📚 Electric Field: Basic Concepts and Key Laws | 电场：基本概念与重要规律
The electric field is a fundamental concept in physics that explains how electric charges exert forces on each other without direct contact. It is a region of space around a charge where another charge experiences a force. This article covers the essential definitions, laws, and problem-solving techniques required for the CIE A-Level Physics syllabus.
电场是物理学中深刻影响我们理解电荷相互作用的核心概念。它是电荷周围存在的、对其他电荷施力的一种空间区域。本文将围绕 CIE A-Level 物理考纲，系统梳理电场的基本定义、重要规律和典型解题方法。
1. Coulomb's Law | 库仑定律
Coulomb's law describes the electrostatic force between two point charges. The magnitude of the force is directly proportional to the product of the charges and inversely proportional to the square of the distance
库仑定律描述了两个点电荷之间的静电力。力的大小与两电荷的电量乘积成正比，与它们之间距离的平方成反比。
F = k Q₁Q₂ / r²
tant. In terms of the permittivity of free space ε₀, k = 1 / (4πε₀) ≈ 8.99 × 10⁹ N·m²·C⁻².
其中 F 为静电力的大小，Q₁、Q₂ 分别为两电荷的电量（单位库仑），r 为距离（单位米），k 为库仑常量。用真空介电常数 ε₀ 表示时，k = 1 / (4πε₀) ≈ 8.99 × 10⁹ N·m²·C⁻²。
The force is attractive for opposite charges and repulsive for like charges. It is important to note that Coulomb's law applies strictly to point charges or spherically symmetric charge distributions when using the centre-to-centre distance.
库仑力的方向遵循“同号相斥、异号相吸”的规律。需注意，库仑定律严格适用于点电荷，或可等效为球对称电荷分布时使用球心间距计算。
2. Electric Field Strength | 电场强度
Electric field strength E at a point is defined as the electric force F experienced by a small positive test charge q divided by the charge itself. It is a vector quantity and its direction is the direction of the force on a positive test charge.
电场强度 E 定义为单位正试探电荷在某点所受的电场力 F，即 E = F / q。电场强度是矢量，其方向与正试探电荷在该点所受力的方向相同。
E = F / q
The SI unit of electric field strength is newtons per coulomb (N/C), which is equivalent to volts per metre (V/m). A stronger field means a larger force per unit charge.
电场强度的国际单位是牛每库仑（N/C），也等价于伏每米（V/m）。电场越强，单位电荷所受的力越大。
For a point charge Q, the electric field strength at a distance r is derived from Coulomb' itive charge the field points radially outward, while for a negative charge it points radially inward.
对点电荷 Q，距离 r 处的场强可由库仑定律推导为 E = kQ / r²。正电荷的电场方向沿径向向外，负电荷的电场方向沿径向向内。
3. Electric Field Lines | 电场线
Electric field lines are pictorial representations used to visualise the electric field. The tangent to a field line at any point gives the direction of the electric field at that point, and the density of lines indicates the relative magnitude of the field strength.
电场线是用图形化方式表示电场的一种手段。电场线上任意一点的切线方向即该点电场强度的方向，电场线的疏密程度则反映场强的大小。
- Field lines start at positive charges and end at negative charges. They may also extend to or originate from infinity.
- Field lines never cross. If they did, a point would have two different field directions.
- The number of lines per unit area is proportional to the electric field strength.
- 电场线从正电荷出发，终止于负电荷；也可延伸到无穷远或来自无穷远。
- 电场线不相交，因为同一点的场强方向是唯一的。
- 单位面积内电场线的条数与电场强度成正比。
For a uniform electric field, the field lines are parallel, equally spaced straight lines. The uniform field is a good approximation between two parallel charged plates.
匀强电场中的电场线彼此平行、间距相等。两块平行带电金属板之间的电场通常可近似视为匀强电场。
4. Uniform Electric Field | 匀强电场
A uniform electric field has constant magnitude and direction throughout a region. The most common example is the field between two large, parallel, oppositely charged conducting plates, neglecting edge effects.
匀强电场是指在某区域内电场强度处处相同（大小相等、方向一致）。最常见的例子是两块平行、带等量异号电荷的导体板之间的电场（忽略边缘效应）。
The potential difference V between two parallel plates separated by distance d is related to the electric field strength E by:
两平行板之间的电势差 V 与板间距离 d 及场强 E 的关系为：
E = V / d
This relationship shows that a stronger field produces a greater potential change over a given distance. The direction of the field is from the plate at higher potential to the plate at lower potential.
该关系表明：场强越大，在单位距离上的电势变化就越大。电场方向从电势高的极板指向电势低的极板。
5. Point Charge Electric Field | 点电荷的电场
For an isolated point charge Q, the electric field at any point in space is directed radically along the line joining the point and the charge. The magnitude is given by:
对于孤立点电荷 Q，其周围任意一点的电场方向沿该点与电荷的连线并具有径向特征。场强大小为：
E = kQ / r²
Here r is the distance from the charge. Since E is inversely proportional to r², the field weakens rapidly as we move away from the charge.
式中 r 为到点电荷的距离。由于 E 与 r² 成反比，当场点远离电荷时，场强衰减得很快。
When dealing with multiple point charges, the total electric field at a point is found by adding the individual field vectors according to the parallelogram rule. This is the superposition principle, which is discussed in more detail later.
当存在多个点电荷时，某点的合电场强度需按平行四边形定则把所有点电荷各自产生的场强矢量相加。这就是后文即将详细讨论的叠加原理。
6. Electric Potential | 电势
The electric potential V at a point in an electric field 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, and its SI unit is the volt (V), equivalent to one joule per coulomb.
电场中某点的电势 V 定义为：将单位正试探电荷从无穷远处移到该点过程中电场力（或外力抗衡电场力）所做的功。电势是标量，国际单位为伏特（V），即 1 V = 1 J/C。
V = Ep / q
Here Ep al at infinity. The potential is positive for a positive charge and negative for a negative charge.
式中 Ep 为电荷 q 在该点的电势能。对于点电荷 Q，取无穷远处电势为零时，距离 r 处的电势为 V = kQ/r。正电荷产生的电势为正，负电荷产生的电势为负。
7. Potential Difference | 电势差
The potential difference (p.d.) between two points A and B is the work done per unit positive charge to move a charge from A to B. It is also called voltage.
两点 A、B 之间的电势差定义为：单位正电荷从 A 移动到 B 时电场力所做的功的大小，也常称为电压。
VAB = VA - VB = WAB / q
Here WAB is the work done by the electric field when a small charge q moves from A to B. Since the electric force is conservative, the work done is independent of the path taken and depends only on the initial and final positions.
式中 WAB 为电荷 q 从 A 移到 B 时电场力做的功。由于电场力是保守力，做功与路径无关，只与初末位置有关。
× d, as described in Section 4, where d is the distance along the direction of the field.
在匀强电场中，两点间的电势差等于 E × d，如第 4 节所述，其中 d 是沿电场方向的距离。
8. Equipotential Surfaces | 等势面
An equipotential surface is a surface on which every point has the same electric potential. Moving a test charge along an equipotential surface requires zero work because the potential difference is zero.
等势面是指电势处处相等的曲面。试探电荷在等势面上移动时，由于电势差为零，电场力不做功。
- For a uniform field, equipotential surfaces are planes perpendicular to the field lines.
- For a point charge, equipotential surfaces are concentric spheres centred on the charge.
- Electric field lines are always perpendicular to equipotential surfaces and point from higher potential to lower potential.
- 匀强电场中的等势面是与电场线垂直的平面。
- 点电荷电场中的等势面是以电荷为球心的一系列同心球面。
- 电场线始终垂直于等势面，并且由高电势指向低电势。
In diagrams, equipotential surfaces are often drawn with equally spaced potential differences. Where the surfaces are closer together, the electric field is stronger.
在示意图中，等势面通常按相等的电势差绘制。若等势面越密集，说明该处电场越强。
9. Electric Potential Energy | 电势能
Electric potential energy Ep is the energy stored in a charge due to its position in an electric field. For a charge q placed at a point where the potential is V, the electric potential energy is:
电势能 Ep 是电荷在电场中因位置不同而具有的能量。电荷 q 在电势为 V 的点处时，其电势能为：
Ep = qV
When a positive charge moves in the direction of the electric field, its potential energy decreases and its kinetic energy increases, if no other forces act. For a negative charge, the opposite is true: moving in the direction of the field increases its potential energy.
当正电荷沿电场方向移动时，它的电势能减少、动能增加（在仅受电场力时）。负电荷则相反：沿电场方向移动时，其电势能增加。
The change in potential energy when moving from point A to point B is ΔEp = q(VB - VA). The work done by the electric field equals the negative of this change: W = -ΔEp.
电荷从 A 点移到 B 点时电势能的变化量为 ΔEp = q(VB - VA)。电场力所做的功等于电势能变化量的相反数，即 W = -ΔEp。
10. Superposition Principle | 场的叠加原理
The superposition principle states that the total electric field due to several charges is the vector sum of the fields produced by each charge independently. The total electric potential, being a scalar, is the algebraic sum of the individual potentials.
叠加原理指出：多个电荷在某点产生的合电场强度等于各电荷单独产生的电场强度的矢量和。由于电势是标量，合电势等于各个电荷单独产生的电势的代数和。
Etotal = E₁ + E₂ + E₃ + ...
For example, to find the electric field at a point between two charges, we first calculate the field due to each charge, then add them as vectors, paying careful attention to directions. For potential, we simply add signed values.
例如，求两个电荷之间某点的合场强时，先分别算出每个电荷在该点产生的场强，再按矢量加法求合矢量，并注意方向；求合电势时，只需将各电势按正负号相加。
This principle greatly simplifies problems involving multiple charged objects, such as electric dipoles or charged rings.
这一原理极大地方便了涉及多个带电物体的计算，例如电偶极子或带电环等问题。
11. Work Done in Moving a Charge | 移动电荷做的功
In an electric field, the work done W in moving a charge q from a point at potential VA to a point at potential VB is:
在电场中，将电荷 q 从电势为 VA 的点移动到电势为 VB 的点，电场力所做的功 W 为：
W = q(VA - VB)
It is important to realise that the result is independent of the path. This is because the electric force is a conservative force. The same work would be done along any path connecting the two points.
注意：做功的结果与路径无关，因为电场力是保守力。沿连接两点的任意路径移动电荷，电场力所做的功都相同。
If the charge is moved in the direction opposite to the electric force, an external agent must do positive work to move the charge, and the stored electric potential energy increases.
若电荷沿与电场力相反的方向移动，则需要外力做正功，此时系统的电势能增加。
12. Relationship Between Field and Potential | 场强与电势的关系
The electric field strength is related to the rate at which electric potential changes with distance. In a uniform electric field, the relationship is simple:
电场强度与电势随距离的变化率密切相关。在匀强电场中，二者的关系最为简单：
E = -ΔV / Δd
Here ΔV is the potential difference between two points separated by a displacement Δd along the field direction. The negative sign indicates that the electric field points in the direction of decreasing electric potential.
式中 ΔV 为沿电场方向相距 Δd 的两点之间的电势差。负号表示电场方向指向电势降低的方向。
In general, for a non-uniform field, the field strength is the negative of the potential gradient. A steep potential gradient implies a strong electric field, while a gentle gradient implies a weak field.
对于非匀强电场，场强等于电势梯度（电势对距离的变化率）的负值。电势梯度越大，电场越强；电势梯度越小，电场越弱。
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