📚 Electric Fields: A Comprehensive A-Level Physics Revision Guide | 电场：A-Level物理全面复习指南
Electric fields are fundamental to understanding how charges interact at a distance. In A-Level Physics, you need to master the principles of forces, energy, potential, and the motion of charged particles in uniform and radial fields. This revision guide breaks down the key concepts with clear explanations, essential formulas, and comparisons.
电场是理解电荷如何远距离相互作用的基础。在 A-Level 物理中，你需要掌握力、能量、电势以及带电粒子在匀强和辐射状电场中的运动原理。本复习指南通过清晰的解释、基本公式和比较，梳理了核心概念。
1. Coulomb’s Law and Point Charges | 库仑定律与点电荷
Coulomb’s law states that the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of their separation. The magnitude of the electrostatic force is given by:
F = k Q₁Q₂ / r²
库仑定律指出，两个点电荷之间的力与它们的电荷乘积成正比，与距离的平方成反比。静电力的大小由下式给出：F = k Q₁Q₂ / r²，其中k为库仑常数，真空中 k = 1/(4π ε₀) ≈ 8.99×10⁹ N m² C⁻²。
This force acts along the line joining the two charges and is attractive for unlike charges, repulsive for like charges. When multiple point charges interact, the net force on any one charge is the vector sum of the individual pairwise forces — this is the principle of superposition.
该力沿两电荷连线方向作用，异种电荷相吸，同种电荷相斥。当多个点电荷相互作用时，任一电荷所受的合力等于各对力的矢量和——这就是叠加原理。
2. Electric Field Strength and Direction | 电场强度与方向
An electric field is a region around a charge where a force is exerted on other charges. The electric field strength E at a point is defined as the force per unit positive charge placed at that point:
E = F / q
电场是电荷周围对其他电荷施加力的区域。空间中某点的电场强度E定义为单位正电荷在该点所受的力：E = F / q，单位为 N C⁻¹ 或 V m⁻¹。
The direction of E is the direction of the force on a small positive test charge. For a point charge Q, the magnitude of the electric field at a distance r is:
E = k Q / r² = (1/(4π ε₀)) Q / r²
电场E的方向与放在该点的小正试探电荷所受力的方向相同。对于点电荷Q，距离r处的电场强度大小为：E = k Q / r² = (1/(4π ε₀)) Q / r²。正电荷的电场径向向外，负电荷的电场径向向内。电场是矢量场，服从叠加原理。
3. Electric Field Lines and Patterns | 电场线与场图案
Electric field lines are a visual tool to represent the direction and strength of the field. Field lines begin on positive charges and end on negative charges. The tangent to a field line at any point gives the direction of E, and the density of lines indicates the field strength.
电场线是表示电场方向和强弱的可视化工具。电场线从正电荷出发，终止于负电荷。线上任一点的切线方向即为该点电场E的方向，而线的疏密程度表示场强的大小。
Key rules for field lines: they never cross, they are continuous, and in uniform fields they are parallel and equally spaced. Common patterns include radial lines for a single point charge, curved lines for a dipole, and parallel lines between oppositely charged parallel plates.
绘制电场线的关键规则：电场线永不相交，是连续的，在匀强电场中它们平行且等间距。常见图样包括单个点电荷的辐射状电场线、电偶极子的弯曲电场线，以及带异种电荷的平行板间的匀强电场线。
4. Electric Potential Energy | 电势能
Electric potential energy U is the work done by an external force to bring a charge from infinity to a point in an electric field without acceleration. For two point charges Q₁ and Q₂ separated by distance r, the mutual potential energy is:
U = k Q₁Q₂ / r
电势能U是将一个电荷从无穷远处无加速地移到电场中某点的过程中外力所做的功。对于相距r的两个点电荷Q₁和Q₂，其相互势能为：U = k Q₁Q₂ / r。同种电荷势能为正（排斥），异种电荷势能为负（吸引）。
When a test charge q moves through a potential difference ΔV, the change in its electric potential energy is ΔEₚ = q ΔV. This relationship is central to understanding energy transfers in electric fields.
当试探电荷q经过电势差ΔV时，其电势能的变化量为 ΔEₚ = q ΔV。这一关系是理解电场中能量转化的核心。
5. Electric Potential | 电势
Electric potential V at a point is the electric potential energy per unit positive charge. It is a scalar quantity measured in volts (J C⁻¹):
V = Eₚ / q
电势V是单位正电荷所具有的电势能，是一个标量，单位为伏特（J C⁻¹）：V = Eₚ / q。
For a point charge Q, the potential at a distance r is V = k Q / r, taking the zero of potential at infinity. The potential due to several point charges is the algebraic sum of the potentials from each charge. The potential difference ΔV between two points is the work done per unit charge in moving a charge between them: W = q ΔV.
对于点电荷Q，距离r处的电势为 V = k Q / r，无穷远处电势取为零。多个点电荷产生的电势等于各电荷电势的代数和。两点间的电势差ΔV是将单位电荷从一点移到另一点时电场力做的功：W = q ΔV。
6. Equipotential Surfaces | 等势面
Equipotential surfaces are surfaces on which every point has the same electric potential. No work is done by the electric field when a charge moves along an equipotential surface because there is no change in potential energy.
等势面是面上各点电势都相等的曲面。电荷沿等势面移动时，由于电势能不变，电场力不做功。
Field lines are always perpendicular to equipotential surfaces. In a uniform field between parallel plates, the equipotentials are planes parallel to the plates. For a point charge, they are concentric spheres. The spacing of equipotentials reflects the electric field strength: closely spaced surfaces indicate a strong field.
电场线总是与等势面垂直。在平行板间的匀强电场中，等势面是平行于板的平面。对于点电荷，等势面为同心球面。等势面的间距反映了电场强度：间距越小，场强越大。
7. Uniform Electric Fields | 匀强电场
A uniform electric field has the same magnitude and direction at all points. The classic example is the field between two oppositely charged parallel plates. The relationship between the potential difference V across the plates and the plate separation d gives the field strength:
E = V / d
匀强电场中各点的场强大小和方向都相同。典型的例子是两带异种电荷的平行板之间的电场。板间电势差V与板间距d的关系给出了场强：E = V / d。
A charged particle in a uniform field experiences a constant force F = qE, leading to constant acceleration a = F / m = qE / m. The work done by the field on a charge moving a distance x parallel to the field is W = q E x = q ΔV. This linear relationship makes calculations straightforward.
匀强电场中的带电粒子受到恒力 F = qE，因而产生恒定加速度 a = F / m = qE / m。电场对平行于场强方向移动距离x的电荷所做的功为 W = q E x = q ΔV。这种线性关系使计算非常直接。
8. Motion of Charged Particles in Uniform Electric Fields | 带电粒子在匀强电场中的运动
When a charged particle enters a uniform electric field perpendicular to the field lines, its motion is analogous to projectile motion in a uniform gravitational field. The component of velocity parallel to the plates remains constant, while the perpendicular component is uniformly accelerated.
当带电粒子垂直射入匀强电场时，其运动类似于抛体在均匀引力场中的运动。平行于极板方向的速度分量保持不变，而垂直于极板方向的速度分量做匀加速运动。
If an electron with initial horizontal velocity vₓ enters a field E across plates of length L, it experiences a vertical acceleration a = eE / m (where e is the elementary charge). The time spent between the plates is t = L / vₓ, and the vertical deflection y = ½ a t² = ½ (eE/m)(L/vₓ)². This principle is exploited in inkjet printers and cathode-ray tubes to steer charged particles.
若电子以水平初速度vₓ进入长度为L的极板间匀强电场E，则竖直方向加速度为 a = eE / m（e为元电荷）。在两板间运动时间为 t = L / vₓ，竖直偏转量 y = ½ a t² = ½ (eE/m)(L/vₓ)²。这一原理被应用于喷墨打印机和阴极射线管中，以实现对带电粒子的偏转。
9. Capacitance and Energy Storage in Electric Fields | 电容与电场中的能量存储
Capacitance C is the measure of a component’s ability to store charge. For any capacitor, C = Q / V, where Q is the charge stored on one plate and V is the potential difference between the plates. The unit of capacitance is the farad (F).
电容C衡量器件储存电荷的能力。对任何电容器，C = Q / V，其中Q为单极板上的电荷量，V为两极板间的电势差，单位是法拉（F）。
For a parallel-plate capacitor with vacuum between the plates, the capacitance depends only on geometry:
C = ε₀ A / d
对于极板间为真空的平行板电容器，电容仅取决于几何结构：C = ε₀ A / d，A为极板面积，d为间距。插入电介质会通过相对介电常数εᵣ增大电容。
The energy stored in a charged capacitor is a consequence of the electric field established between the plates:
U = ½ Q V = ½ C V² = ½ Q² / C
充电电容器储存的能量来自极板间建立的电场：U = ½ Q V = ½ C V² = ½ Q² / C。这一能量可以看作储存在电场中，电场能量密度为 u = ½ ε₀ E²。
10. Comparison of Electric and Gravitational Fields | 电场与引力场的比较
Electric and gravitational fields share many formal similarities: both are vector fields, forces follow an inverse-square law for point sources, and the concepts of potential and potential energy are analogous. However, crucial differences exist, especially concerning the nature of charges versus masses.
电场与引力场在形式上有很多相似之处：两者都是矢量场，点源的力均遵循平方反比定律，且电势与引力势的概念类似。但二者也存在关键差异，尤其是电荷与质量的本性不同。
| Property | Electric Field | Gravitational Field |
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