📚 AS Physics: Electric Fields – Key Points | AS 物理：电场 考点精讲
Electric fields are fundamental in AS Physics, describing the influence of electric charges on their surroundings. Mastering this topic is essential for understanding circuits, capacitors, and particle dynamics. This guide comprehensively covers key concepts, formulas, and exam tips.
电场是 AS 物理中的基础内容，描述了电荷对其周围环境的影响。掌握这一主题对于理解电路、电容器以及粒子动力学至关重要。本文全面覆盖核心概念、公式与应试技巧。
1. Electric Charge and Coulomb’s Law | 电荷与库仑定律
There are two types of electric charge: positive and negative. Like charges repel, opposite charges attract. Charge is quantised, with the elementary charge e = 1.60 × 10⁻¹⁹ C. The force between two point charges is given by Coulomb’s law: F = (1/(4πε₀)) Q₁Q₂ / r², where ε₀ = 8.85 × 10⁻¹² F m⁻¹ is the permittivity of free space. Often the constant k = 1/(4πε₀) ≈ 8.99 × 10⁹ N m² C⁻² is used.
电荷分为正电荷和负电荷两种。同种电荷相互排斥，异种电荷相互吸引。电荷是量子化的，基本电荷 e = 1.60 × 10⁻¹⁹ C。两个点电荷之间的力由库仑定律给出：F = (1/(4πε₀)) Q₁Q₂ / r²，其中 ε₀ = 8.85 × 10⁻¹² F m⁻¹ 是真空介电常数。常数 k = 1/(4πε₀) ≈ 8.99 × 10⁹ N m² C⁻² 也经常使用。
Coulomb force is a vector quantity; for multiple charges, use vector addition. The force acts along the line joining the centres of the two charges. If the charges have the same sign, the force is repulsive and positive; if opposite, attractive and negative in sign when considering direction.
库仑力是矢量；对于多个电荷，需要使用矢量加法。力的方向沿两个电荷中心的连线。同号电荷之间为排斥力，异号之间为吸引力，矢量的正负号需要结合方向来定。
2. Electric Field and Electric Field Strength | 电场与电场强度
An electric field is a region of space where a charged particle experiences an electric force. The electric field strength E at a point is defined as the force per unit positive charge: E = F / q. It is a vector, with units N C⁻¹ or equivalently V m⁻¹.
电场是空间中电荷受到电场力的区域。电场强度 E 定义为单位正电荷在该点受到的力：E = F / q。它是矢量，单位为 N C⁻¹，也可用 V m⁻¹ 表示。
For a point charge Q, the field strength at a distance r is given by E = (1/(4πε₀)) Q / r². The direction of E is radially outward from a positive charge and radially inward toward a negative charge.
对于点电荷 Q，距离 r 处的电场强度为 E = (1/(4πε₀)) Q / r²。E 的方向从正电荷向外辐射，指向负电荷向内。
The principle of superposition applies: the resultant electric field at a point is the vector sum of the fields due to individual charges.
电场叠加原理适用：某点的合电场是各个电荷产生的电场的矢量和。
3. Electric Field Lines and Field Patterns | 电场线与场型
Electric field lines are a visual tool to represent electric fields. The rules are: lines start on positive charges and end on negative charges; the tangent at any point gives the direction of E; the density of lines indicates the field strength – closer spacing means stronger field. Field lines never cross.
电场线是表示电场的可视化工具。规则是：电场线起始于正电荷，终止于负电荷；任意点的切线方向即为 E 的方向；线的疏密表示场强大小——越密场越强。电场线永不相交。
Common patterns to know: isolated positive charge (radial outwards), isolated negative charge (radial inwards), dipole (from positive to negative, curving), two like charges (null point between them where E = 0), and uniform field between parallel plates (straight, parallel, equally spaced).
需要掌握的常见场型：孤立正电荷（径向向外）、孤立负电荷（径向向内）、电偶极子（从正到负，弯曲）、两个同号电荷（中间存在 E = 0 的中性点）、平行板间的匀强电场（等距直线）。
4. Uniform Electric Fields and Parallel Plates | 均匀电场与平行板
A uniform electric field is one in which the field strength E is constant in magnitude and direction. It is produced by two oppositely charged parallel plates. The field lines are straight, parallel, and evenly spaced, directed from the positive to the negative plate.
匀强电场中场强 E 的大小和方向处处相同。两个带等量异号电荷的平行板之间可以产生匀强电场。电场线为等距的平行直线，方向从正极板指向负极板。
The relationship between the potential difference V across the plates and the separation d is E = V / d. This holds only if the field is uniform. The force on a charge q in such a field is F = qE, which is constant in magnitude and direction.
板间电势差 V 与板间距 d 之间的关系为 E = V / d（仅适用于匀强电场）。处于该场中的电荷 q 受到的力为 F = qE，该力大小和方向均恒定。
5. Electric Potential Energy | 电势能
Electric potential energy U is the work done to bring a charge from infinity to a point in an electric field. For two point charges Q and q separated by distance r, U = (1/(4πε₀)) Qq / r. The sign of U depends on the signs of the charges: positive for repulsive configuration (like charges), negative for attractive (opposite charges).
电势能 U 是将一个电荷从无穷远处移动到电场中某点所做的功。对于两个点电荷 Q 和 q，相距 r 时，U = (1/(4πε₀)) Qq / r。U 的符号取决于电荷符号：同号电荷为正（排斥情况），异号电荷为负（吸引情况）。
In a uniform field, change in potential energy when a charge q moves through a potential difference ΔV is ΔU = qΔV. If the charge moves in the direction of the field, its potential energy decreases (if positive charge).
在匀强电场中，电荷 q 经过电势差 ΔV 时电势能的变化为 ΔU = qΔV。若正电荷沿电场线方向运动，电势能减小。
6. Electric Potential | 电势
Electric potential V is the electric potential energy per unit charge: V = U / q. It is a scalar quantity, measured in volts (V = J C⁻¹). For a point charge Q, V = (1/(4πε₀)) Q / r. The potential is zero at infinity.
电势 V 定义为单位电荷具有的电势能：V = U / q。它是标量，单位为伏特 (V = J C⁻¹)。对于点电荷 Q，V = (1/(4πε₀)) Q / r。无穷远处电势为零。
Equipotential surfaces are surfaces of constant potential. No work is done moving a charge along an equipotential surface. In radial fields, they are concentric spheres; in uniform fields, they are planes perpendicular to the field lines.
等势面是电势恒定的面。沿等势面移动电荷不做功。在径向场中，等势面为同心球面；在匀强电场中，等势面为垂直于电场线的一系列平面。
7. Potential Difference and Work Done | 电势差与做功
The potential difference (p.d.) V between two points is the work done per unit charge moving a positive charge between them: V = W / q. In a uniform field, V = Ed, where d is the distance moved along the field lines.
两点之间的电势差 V 是将单位正电荷从一点移动到另一点所做的功：V = W / q。在匀强电场中，V = Ed，其中 d 是沿电场线方向移动的距离。
When a charge q moves through a p.d. V, the change in kinetic energy can be found using conservation of energy: ½mv² = qV (if initial KE is zero). This is the basis for calculations involving accelerated charged particles.
当电荷 q 经过电势差 V 时，可利用能量守恒求出动能变化：½mv² = qV（假设初动能为零）。这是计算带电粒子加速问题的基础。
8. Motion of Charged Particles in Uniform Electric Fields – Parallel to Field | 带电粒子在匀强电场中的运动 – 平行于电场方向
When a charged particle moves parallel to a uniform electric field, it experiences a constant force F = qE, and thus a constant acceleration a = F/m = qE/m. The motion is linear and can be analysed using SUVAT equations.
当带电粒子沿匀强电场方向运动时，受到恒力 F = qE，加速度 a = F/m = qE/m 也恒定。运动是直线运动，可用 SUVAT 方程分析。
If a particle starts from rest and accelerates through a p.d. V, the final speed is given by v = √(2qV/m). This is derived from ½mv² = qV. Note that the result is independent of the distance travelled, only on the p.d.
若粒子从静止开始经电势差 V 加速，末速度为 v = √(2qV/m)，由 ½mv² = qV 推导得出。注意结果只与电势差有关，与运动距离无关。
9. Motion of Charged Particles – Perpendicular to Field (Projectile-like) | 带电粒子的类抛体运动
If a charged particle enters a uniform electric field at right angles with initial speed v₀, its trajectory becomes parabolic, similar to a projectile. The velocity parallel to the plates (x-direction) remains constant, while the velocity perpendicular (y-direction) accelerates due to the electric force.
若带电粒子以初速度 v₀ 垂直进入匀强电场，其轨迹变为抛物线，类似于抛体运动。平行于极板的速度（x 方向）保持不变，垂直方向（y 方向）因电场力而加速。
Let the field be along y, E = V/d. Acceleration in y: a_y = qE/m = qV/(md). Time in field: t = L / v₀, where L is the length of the plates. Vertical displacement: y = ½ a_y t² = ½ (qV/(md)) (L/v₀)². Vertical velocity component at exit: v_y = a_y t.
设电场沿 y 方向，E = V/d。y 方向加速度为 a_y = qE/m = qV/(md)。在电场中的时间 t = L / v₀，其中 L 为板长。竖直位移为 y = ½ a_y t² = ½ (qV/(md)) (L/v₀)²。出口处的竖直分速度为 v_y = a_y t。
The deflection angle θ upon exit is given by tan θ = v_y / v₀. These equations allow calculation of coordinates on a screen placed beyond the plates.
出口处的偏转角 θ 满足 tan θ = v_y / v₀。利用这些方程可计算粒子在屏幕上的落点。
10. Comparison between Electric and Gravitational Fields | 电场与引力场的比较
Both electric and gravitational fields are described by inverse-square laws for point sources, but electric fields can be attractive or repulsive, while gravity is always attractive. The gravitational field strength g is force per unit mass, while E is force per unit charge.
电场和引力场对于点源均满足平方反比律，但电场有吸引和排斥之分，而引力总是吸引的。引力场强度 g 定义为单位质量的力，而 E 为单位电荷的力。
Gravitational potential V_g = -GM/r, and electric potential V_e = (1/(4πε₀)) Q/r. The negative sign in gravitational potential corresponds to the always-attractive nature. Both potentials are zero at infinity.
引力势 V_g = -GM/r，电势 V_e = (1/(4πε₀)) Q/r。引力势的负号反映了引力的吸引性。两者在无穷远处均为零。
For charges, there exists the concept of a neutral point where net field is zero (between two like charges), while in gravity, an analogous point exists between two masses (Lagrange points are a related but more complex concept).
对于电荷，存在合场强为零的中性点（在同号电荷之间）；而在引力场中，类似点存在于两质量之间（如拉格朗日点，但概念更复杂）。
11. Key Formulas Summary | 关键公式总结
| Quantity / 物理量 | Formula / 公式 | Notes / 备注 |
|---|---|---|
| Coulomb’s law / 库仑定律 | F = (1/(4πε₀)) Q₁Q₂ / r² | For point charges / 点电荷 |
| Electric field strength / 电场强度 | E = F / q | Definition / 定义 |
| Field of a point charge / 点电荷的场强 | E = (1/(4πε₀)) Q / r² | Radial field / 径向场 |
| Uniform field / 匀强电场 | E = V / d | Between parallel plates / 平行板间 |
| Electric potential energy / 电势能 | U = (1/(4πε₀)) Qq / r | Two point charges / 两点电荷 |
| Electric potential / 电势 | V = (1/(4πε₀)) Q / r | Scalar / 标量 |
| Work and p.d. / 功与电势差 | W = qV | ΔU = qΔV |
| Kinetic energy from p.d. / 由电势差获得的动能 | ½mv² = qV | Particle starting from rest / 从静止加速 |
| Deflection in uniform field / 匀强电场中的偏转 | y = ½ (qV/(md)) (L/v₀)² | Perpendicular entry / 垂直入射 |
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