📚 Edexcel Physics: Electric Fields Key Points | Edexcel 物理：电场 考点精讲
In Edexcel A Level Physics, electric fields form a core part of Topic 7: Electric and Magnetic Fields. Understanding the behaviour of charged particles, field lines, potential, and capacitance is essential for success. This article distils the key concepts, formulas, and graphical relationships you must master.
在 Edexcel A Level 物理中，电场是第七单元“电场与磁场”的核心内容。掌握电荷行为、电场线、电势和电容是考试成功的关键。本文提炼了你必须掌握的核心概念、公式和图像关系。
1. Electric Fields and Field Strength | 电场与电场强度
An electric field is a region around a charged object where a force is exerted on other charged objects. Field lines indicate the direction a positive test charge would move; they point away from positive charges and towards negative charges.
电场是带电物体周围对其他电荷施加力的区域。电场线表示正检验电荷的运动方向：电场线从正电荷发出，指向负电荷。
The electric field strength E at a point is defined as the electrostatic force F experienced per unit positive charge q₀ placed at that point. It is a vector quantity, with units N C⁻¹ or equivalently V m⁻¹.
电场强度 E 在某点的定义是：放在该点的单位正电荷 q₀ 所受的静电力 F。E 是矢量，单位是 N C⁻¹，也等价于 V m⁻¹。
E = F / q₀
For a uniform field produced by parallel plates, the field strength is constant in magnitude and direction. For a radial field around a point charge, the magnitude varies with distance, and the direction is radially outward (for positive charge) or inward (for negative charge).
对于平行板产生的匀强电场，场强大小和方向处处相同。对于点电荷周围的径向电场，场强大小随距离变化，方向沿径向向外（正电荷）或向内（负电荷）。
2. Coulomb’s Law | 库仑定律
Coulomb’s law describes the electrostatic force between two point charges. The force F is directly proportional to the product of the charges and inversely proportional to the square of the distance r between their centres.
库仑定律描述两个点电荷之间的静电力。力 F 与两电荷量的乘积成正比，与它们中心距离 r 的平方成反比。
F = k Q₁ Q₂ / r²
Here k = 1/(4πε₀), where ε₀ is the permittivity of free space (≈ 8.85 × 10⁻¹² F m⁻¹). The constant k has a value of approximately 8.99 × 10⁹ N m² C⁻². Like charges repel, opposite charges attract; the sign of the product Q₁Q₂ determines whether the force is attractive or repulsive.
式中 k = 1/(4πε₀)，ε₀ 为真空介电常数（≈ 8.85 × 10⁻¹² F m⁻¹），k 值约为 8.99 × 10⁹ N m² C⁻²。同种电荷相斥，异种电荷相吸；Q₁Q₂ 的符号决定力是吸引力还是排斥力。
Coulomb’s law applies strictly to point charges, but it also holds for spherically symmetric charge distributions as if all the charge were concentrated at the centre. This is analogous to Newton’s law of gravitation.
库仑定律严格适用于点电荷，但对于球对称的电荷分布也成立，相当于所有电荷集中在球心。这与万有引力定律类似。
3. Radial Electric Fields | 径向电场
A point charge Q produces a radial electric field. Using Coulomb’s law, the field strength E at a distance r from the charge is given by:
点电荷 Q 产生径向电场。根据库仑定律，距离电荷 r 处的场强 E 为：
E = k Q / r²
The field lines radiate outwards from a positive Q and inwards towards a negative Q. The magnitude E decreases with the square of distance; when r doubles, E falls to one quarter.
电场线从正电荷发出，汇聚于负电荷。E 的大小随距离的平方递减；r 加倍时，E 变为原来的四分之一。
Radial fields are key to understanding the motion of charged particles in orbits around nuclei (Rutherford scattering) and in electrostatic precipitators. The force on a charge q in this field is simply F = qE = k Q q / r².
径向电场对理解带电粒子绕核运动（卢瑟福散射）以及静电除尘器中的运动至关重要。电荷 q 在该场中受力为 F = qE = k Q q / r²。
4. Uniform Electric Fields | 匀强电场
A uniform electric field is produced between two parallel conducting plates separated by distance d when a potential difference V is applied across them. The field is perpendicular to the plates, and its magnitude is constant.
当两平行导电板间距 d 并加以电势差 V 时，板间产生匀强电场。场的方向垂直于板面，大小处处相等。
E = V / d
This equation is derived from the relation between work done and potential difference: W = qV, and also W = F d = qE d, hence V = E d. The unit V m⁻¹ is identical to N C⁻¹.
该式源自功与电势差的关系：W = qV，同时 W = F d = qE d，从而 V = E d。单位 V m⁻¹ 与 N C⁻¹ 相同。
In practical problems, remember that the potential difference is the voltage between the plates. If an electron moves through 1 V, it gains kinetic energy of 1 eV (electronvolt): 1 eV = 1.6 × 10⁻¹⁹ J.
在实际问题中，注意电势差就是板间电压。若电子经过 1 V 电势差，将获得 1 eV 动能：1 eV = 1.6 × 10⁻¹⁹ J。
5. Electric Potential | 电势
Electric potential V at a point in an electric field is defined as the work done per unit positive charge in bringing a small test charge from infinity to that point, without any change in kinetic energy.
电场中某点的电势 V 定义为：将单位正检验电荷从无穷远处移到该点所做的功（无动能变化）。
V = k Q / r
For a point charge Q, potential is a scalar quantity. It is positive around a positive charge and negative around a negative charge. At infinity, V is taken as zero. Potential difference between two points A and B is ΔV = V_B − V_A.
对于点电荷 Q，电势是标量。正电荷周围电势为正，负电荷周围为负。无穷远处电势取为零。A、B 两点间电势差 ΔV = V_B − V_A。
In a uniform field, potential changes linearly with distance along the field direction. In a radial field, it follows an inverse relationship with r.
在匀强电场中，沿场方向电势随距离线性变化。在径向场中，电势与 r 成反比。
6. Equipotential Surfaces | 等势面
Equipotential surfaces are imaginary surfaces where every point has the same electric potential. No work is done when moving a charge along an equipotential, because the change in potential is zero.
等势面是假想的、所有点电势相同的面。沿等势面移动电荷不做功，因为电势变化为零。
Field lines are always perpendicular to equipotential surfaces. For a point charge, equipotentials are concentric spheres; for a uniform field, they are planes parallel to the plates.
电场线始终与等势面垂直。点电荷的等势面是同心球面；匀强电场的等势面是与板平行的平面。
Drawing and interpreting equipotential lines (in 2D) is a common graphical skill. The closer the equipotentials, the stronger the electric field.
绘制和解释二维等势线是常见的作图技能。等势面（线）越密集，电场越强。
7. Motion of Charged Particles in Electric Fields | 带电粒子在电场中的运动
A charged particle in an electric field experiences a force F = qE, causing constant acceleration a = qE/m if the field is uniform and other forces (like gravity) are negligible.
带电粒子在电场中受力 F = qE，若电场均匀且其他力（如重力）可忽略，粒子将做匀加速运动，加速度 a = qE/m。
In an electron gun, electrons are accelerated from rest through a potential difference V. Their final kinetic energy ½ m v² = e V. The speed v can be found as:
在电子枪中，电子从静止经电势差 V 加速，末动能 ½ m v² = e V。速度 v 可用下式求出：
v = √(2 e V / m)
If a charged particle enters a uniform field at right angles, it follows a parabolic path, similar to projectile motion under gravity. The horizontal velocity remains constant, while the vertical velocity increases linearly, resulting in parabolic deflection.
若带电粒子垂直进入匀强电场，它将沿抛物线运动，类似于重力场中的抛体运动。水平方向速度不变，竖直方向速度线性增加，形成抛物线轨迹。
8. Capacitance | 电容
A capacitor is a device that stores charge and energy. The capacitance C of a capacitor is defined as the charge Q stored per unit potential difference V across its plates.
电容器是储存电荷和能量的器件。电容 C 的定义是：极板间电势差 V 与所存储电荷 Q 之比。
C = Q / V
The unit of capacitance is the farad (F): 1 F = 1 C V⁻¹. Most practical capacitors have capacitances in microfarads (μF, 10⁻⁶ F), nanofarads (nF, 10⁻⁹ F) or picofarads (pF, 10⁻¹² F).
电容的单位是法拉（F）：1 F = 1 C V⁻¹。实际电容器的电容多为微法（μF, 10⁻⁶ F）、纳法（nF, 10⁻⁹ F）或皮法（pF, 10⁻¹² F）。
For a parallel-plate capacitor, capacitance is proportional to the plate area A and inversely proportional to the separation d: C = ε₀ εᵣ A / d, where εᵣ is the relative permittivity (dielectric constant) of the material between the plates.
平行板电容器的电容与极板面积 A 成正比，与间距 d 成反比：C = ε₀ εᵣ A / d，式中 εᵣ 是板间材料的相对介电常数。
9. Capacitors in Series and Parallel | 电容的串联和并联
When capacitors are combined, the total capacitance changes according to simple rules, opposite to those for resistors.
当电容器组合时，总电容遵循简单的规则，但与电阻的规则相反。
For capacitors in parallel, the total capacitance is the sum of individual capacitances, because the potential difference is the same across each, and charges add up.
电容器并联时，总电容等于各电容之和，因为每个电容两端电势差相同，电荷相加。
C_total = C₁ + C₂ + C₃ + …
For capacitors in series, the reciprocal of the total capacitance is the sum of the reciprocals of individual capacitances, because the charge stored is the same on each capacitor, and the voltages add up.
电容器串联时，总电容的倒数等于各电容倒数之和，因为每个电容器上储存的电荷相同，电压相加。
1 / C_total = 1 / C₁ + 1 / C₂ + 1 / C₃ + …
| Configuration / 组合方式 | Total Capacitance / 总电容 |
|---|---|
| Series / 串联 | Reduced; less than smallest individual capacitor / 减小；小于最小的单个电容 |
| Parallel / 并联 | Increased; sum of all / 增大；为各电容之和 |
10. Energy Stored in a Capacitor | 电容器储存的能量
A charged capacitor stores electrical potential energy. The energy U stored is equal to the work done in moving charge against the increasing potential difference. This leads to three equivalent expressions.
已充电电容器储存电势能。储存的能量 U 等于在不断增加的电势差下移动电荷所做的功，由此得到三个等价表达式。
U = ½ Q V = ½ C V² = ½ Q² / C
The graph of Q against V for a capacitor is a straight line through the origin (since C = Q/V). The area under the Q–V graph represents the energy stored (½ base × height). This triangular area derivation is examinable.
电容器的 Q–V 图线是一条过原点的直线（因为 C = Q/V）。Q–V 图线下的面积代表储存的能量（½ 底 × 高）。该三角形面积推导是考点。
When discharging, this stored energy is converted into other forms, typically heat in a resistor or light in a camera flash.
放电时，这些储存的能量转化为其他形式，通常是电阻中的热量或照相机闪光灯的光能。
11. Charging and Discharging of Capacitors | 电容器的充放电
When a capacitor charges through a resistor from a battery of emf ε, the voltage across the capacitor V_C rises exponentially, approaching ε asymptotically. The equation is:
当电容器通过电阻由电动势 ε 的电源充电时，电容器两端电压 V_C 按指数规律上升，渐近趋于 ε。方程为：
V_C = ε (1 − e⁻t/RC)
During charging, current starts at ε/R and decays exponentially to zero. Charge on the capacitor follows a similar exponential curve.
充电过程中，电流从 ε/R 开始按指数衰减至零。电容器上的电荷也遵循类似的指数曲线。
For discharging, the capacitor voltage, charge and current all decay exponentially from their initial values according to:
放电时，电容器电压、电荷和电流均从初始值按指数衰减：
V = V₀ e⁻t/RC, Q = Q₀ e⁻t/RC, I = I₀ e⁻t/RC
These exponential relationships are fundamental. Sketching and interpreting V–t, Q–t, and I–t graphs is a regular exam requirement.
这些指数关系是基础。绘制和解释 V–t、Q–t、I–t 图像是常见的考试要求。
12. Time Constant and Exponential Decay | 时间常数与指数衰减
The product RC, called the time constant τ, has the unit of seconds (Ω × F = s). It determines how quickly the capacitor charges or discharges: after time τ, the voltage (or charge) during charging reaches about 63% of its final value, and during discharge falls to about 37% of its initial value.
乘积 RC 称为时间常数 τ，单位是秒（Ω × F = s）。它决定电容器充放电的快慢：经过时间 τ，充电时电压（或电荷）达到最终值的约 63%，放电时电压降至初始值的约 37%。
τ = R C
Graphically, τ can be found from a V–t graph as the time at which V has changed by approximately 63%; it can also be determined from the initial gradient of the decay curve or from logarithmic plots.
从图像上看，τ 可在 V–t 图中找到对应变化约 63% 的时间；也可通过衰减曲线的初始斜率或对数图像确定。
Taking natural logarithms of the discharge equation gives ln V = ln V₀ − t/RC. A graph of ln V against t yields a straight line with gradient −1/τ, enabling experimental determination of the time constant.
对放电方程取自然对数得 ln V = ln V₀ − t/RC。ln V 随 t 变化的图像是一条斜率为 −1/τ 的直线，可用于实验测定时间常数。
The concept of half-life t½ is also useful: t½ = RC ln 2 ≈ 0.693 RC. In each half‑life, the charge or voltage halves.
半衰期 t½ 的概念也很有用：t½ = RC ln 2 ≈ 0.693 RC。每经过一个半衰期，电荷或电压减半。
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