📚 A-Level Physics: Electric Field Concepts Explained | A-Level 物理：电场概念精讲
Electric fields are one of the most important topics in A-Level Physics. They connect force, energy, and motion in a way that appears repeatedly across the CIE syllabus. In this article, we will break down every core idea step by step, from Coulomb's law to equipotential surfaces, so that you can master both the concepts and the calculations.
电场是 A-Level 物理中最重要的主题之一。它将力、能量和运动联系在一起，在 CIE 考纲中反复出现。在本文中，我们将逐步拆解每一个核心概念，从库仑定律到等势面，帮助你同时掌握物理思想与计算方法。
1. What Is an Electric Field? | 什么是电场？
An electric field is a region of space around a charged object in which another electric charge experiences a force. Even when two charges are not touching, they can still interact because the field carries the effect of one charge to the other.
电场是带电物体周围的一个空间区域，在这个区域内，另一个电荷会受到力的作用。即使两个电荷没有接触，它们仍然可以相互作用，因为电场将一个电荷的影响传递给另一个电荷。
The electric field at any point is a vector quantity. Its direction is defined as the direction of the force that would act on a small positive test charge placed at that point. The magnitude of the field is the force per unit positive charge.
电场中任意一点的电场强度是一个矢量。其方向定义为：将一个小正试探电荷放在该点时，它所受力的方向。电场强度的大小等于单位正电荷所受的力。
2. Coulomb's Law | 库仑定律
Coulomb's law describes the electric force between two point charges. For two charges Q₁ and Q₂ separated by a distance r, the magnitude of the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
库仑定律描述了两个点电荷之间的静电力。对于相距为 r 的两个点电荷 Q₁ 和 Q₂，力的大小与两电荷电荷量的乘积成正比，与它们之间距离的平方成反比。
F = k Q₁Q₂ / r²
Here, k is Coulomb' 9 × 10⁹ N m² C⁻². In terms of the permittivity of free space ε₀, we write k = 1 / (4π ε₀).
其中，k 是库仑常数，约等于 8.99 × 10⁹ N·m²·C⁻²。用真空介电常数 ε₀ 表示时，k = 1 / (4π ε₀)。
If the two charges have the same sign, the force is repulsive and acts along the line joining them. If the signs are opposite, the force is attractive. When more than two charges are present, the resultant force on any one charge is the vector sum of the individual forces from each other charge. This is called the principle of superposition.
如果两个电荷同号，则力为斥力，方向沿两电荷连线；如果异号，则为引力。当存在多个电荷时，某个电荷所受合力等于其他各个电荷单独作用力的矢量和，这称为叠加原理。
3. Electric Field Strength | 电场强度
Electric field strength E at a point is defined as the force experienced per unit positive charge placed at that point. Mathematically,
电场强度 E 定义为：放在该点的单位正电荷所受的力。数学表达为：
E = F / q
The unit of electric field strength is N C⁻¹, which is equivalent to V m⁻¹. Notice that E is a vector: its direction is the same as the direction of the force on a positive charge.
电场强度的单位是 N·C⁻¹，也等价于 V·m⁻¹。注意 E 是矢量，其方向与正电荷所受力的方向相同。
For the field created by a single point charge Q, we combine Coulomb's law with the definition of E. The field strength at a distance r from Q is:
对于单个点电荷 Q 产生的电场，我们将库仑定律与 E 的定义结合，得到距离 Q 为 r 处的电场强度为：
E = k Q / r²
This formula shows that the field strength decreases with the square of the distance. It also shows that a positive charge produces a field pointing away from it, while a negative charge produces a field pointing toward it.
该公式表明电场强度随距离的平方而减小。同时，正电荷产生的电场方向背离电荷，负电荷产生的电场方向指向电荷。
4. Electric Field Lines | 电场线
Electric field lines, also called field lines, are a visual way of representing an electric field. They help us see the direction and relative strength of the field.
电场线是用来形象化表示电场的一种方法。它能帮助我们直观地看出电场的方向和相对强弱。
There are several rules for drawing field lines:
绘制电场线需要遵循以下几条规则：
Field lines start on positive charges and end on negative charges. If there is only one charge, the lines extend to infinity or come from infinity.
电场线从正电荷出发，终止于负电荷。如果只有一个电荷，电场线延伸到无穷远或从无穷远来。
The direction of the field at any point is tangent to the field line at that point.
电场中某点的场强方向，就是该点电场线的切线方向。
The closer the lines are together, the stronger the electric field. In a uniform field, the lines are parallel and equally spaced.
电场线越密，表示该处电场越强。在匀强电场中，电场线是平行且等间距的。
Field lines never cross each other, because the field has only one direction at any given point.
电场线永不相交，因为在任意一点处电场方向只有一个。
For a positive point charge, the lines radiate outward symmetrically. For a negative point charge, they point inward. Between two opposite charges, the lines curve from the positive charge to the negative charge, showing an attractive field pattern.
对于正点电荷，电场线对称地向外辐射；对于负点电荷，电场线向内汇聚。在两个异种电荷之间，电场线从正电荷弯曲指向负电荷，展现出吸引的场分布。
5. Uniform Electric Fields | 匀强电场
A uniform electric field has the same magnitude and direction at every point. In the laboratory, this is often produced by two parallel metal plates connected to a battery. One plate becomes positive and the other becomes negative, creating a constant field between them.
匀强电场在每一点的场强大小和方向都相同。在实验室中，通常用两块连接到电池上的平行金属板来产生匀强电场：一块板带正电，另一块带负电，从而在两板之间形成恒定电场。
If the potential difference between the plates is V and the separation is d, the magnitude of the uniform field is given by:
如果两板之间的电势差为 V，板间距为 d，则匀强电场的大小为：
E = V / d
This equation is extremely useful. It also shows that the units V m⁻¹ and N C⁻¹ are truly equivalent.
这个公式非常有用，同时也说明 V·m⁻¹ 与 N·C⁻¹ 这两个单位本质上是等价的。
The direction of the field is from the positive plate to the negative plate. Since electric field direction is defined from high potential to low potential, the positive plate is at a higher potential than the negative plate.
电场的方向从正极板指向负极板。由于电场方向定义为从高电势指向低电势，因此正极板的电势高于负极板。
6. Electric Potential | 电势
Electric potential V at a point is defined as the work done per unit positive charge in bringing a small positive test charge from infinity to that point. It is a scalar quantity, unlike electric field strength, which is a vector.
电场中某点的电势 V 定义为：将一个小正试探电荷从无穷远处移动到该点过程中，外力所做的功与电荷量之比。电势是标量，而电场强度是矢量。
V = W / q
lectric potential is the volt, V. One volt equals one joule per coulomb, so 1 V = 1 J C⁻¹.
电势的单位是伏特 (V)。1 伏特等于 1 焦耳每库仑，即 1 V = 1 J·C⁻¹。
For a point charge Q, the potential at a distance r is:
对于点电荷 Q，距离 r 处的电势为：
V = k Q / r
Because potential is a scalar, the total potential at a point due to multiple charges is simply the algebraic sum of the individual potentials. If Q is positive, V is positive; if Q is negative, V is negative.
由于电势是标量，多个电荷在某点产生的总电势等于各个电荷单独产生的电势的代数和。如果 Q 为正，则 V 为正；如果 Q 为负，则 V 为负。
7. Potential Difference and Work | 电势差与做功
e work done per unit charge when a charge moves from A to B. It is often written as ΔV = VA − VB, but we are usually concerned with its magnitude and sign.
两点 A 和 B 之间的电势差，是指电荷从 A 移动到 B 时，单位电荷所做的功。通常写成 ΔV = VA − VB，但我们更常关心它的大小和正负。
When a charge q moves through a potential difference ΔV, the work done on the charge by the electric field is:
当电荷 q 经过电势差 ΔV 时，电场对电荷做的功为：
W = q ΔV
This work can be positive or negative. If a positive charge moves from a high potential to a low potential, the field does positive work on it, and it gains kinetic energy. If a positive charge moves from low to high potential, the field does negative work, meaning external work is required.
这个功可以是正功，也可以是负功。如果正电荷从高电势向低电势移动，电场对其做正功，电荷动能增加；如果正电荷从低电势向高电势移动，电场做负功，需要外力做功。
a special case of the general relationship between electric field and potential gradient.
在匀强电场中，沿电场方向相距 d 的两点之间的电势差与场强之间满足 E = V / d。这个公式实际上是电场与电势梯度之间普遍关系的特例。
8. Equipotential Surfaces | 等势面
An equipotential surface is a surface on which every point has the same electric potential. No work is required to move a charge along an equipotential surface, because the potential does not change.
等势面是指电势处处相等的曲面。由于电势不变，将电荷沿着等势面移动时不需要做功。
Field lines are always perpendicular to equipotential surfaces. This is why, in diagrams, equipotential lines are drawn at right angles to electric field lines.
电场线总是垂直于等势面。因此，在图中，等势线总是与电场线成直角相交。
For a uniform field, the equipotential surfaces are parallel planes, equally spaced if the field is constant. For a point charge, the equipotential surfaces are concentric spheres centered on the charge.
对于匀强电场，等势面是彼此平行的平面；如果场强恒定，等势面间距相等。对于点电荷，等势面是以电荷为球心的同心球面。
When drawing equipotential lines, remember that the potential gradient is greatest where the field is strongest. Closely spaced equipotential lines indicate a strong field, just as closely spaced field lines do.
绘制等势线时需要记住：电势梯度在电场最强处最大。等势线间距越小，表示电场越强，这与电场线密度表示场强大小是类似的。
9. Motion of Charged Particles in Electric Fields | 带电粒子在电场中的运动
harged particle is placed in an electric field, it experiences a force F = qE. From Newton's second law, its acceleration is:
当带电粒子处于电场中时，它受到力 F = qE 的作用。根据牛顿第二定律，它的加速度为：
a = F / m = qE / m
If the particle starts from rest and moves through a potential difference V, the work done by the field is qV. By the work-energy theorem, this equals the gain in kinetic energy:
如果粒子从静止开始经过电势差 V，电场做的功为 qV。根据动能定理，这等于动能的增加量：
qV = ½ m v²
This equation is often used to find the speed of an electron or proton after acceleration. It works for uniform and non-uniform fields as long as the starting and ending points have a known potential difference.
这个公式常用于计算电子或质子经过加速后的速度。只要起点和终点的电势差已知，该公式对匀强和非匀强电场都适用。
If a charged particle enters a uniform electric field at right angles to the field, its motion is similar to projectile motion. It has constant velocity perpendicular to the field, but constant acceleration parallel to the field. The result is a parabolic trajectory.
如果带电粒子垂直于匀强电场方向进入电场，它的运动类似于抛体运动：在垂直于电场方向上做匀速运动，在平行于电场方向上做匀加速运动，轨迹为抛物线。
This principle is used in cathode ray tubes and oscilloscopes. By adjusting the electric field between deflection plates, the electron beam can be moved precisely across a screen.
这个原理应用于阴极射线管和示波器中。通过调节偏转板之间的电场，可以精确地控制电子束在屏幕上的位置。
10. Electric Field vs Gravitational Field | 电场与引力场的对比
Electric fields and gravitational fields share many mathematical similarities, but they also have important differences. Understanding these helps you remember formulas and avoid confusion.
电场与引力场在数学上有很多相似之处，但二者也存在重要差异。理解这些异同有助于记忆公式并避免混淆。
| Property | Electric Field | Gravitational Field |
| Source | Electric charge (positive or negative) | Mass (positive only) |
| Force | F = k Q₁Q₂ / r² | F = G m₁m₂ / r² |
| Field strength | E = k Q / r² | g = G M / r² |
| Potential | V = k Q / r (can be + or −) | V = − G M / r (always negative) |
| Direction of force | Toward opposite charge; away from like charge | Always attractive, toward the mass |
Like gravitational fields, electric fields obey an inverse square law for point sources. However, because electric charges can be positive or negative, the force can be either attractive or repulsive, while gravity is always attractive.
与引力场相似，点电荷的电场也遵循平方反比定律。但由于电荷有正有负，电力可以是引力也可以是斥力，而引力永远表现为吸引。
Another important comparison is the convention for potential. In gravitational fields, potential is taken as zero at infinity and is always negative. In electric fields, potential at infinity is also zero, but it can be positive around positive charges and negative around negative charges.
另一个重要对比是电势的符号约定。在引力场中，无穷远处电势为零，且各点势能通常为负。在电场中，无穷远处电势也为零，但正电荷周围的电势为正，负电荷周围的电势为负。
11. Exam Tips and Common Mistakes | 考试技巧与常见错误
To do well in CIE A-Level questions on electric fields, keep these key points in mind:
要想在 CIE A-Level 电场题目中取得好成绩，请注意以下要点：
The direction of the electric field is the direction of force on a positive test charge. Never use the force on a negative charge to define the field direction.
电场方向是正试探电荷所受力的方向。绝不要用负电荷所受力的方向来定义场强方向。
Electric field strength is a vector, so when adding fields from multiple charges, you must use vector addition. Electric potential is a scalar, so it is added algebraically.
电场强度是矢量，多个电荷产生的场强相加时必须使用矢量加法。电势是标量，因此可以直接进行代数相加。
= V / d. In a radial field of a point charge, use E = k Q / r². Do not mix these formulas.
在匀强电场中，使用 E = V / d。在点电荷的径向电场中，使用 E = k Q / r²。不要混淆这两个公式。
Remember that a negative charge moves opposite to the direction of the electric field, while a positive charge moves in the same direction as the field.
记住：负电荷沿电场反方向运动，正电荷沿电场方向运动。
m high to low potential, W = qΔV is positive; if it moves from low to high, W is negative.
计算功时要注意正负号。正电荷从高电势移到低电势时，W = qΔV 为正；从低电势移到高电势时，W 为负。
². This avoids mistakes with the sign of the charge, because both q and V can be negative.
对于从静止开始经过电势差 V 加速的带电粒子，始终从 qV = ½ m v² 出发。由于 q 和 V 都可能为负，这样处理可以避免符号错误。
Finally, always include units in your final answer. Electric field strength is measured in N C⁻¹ or V m⁻¹, potential is measured in volts, and force is measured in newtons.
最后，最终答案中一定要包含单位。电场强度的单位是 N·C⁻¹ 或 V·m⁻¹，电势的单位是伏特，力的单位是牛顿。
12. Summary of Key Formulas | 核心公式总结
Below is a quick revision list of the most important equations from this topic. You should be able to use each one confidently and explain what every symbol means.
下面列出本主题最重要的公式作为快速复习清单。你应该能够熟练使用每个公式，并解释每个符号的含义。
F = k Q₁Q₂ / r²
E = F / q
E = k Q / r²
E = V / d
V = W / q
V = k Q / r
W = q ΔV
qV = ½ m v²
Master these definitions and formulas, and then practise applying them to past paper questions
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