📚 Electric Fields: IB & OCR Physics Revision Guide | 电场考点精讲
This revision guide covers the essential concepts of electric fields for IB and OCR Physics. It is designed to reinforce your understanding of Coulomb’s law, field strength, potential, and the motion of charged particles, with a clear focus on exam-style reasoning and calculations.
本考点精讲涵盖IB和OCR物理电场部分的核心概念。旨在帮助你巩固库仑定律、电场强度、电势以及带电粒子运动等内容，并紧扣考试中常见的分析思路与计算。
1. Coulomb’s Law | 库仑定律
The magnitude of the electrostatic force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of their separation.
两个点电荷之间静电力的大小与电荷量的乘积成正比，与它们之间距离的平方成反比。
F = k |q₁ q₂| / r²
Here k = 8.99 × 10⁹ N m² C⁻² in a vacuum, often written as 1/(4πε₀). ε₀ is the permittivity of free space, ε₀ = 8.85 × 10⁻¹² F m⁻¹.
真空中 k = 8.99 × 10⁹ N m² C⁻²，常写作 1/(4πε₀)。ε₀ 为真空介电常数，ε₀ = 8.85 × 10⁻¹² F m⁻¹。
Forces are attractive if charges have opposite signs and repulsive if they have the same sign. The direction of the force is along the line joining the centres of the two charges.
异种电荷相互吸引，同种电荷相互排斥。力的方向沿两个电荷中心的连线。
2. Electric Field Strength | 电场强度
Electric field strength E at a point is defined as the force per unit positive charge acting on a small test charge placed at that point.
电场强度 E 定义为放置在电场中某点的单位正电荷所受到的力。
E = F / q
It is a vector field, meaning it has both magnitude and direction. The SI unit is N C⁻¹, which is equivalent to V m⁻¹.
电场是矢量场，既有大小又有方向。国际单位是 N C⁻¹，等同于 V m⁻¹。
The direction of E is the direction of the force on a positive test charge. Field strength around a point charge decreases with distance according to the inverse square law.
E 的方向与正检验电荷所受力的方向一致。点电荷周围的场强随距离按平方反比规律减弱。
3. Electric Field Lines and Patterns | 电场线与场型
Field lines provide a visual representation of the electric field. The lines start on positive charges and end on negative charges, or go to infinity if the charge is isolated.
电场线可直观表示电场。电场线起始于正电荷，终止于负电荷；若为孤立电荷则可延伸至无穷远。
The density of lines indicates field strength: closer lines mean a stronger field. Lines never cross, and the tangent at any point gives the direction of E.
电场线的疏密表示场强：线越密，场越强。电场线永不相交，任一点的切线方向即为该点电场强度 E 的方向。
Common patterns include radial lines around a point charge, parallel lines for a uniform field, and the dipole pattern joining opposite charges.
常见场型包括点电荷周围的辐射状电场线、匀强电场的平行线，以及连接异种电荷的电偶极子场型。
4. Electric Field Due to a Point Charge | 点电荷的电场
For a point charge Q, the electric field strength at a distance r from the charge is given by:
对于点电荷 Q，在距离 r 处产生的电场强度为：
E = k |Q| / r² or E = |Q| / (4πε₀ r²)
The field is radial: away from a positive charge and towards a negative charge. The magnitude follows the inverse square relationship, so doubling the distance reduces the field to one quarter.
电场呈辐射状：正电荷的电场向外辐射，负电荷的电场向里汇聚。场强遵循平方反比关系，距离加倍则场强减为原来的四分之一。
When combining fields from multiple point charges, the principle of superposition applies: the resultant field is the vector sum of individual fields.
多个点电荷的电场叠加时，遵循叠加原理：合场强为各分场强的矢量和。
5. Uniform Electric Fields | 匀强电场
A uniform electric field exists between two parallel charged plates separated by a small distance. The field strength is constant in magnitude and direction between the plates, except near the edges.
两块带电平行板间距较小时，板间可产生匀强电场。除边缘区域外，板间电场的大小和方向处处相同。
E = V / d
Where V is the potential difference between the plates and d is their separation. This relation shows that E can be expressed in V m⁻¹, which is more practical for uniform fields.
式中 V 为板间电势差，d 为板间距离。该关系式表明 E 可用 V m⁻¹ 表示，在匀强电场中更为实用。
The force on a charge q in this field is constant: F = qE, leading to constant acceleration along the field direction (or opposite, depending on sign).
匀强电场中电荷 q 所受的力恒定：F = qE，因此电荷将沿电场方向（或相反方向）获得恒定加速度。
6. Electric Potential Energy | 电势能
Electric potential energy (U) is the energy a charge possesses due to its position in an electric field. For two point charges, the potential energy is given by:
电势能 (U) 是电荷因处于电场中而具有的能量。两点电荷系统的电势能为：
U = k q₁ q₂ / r or U = q₁ q₂ / (4πε₀ r)
The zero of potential energy is usually taken at infinite separation. Positive work must be done to bring like charges closer together, increasing their potential energy.
通常取无穷远处为零势能点。使同种电荷相互靠近需做正功，电势能增加。
In a uniform field, the change in electric potential energy when a charge q moves through a potential difference V is ΔU = qV. A positive charge loses potential energy when moving in the direction of the electric field.
在匀强电场中，电荷 q 移动经过电势差 V 时电势能的变化为 ΔU = qV。正电荷沿电场方向移动时电势能减少。
7. Electric Potential | 电势
Electric potential (V) at a point is the work done per unit positive charge to bring a test charge from infinity to that point without changing its kinetic energy.
电势 (V) 定义为将单位正电荷从无穷远处移到该点过程中外力所做的功（不改变其动能）。
V = k Q / r (for a point charge)
Potential is a scalar quantity, measured in volts (V) where 1 V = 1 J C⁻¹. The potential near a positive charge is positive; near a negative charge it is negative.
电势是标量，单位为伏特 (V)，1 V = 1 J C⁻¹。正电荷周围电势为正，负电荷周围电势为负。
Equipotential surfaces are surfaces of constant potential. No work is done moving a charge along an equipotential surface because the force is perpendicular to the surface.
等势面是电势处处相等的面。沿等势面移动电荷不做功，因为电场力总是垂直于等势面。
8. Potential Difference and Energy | 电势差与能量
Potential difference (p.d.) between two points is the work done per unit charge in moving a positive charge from one point to the other.
两点间的电势差 (p.d.) 等于将单位正电荷从一点移至另一点过程中所做的功。
V = W / q or ΔV = ΔU / q
In many exam problems, especially those involving the motion of electrons or ions, the kinetic energy gained is linked to the accelerating voltage by: qV = ½ m v².
考试中多数涉及电子或离子运动的问题，均可利用加速电压与动能的关系：qV = ½ m v²。
For an electron accelerated through a p.d. of V, the speed is v = √(2eV / mₑ), assuming initial speed is zero and non-relativistic speeds.
对于一个经电势差 V 加速的电子，若初速为零且不考虑相对论效应，则速率 v = √(2eV / mₑ)，其中 mₑ 为电子质量。
9. Relationship between Field and Potential | 场与电势的关系
In general, the electric field is the negative gradient of the potential. For a uniform field, this simplifies to E = ΔV / d, where ΔV is the potential difference over distance d.
一般而言，电场强度等于电势的负梯度。在匀强电场中，简化为 E = ΔV / d，其中 ΔV 为沿电场方向距离 d 上的电势差。
E = – (dV / dr) (for radial fields)
This is why field lines point from high potential to low potential. The spacing of equipotential lines indicates the field strength: closer equipotentials mean a larger E.
这就是电场线由高电势指向低电势的原因。等势线的疏密可反映场强大小：等势线越密，E 越大。
Understanding this relation is essential for interpreting graphical questions, e.g., determining E from a V–r graph by finding the slope.
理解该关系对于解读图像题至关重要，例如通过 V–r 图求斜率来得出电场强度 E。
10. Motion of Charged Particles in Uniform Fields | 带电粒子在匀强场中的运动
A charged particle entering a uniform electric field perpendicularly will experience a parabolic trajectory, similar to a projectile in a gravitational field.
带电粒子垂直射入匀强电场时将做抛物线运动，类似于匀强重力场中的抛体运动。
The constant acceleration a = qE / m acts perpendicular to the initial velocity. The horizontal motion remains uniform, while the vertical motion is uniformly accelerated.
恒定加速度 a = qE / m 方向与初速度垂直。水平方向为匀速直线运动，竖直方向为匀加速运动。
This principle is widely used in cathode ray tubes and inkjet printers to deflect beams of electrons or charged droplets. Exam questions frequently ask for the deflection y = (qE L²) / (2 m v₀²) where L is the length of the field region.
该原理广泛应用于阴极射线管和喷墨打印机中，以偏转电子束或带电墨滴。考试常要求计算偏转量 y = (qE L²) / (2 m v₀²)，其中 L 为电场区域长度。
11. Millikan’s Oil Drop Experiment | 密立根油滴实验
Millikan’s experiment determined the elementary charge e by suspending tiny charged oil droplets in a uniform field between parallel plates.
密立根通过在两平行板间的匀强电场中悬浮带电小油滴，测定了元电荷 e。
When the droplet is stationary, the electric force qE balances the weight mg minus the upthrust. With the field off, the terminal velocity under viscosity gives the droplet radius.
当油滴静止时，电场力 qE 与重力 mg 减去浮力相平衡。关闭电场后，通过测量粘滞阻力下的终端速度可得到油滴半径。
The charge q was always found to be an integer multiple of e = 1.60 × 10⁻¹⁹ C, proving the quantisation of charge. This is a classic OCR experiment combining mechanics, electricity and data analysis.
实验发现油滴的电荷量总是元电荷 e = 1.60 × 10⁻¹⁹ C 的整数倍，证明了电荷的量子化。这是OCR考试中经典的力学、电学与数据分析综合实验。
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