allows an object to escape the gravitational field, given by
v_esc = √(2GM/R)
从质量为 M、半径为 R 的行星逃逸的最小速度由下式给出
v_esc = √(2GM/R)
4. Electric Fields | 电场
An electric field exists around any electric charge. The force between two point charges Q and q separated by distance r is given by Coulomb’s law:
F = kQq/r² = Qq/(4πε₀r²)
where ε₀ ≈ 8.85 × 10⁻¹² F m⁻¹ is the permittivity of free space.
电场存在于任何电荷周围。两相距 r 的点电荷 Q 与 q 之间的力由库仑定律给出：
F = kQq/r² = Qq/(4πε₀r²)
其中 ε₀ ≈ 8.85 × 10⁻¹² F m⁻¹ 是真空介电常数。
Electric field strength E is defined as the force per unit positive charge: E = F/q. For a point charge Q,
E = Q/(4πε₀r²)
directed radially outward from a positive charge. For a uniform electric field between parallel plates separated by distance d and potential difference V,
E = V/d
电场强度 E 定义为每单位正电荷所受的力：E = F/q。对于点电荷 Q，
E = Q/(4πε₀r²)
方向从正电荷径向向外。平行板间相距 d、电势差 V 的匀强电场中，
E = V/d
5. Electric Potential and Energy | 电势与电势能
Electric potential V_e at a point is the work done per unit positive charge to bring a small test charge from infinity to that point. For a point charge Q,
V_e = Q/(4πε₀r)
Electric potential energy of a charge q placed in the potential is U = qV_e = Qq/(4πε₀r). The electronvolt (eV) is a convenient unit of energy: 1 eV = 1.60 × 10⁻¹⁹ J, equal to the energy an electron gains when accelerated through a potential difference of 1 V.
电势 V_e 是把单位正电荷从无穷远移到该点所做的功。对于点电荷 Q，
V_e = Q/(4πε₀r)
电荷 q 在该势中的电势能为 U = qV_e = Qq/(4πε₀r)。电子伏特 (eV) 是能量单位：1 eV = 1.60 × 10⁻¹⁹ J，等于一个电子经 1 V 电势差加速获得的能量。
6. Motion of Charged Particles in Electric Fields | 带电粒子在电场中的运动
When a charged particle (charge q, mass m) enters a uniform electric field with initial velocity v perpendicular to the field lines, it experiences a constant force qE and undergoes projectile-like motion. The acceleration is a = qE/m, and the parabolic deflection y after travelling a horizontal distance L is
y = ½ a t² = ½ (qE/m)(L/v)²
处于匀强电场中，以初速度 v 垂直电场射入的带电粒子（电荷 q，质量 m）受恒力 qE 作用，做类平抛运动。加速度 a = qE/m，水平运动 L 后的抛物线偏转 y 为
y = ½ a t² = ½ (qE/m)(L/v)²
This principle is applied in cathode-ray oscilloscopes and inkjet printers to deflect beams of charged particles.
该原理应用于阴极射线示波器和喷墨打印机中带电粒子束的偏转。
7. Magnetic Fields | 磁场
Magnetic fields are produced by moving charges (electric currents) or permanent magnets. Magnetic flux density B, measured in tesla (T), represents the strength of a magnetic field. The direction of the field at a point is the direction the north pole of a small compass needle would point.
磁场由运动电荷（电流）或永磁体产生。磁通量密度 B，单位为特斯拉 (T)，代表磁场强度。某点的磁场方向是小磁针 N 极所指的方向。
A charge q moving with velocity v in a magnetic field B experiences a magnetic force
F = q v B sinθ
where θ is the angle between v and B. This force is always perpendicular to both v and B. Fleming’s left-hand rule (or the right-hand rule for positive charges) gives the direction. For a straight conductor of length L carrying current I, the force is F = B I L sinθ.
以速度 v 在磁场 B 中运动的电荷 q 受到的磁场力为
F = q v B sinθ
其中 θ 是 v 与 B 的夹角。此力始终垂直于 v 和 B。弗莱明左手定则（正电荷可用右手定则）给出力的方向。对于长度为 L 载流 I 的直导线，受力 F = B I L sinθ。
8. Charged Particles in Magnetic Fields: Circular Motion | 电荷在磁场中的圆周运动
When a charged particle moves perpendicular to a uniform magnetic field, the magnetic force acts as a centripetal force, causing the particle to travel in a circular path. Equating qvB = mv²/r gives the radius:
r = mv/(qB)
The period T of the circular motion is independent of speed:
T = 2πr/v = 2πm/(qB)
This principle underlies mass spectrometers and cyclotron accelerators, where knowing B and measuring r allows determination of specific charge q/m.
当带电粒子垂直进入匀强磁场，磁场力充当向心力，粒子做匀速圆周运动。由 qvB = mv²/r 得半径：
r = mv/(qB)
圆周运动的周期与速度无关：
T = 2πr/v = 2πm/(qB)
这一原理是质谱仪和回旋加速器的基础，知道 B 并测量 r 即可测定比荷 q/m。
9. Velocity Selector | 速度选择器
A velocity selector uses perpendicular electric and magnetic fields to filter charged particles moving at a specific speed. The electric force qE is directed opposite to the magnetic force qvB. When they balance, the net force is zero and particles travel straight:
qE = qvB ⇒ v = E/B
速度选择器利用相互垂直的电场和磁场筛选特定速度的带电粒子。电场力 qE 与磁场力 qvB 方向相反。当二者平衡，合力为零，粒子直线通过：
qE = qvB ⇒ v = E/B
In a Bainbridge mass spectrometer, a velocity selector first picks ions of a given speed; they then enter a uniform magnetic field where their circular radius gives m/q. This allows isotopic mass measurements.
在班布里奇质谱仪中，速度选择器先选出特定速度的离子；然后离子进入匀强磁场，通过圆周半径测定 m/q，用于同位素质量测量。
10. Hall Effect | 霍尔效应
When a current-carrying conductor (or semiconductor) is placed in a magnetic field perpendicular to the current, a voltage (Hall voltage V_H) develops across the conductor perpendicular to both current and field. For a rectangular slab of thickness t, charge carrier concentration n, and electron charge e, the Hall voltage is
V_H = (B I)/(n e t)
当通电导体（或半导体）置于垂直于电流方向的磁场中时，在同时垂直于电流和磁场的方向上产生电压——霍尔电压 V_H。对于厚度为 t、载流子浓度为 n、电子电荷为 e 的矩形片，
V_H = (B I)/(n e t)
The sign of V_H reveals whether the charge carriers are positive or negative. The Hall effect is used to measure magnetic field strength, determine carrier density, and in position sensors.
V_H 的正负揭示了载流子是正还是负。霍尔效应用于测量磁场强度、测定载流子浓度以及位置传感器中。
11. Comparison of Gravitational, Electric and Magnetic Fields | 三种场的比较
Gravitational and electric fields share many formal similarities: both are radial inverse-square fields for point sources, have associated scalar potentials, and exert conservative forces. Magnetic fields, however, only act on moving charges and are non-conservative. The following table highlights key comparisons.
重力场和电场在形式上非常相似：对于点源都是径向平方反比场，都有相应的标量势，都是保守力。但磁场只作用于运动电荷，且为非保守场。下表总结了关键比较。
| Property | Gravitational | Electric | Magnetic |
| Source | Mass | Charge | Moving charge / current |
| Force law | F = Gm₁m₂/r² | F = Qq/(4πε₀r²) | F = qvB sinθ |
| Field strength | g = GM/r² | E = Q/(4πε₀r²) | B = F/(qv sinθ) |
| Potential | V_g = −GM/r | V_e = Q/(4πε₀r) | (not scalar potential in same sense) |
| Conservative? | Yes | Yes | No |
| Acts on | Mass | Stationary or moving charge | Moving charge only |
Understanding these differences helps tackle problems involving combined fields, such as crossed E and B fields in a velocity selector or the motion of particles in Earth’s gravitational and magnetic fields.
理解这些差异有助于处理复合场问题，例如速度选择器中的正交电磁场，或粒子在地球重力和磁场中的运动。
12. Key Formulas and Summary | 关键公式与总结
Mastering fields in A-Level Physics requires a solid grasp of the core equations and their applications. Below is a succinct list of the most essential formulas:
掌握A-Level物理中的场需要对核心方程及其应用有扎实的理解。以下是必须掌握的关键公式：
- Gravity: F = Gm₁m₂/r² ; g = GM/r² ; V_g = −GM/r ; v_esc = √(2GM/R)
- Electric: F = Qq/(4πε₀r²) ; E = Q/(4πε₀r²) (point) ; E = V/d (uniform) ; V_e = Q/(4πε₀r)
- Magnetic: F = qvB sinθ ; F = BIL sinθ ; r = mv/(qB) ; T = 2πm/(qB) ; V_H = BI/(net)
- Velocity selector: v = E/B
- Energy: eV = 1.60 × 10⁻¹⁹ J
Always pay attention to vector directions using appropriate hand rules, distinguish between field and force, and remember that gravitational force is always attractive while electric and magnetic forces can be attractive or repulsive (or deflecting). Regular practice with numerical and graph-based questions involving these three fields will build the confidence needed for exam success.
务必使用对应的手性定则判断矢量方向，区分场和力，并牢记重力始终是吸引力，而电
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