Electric Fields | 电场

📚 Electric Fields | 电场

Electric fields are one of the core ideas in A-Level Physics. They describe how a charged object can exert a force on another charge without direct contact. In CIE examinations, you need to understand field strength, potential, energy changes, and the motion of charged particles in uniform fields.

电场是 A-Level 物理的核心概念之一。它描述带电物体如何在不直接接触的情况下对另一个电荷施加作用力。在 CIE 考试中，你需要理解电场强度、电势、能量变化以及带电粒子在匀强电场中的运动。

1. The Concept of an Electric Field | 电场的基本概念

An electric field is a region of space around a charged particle or object in which another charge experiences an electric force. The field exists even if there is no test charge present to feel the force. Field lines are used to represent the direction and strength of the field, where the tangent to a field line gives the direction of the force on a small positive test charge.

电场是带电粒子或物体周围空间中使另一个电荷受到电场力的区域。即使没有检验电荷存在，电场依然存在。电场线用来表示电场的方向和强弱，电场线上某点的切线方向就是放置在该点的正检验电荷所受电场力的方向。

The strength of an electric field can be uniform or non-uniform. A uniform field has the same magnitude and direction at every point, while a non-uniform field changes from point to point. The field around a point charge is radial and therefore non-uniform, whereas the field between two oppositely charged parallel plates is approximately uniform in the central region.

电场的强弱可以是均匀的，也可以是不均匀的。匀强电场中各点的大小和方向都相同，而非匀强电场中各点不同。点电荷周围的电场是径向的，因此是非匀强电场；而两块带异种电荷的平行板之间的电场在中央区域近似为匀强电场。

2. Electric Field Strength | 电场强度

Electric field strength is defined as the force per unit positive charge acting on a small test charge placed at a point in the field. The defining equation is:

电场强度定义为放置在电场中某点的小正检验电荷所受的电场力与其电荷量之比。定义式为：

E = F / q

In this equation, E is the electric field strength in newtons per coulomb (N C⁻¹), F is the electric force in newtons (N), and q is the charge in coulombs (C). Since force is a vector, electric field strength is also a vector, and it points in the direction of the force on a positive charge.

在这个公式中，E 是电场强度，单位是牛每库仑（N C⁻¹）；F 是电场力，单位是牛（N）；q 是电荷量，单位是库仑（C）。由于力是矢量，电场强度也是矢量，其方向与正电荷所受电场力的方向相同。

If a charge q is placed in a known electric field E, the force on that charge is given by F = qE. This applies to any charge, positive or negative. A positive charge experiences a force in the direction of E, while a negative charge experiences a force in the opposite direction.

如果把电荷 q 放入已知电场 E 中，则该电荷所受电场力为 F = qE。这适用于正电荷和负电荷。正电荷所受电场力方向与 E 的方向相同，而负电荷所受电场力方向与 E 的方向相反。

3. Electric Field due to a Point Charge | 点电荷产生的电场

The electric field around an isolated point charge Q is radial. Its magnitude depends on the distance r from the charge and is given by Coulomb’s law in field form:

孤立点电荷 Q 周围的电场是径向的。其大小与距电荷的距离 r 有关，由库仑定律的场强形式给出：

E = Q / (4πε₀r²)

Here, ε₀ is the permittivity of free space, with a value of 8.85 × 10⁻¹² F m⁻¹. This equation shows that the electric field strength from a point charge follows an inverse square law: if the distance is doubled, the field strength decreases by a factor of four.

其中 ε₀ 是真空介电常数，数值为 8.85 × 10⁻¹² F m⁻¹。该公式表明，点电荷产生的电场强度遵循平方反比定律：如果距离加倍，电场强度会减小到原来的四分之一。

For several point charges, the resultant electric field at any point is the vector sum of the fields due to each charge individually. This is the principle of superposition. It means you must add the individual field vectors, taking into account both their magnitudes and directions.

对于多个点电荷，任意点的合电场强度等于各个电荷单独产生的电场强度的矢量和。这就是叠加原理。这意味着必须将各个电场矢量相加，同时考虑它们的大小和方向。

4. Uniform Electric Fields and Parallel Plates | 匀强电场与平行板

A uniform electric field can be produced between two large, oppositely charged parallel plates. In the central region, far from the edges, the field is uniform: the field lines are parallel, equally spaced, and point from the positive plate to the negative plate.

在两块带异种电荷的大平行板之间可以产生匀强电场。在远离边缘的中央区域，电场是均匀的：电场线平行、等间距，方向从正极板指向负极板。

For a potential difference V between two plates separated by a distance d, the electric field strength between the plates is given by:

当两板之间的电势差为 V、板间距离为 d 时，板间电场强度为：

E = V / d

The unit of E can therefore also be written as volts per metre (V m⁻¹), which is equivalent to N C⁻¹. This equation applies only when the field is uniform. It shows that a smaller plate separation produces a stronger field for the same potential difference.

因此 E 的单位也可以写成伏每米（V m⁻¹），它与 N C⁻¹ 是等价的。该公式只适用于匀强电场。它表明，在相同电势差下，板间距离越小，电场越强。

In a uniform field, a charged particle experiences a constant force F = qE. This is very useful because constant force produces constant acceleration, allowing standard kinematic equations to be used when analysing motion.

在匀强电场中，带电粒子受到恒定的电场力 F = qE。这一点非常有用，因为恒力产生恒定加速度，因此分析运动时可以使用标准运动学方程。

5. Electric Potential Energy | 电势能

Electric potential energy is the work done in bringing a charge from infinity to a point in an electric field without acceleration. For a charge q at a point where the electric potential is V, the electric potential energy U is given by:

电势能是将电荷从无穷远处不加速地移到电场中某一点所做的功。对于处于电势为 V 的点的电荷 q，其电势能 U 为：

U = qV

Potential energy is a scalar quantity, even though it is derived from force and displacement. Its unit is the joule (J). When a positive charge moves in the direction of the electric field, its electric potential energy decreases because the field does positive work on the charge.

电势能是标量，尽管它由力和位移推导而来。其单位是焦耳（J）。当正电荷沿电场方向运动时，其电势能减小，因为电场对电荷做正功。

Conversely, moving a positive charge against the electric field requires external work and increases its potential energy. This analogy is often made with lifting a mass in a gravitational field, where gravitational potential energy increases as height increases.

相反，将正电荷逆着电场方向移动需要外力做功，其电势能增加。这一类比常与在重力场中提升重物相比较：高度增加时，重力势能增加。

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 test charge from infinity to that point. It is a scalar quantity measured in volts (V), where 1 V = 1 J C⁻¹.

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。它是标量，单位为伏特（V），其中 1 V = 1 J C⁻¹。

For a point charge Q, the electric potential at a distance r from the charge is:

对于点电荷 Q，在距离 r 处的电势为：

V = Q / (4πε₀r)

Unlike the electric field due to a point charge, which falls off as 1/r², the potential falls off as 1/r. Potential can be positive or negative depending on the sign of Q. For a positive charge, the potential is positive everywhere, and for a negative charge it is negative everywhere, taking infinity as the zero reference.

与点电荷电场按 1/r² 衰减不同，电势按 1/r 衰减。电势可正可负，取决于 Q 的符号。若取无穷远处为零电势参考点，正电荷的电势处处为正，负电荷的电势处处为负。

Because electric potential is a scalar, the total potential at a point due to several charges is simply the algebraic sum of the individual potentials. This makes potential calculations easier than electric field calculations, which require vector addition.

由于电势是标量，多个电荷在某点产生的总电势就是各个电势的代数和。这使得电势的计算比需要矢量合成的电场计算更简单。

7. Potential Difference and Work Done | 电势差与做功

The potential difference between two points A and B is the work done per unit positive charge when moving a charge from A to B. It is written as ΔV = V_B − V_A. The work done W on a charge q moving through a potential difference ΔV is:

两点 A 和 B 之间的电势差是将单位正电荷从 A 移到 B 所做的功。它写作 ΔV = V_B − V_A。电荷 q 通过电势差 ΔV 时所做的功 W 为：

W = qΔV

This equation links the idea of potential difference directly to energy transfer. If a charge moves through a potential difference of 1 V, the work done on or by the charge is 1 joule per coulomb of charge. For an electron moving through 1 V, this defines a very useful energy unit called the electronvolt.

该公式将电势差与能量传递直接联系起来。如果电荷通过 1 V 的电势差，则每库仑电荷所做或所获得的功为 1 焦耳。电子通过 1 V 电势差时所获得的能量定义了一个非常有用的能量单位——电子伏特。

One electronvolt is equal to the energy gained by an electron when it is accelerated through a potential difference of 1 volt:

1 电子伏特等于一个电子在 1 伏特电势差中被加速时所获得的能量：

1 eV = 1.60 × 10⁻¹⁹ J

This unit is convenient for atomic and nuclear physics because energies at that scale are very small in joules. In A-Level problems, always convert eV to joules if you need to use mass and speed in SI units.

这个单位在原子物理和核物理中非常方便，因为该尺度下的能量用焦耳表示非常小。在 A-Level 题目中，如果需要在国际单位制下使用质量和速度，请务必先将 eV 转换为焦耳。

8. Motion of Charged Particles along the Field | 带电粒子沿电场方向的运动

When a charged particle is released from rest in a uniform electric field, it experiences a constant force F = qE. By Newton’s second law, its acceleration is constant and is given by:

当带电粒子在匀强电场中从静止释放时，它受到恒定的电场力 F = qE。根据牛顿第二定律，其加速度恒定，表示为：

a = F / m = qE / m

Because acceleration is constant, the standard kinematic equations can be used to find velocity, displacement, and time. For example, after travelling a distance s from rest in a uniform field, the speed v of the particle is found from v² = 2as.

由于加速度恒定，可以用标准运动学方程来求速度、位移和时间。例如，在匀强电场中从静止运动一段距离 s 后，粒子的速度 v 可由 v² = 2as 求得。

An alternative energy method is to use the work done by the electric field. The kinetic energy gained equals the loss of electric potential energy, so ½mv² = qΔV. This is often quicker than using kinematics when the potential difference is known.

另一种能量方法是利用电场做功。获得的动能等于电势能的减少量，即 ½mv² = qΔV。当已知电势差时，这通常比用运动学方法更快。

9. Motion of Charged Particles Perpendicular to the Field | 带电粒子垂直进入电场的运动

If a charged particle enters a uniform electric field perpendicular to the field lines, its motion is parabolic, similar to projectile motion in a gravitational field. The velocity component parallel to the plates remains constant, while the component perpendicular to the plates accelerates uniformly.

如果带电粒子垂直于电场线进入匀强电场，其运动轨迹为抛物线，类似于重力场中的抛体运动。平行于极板的速度分量保持不变，而垂直于极板的速度分量做匀加速运动。

For a particle of charge q and mass m entering midway between two plates with initial horizontal speed v, the vertical acceleration is a = qE/m = qV/(md). The horizontal displacement is x = vt, so the time spent in the field is t = x/v. The vertical deflection y is then:

对于电荷量为 q、质量为 m 的粒子以初始水平速度 v 从两板中间进入电场，其竖直加速度为 a = qE/m = qV/(md)。水平位移为 x = vt，因此在电场中运动的时间为 t = x/v。竖直偏转量 y 为：

y = ½at² = (qE / 2m)(x / v)²

This equation shows that the deflection is proportional to the charge and the field strength, and inversely proportional to the mass and the square of the initial speed. The path is a parabola because the vertical displacement is proportional to the square of horizontal displacement.

该公式表明，偏转量与电荷量和电场强度成正比，与质量和初速度的平方成反比。由于竖直位移与水平位移的平方成正比，其路径为抛物线。

10. Field Lines and Equipotential Surfaces | 电场线与等势面

Electric field lines are a visual tool for representing electric fields. They start on positive charges and end on negative charges. The density of field lines indicates field strength: closer lines mean a stronger field. Field lines never cross because the electric field has a unique direction at any point.

电场线是表示电场的一种可视化工具。它们从正电荷出发，终止于负电荷。电场线的疏密表示电场强度：线越密，电场越强。电场线永不相交，因为电场在任意点都有唯一的方向。

Equipotential surfaces are surfaces on which the electric potential is constant. No work is required to move a charge along an equipotential surface because there is no potential difference along it. Equipotential surfaces are always perpendicular to electric field lines.

等势面是电势保持恒定的曲面。沿着等势面移动电荷不需要做功，因为沿等势面没有电势差。等势面总是与电场线垂直。

For a uniform field, equipotential surfaces are planes perpendicular to the field lines and are equally spaced. For a point charge, equipotential surfaces are concentric spheres centred on the charge. Drawing both field lines and equipotentials helps to visualise how potential changes in space.

在匀强电场中，等势面是垂直于电场线的平面，且等间距分布。对于点电荷，等势面是以电荷为中心的同心球面。同时画出电场线和等势面有助于直观理解电势在空间中的变化。

11. Conductors in Electrostatic Fields | 静电场中的导体

In electrostatic equilibrium, the electric field inside a conductor is zero. If there were an electric field inside the conductor, free electrons would move under its influence, creating a current. Since electrostatic equilibrium means no net motion of charge, the internal field must be zero.

在静电平衡状态下，导体内部的电场为零。如果导体内部存在电场，自由电子就会在电场作用下运动，形成电流。由于静电平衡意味着电荷没有净运动，因此内部电场必须为零。

Any excess charge on a conductor resides entirely on its outer surface. The electric field just outside a charged conductor is perpendicular to the surface, and its magnitude depends on the local surface charge density. Sharp points have higher charge density and therefore stronger external fields.

导体上的任何多余电荷都分布在其外表面。带电导体外表面附近的电场方向垂直于表面，其大小取决于局部表面电荷密度。尖端处电荷密度更大，因此外部电场更强。

This property explains electrostatic shielding: a hollow conductor can shield its interior from external electric fields because the charges redistribute on the conductor surface to cancel the field inside. This principle is used in devices such as coaxial cables and Faraday cages.

这一性质解释了静电屏蔽现象：空心导体可以屏蔽其内部免受外部电场影响，因为电荷在导体表面重新分布以抵消内部电场。同轴电缆和法拉第笼等装置就应用了这一原理。

12. Key Equations and Exam Tips | 关键公式与考试技巧

The most important equations for CIE A-Level electric fields are summarised below. You should be able to select the right equation based on whether the field is uniform or radial, and whether you are dealing with force, energy, or potential.

下面总结了 CIE A-Level 电场中最重要的公式。你应当能够根据电场是匀强电场还是径向电场，以及所处理的是力、能量还是电势，选择正确的公式。

Electric field strength: E = F / q
Uniform field between plates: E = V / d
Point charge field: E = Q / (4πε₀r²)
Force on a charge: F = qE
Potential of a point charge: V = Q / (4πε₀r)
Work done: W = qΔV
Kinetic energy change: ½mv² = qΔV

电场强度： E = F / q
平行板间匀强电场： E = V / d
点电荷电场： E = Q / (4πε₀r²)
电荷所受电场力： F = qE
点电荷电势： V = Q / (4πε₀r)
电场做功： W = qΔV
动能变化： ½mv² = qΔV

When solving problems, always identify the sign of the charge and the direction of the field first. Remember that electric potential is a scalar, so signs add algebraically. Use energy methods for speed calculations when possible, and use kinematics only for uniform fields. Finally, check whether your answer is physically reasonable in terms of direction, magnitude, and units.

解题时，首先要确定电荷的符号和电场的方向。记住电势是标量，因此符号按代数相加。在可能的情况下优先使用能量方法求速度，只有对匀强电场才使用运动学。最后检查答案在方向、大小和单位方面是否物理合理。

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Electric Fields | 电场