📚 Comparing Gravitational and Electric Fields | 引力场与电场比较
Gravitational fields and electric fields are taught side by side at A-level because they share an almost identical mathematical skeleton: both obey an inverse-square force law, both are described by field strength and potential, and both can be pictured with field lines and potential-distance graphs. What differs is the source - mass in one case, charge in the other. Because mass comes in only one sign while charge comes in two, the two fields behave very differently in direction, sign, shielding and the motion of the objects placed inside them. This TutorHao revision guide compares the two fields point by point with CIE-style definitions and exam-focused comparisons.
在 A-level 课程中，引力场与电场通常一起学习，因为二者拥有几乎相同的数学骨架：都遵循平方反比力律，都用场强和势来描述，也都可以用场线和"势—距离"图像表示。区别在于场源：一个是质量，一个是电荷。由于质量只有一种"符号"，而电荷有两种，两种场在方向、符号、屏蔽以及场内物体运动方面表现出重要差异。本 TutorHao 复习指南按 CIE 的定义方式，对两种场进行逐点比较，并给出面向考试的对比要点。
1. Field Concept and Sources | 场的概念与场源
A field is a region of space in which a suitably chosen test object experiences a non-contact force. A gravitational field surrounds any object with mass; an electric field surrounds any object with charge. Both fields act through a vacuum, both weaken with distance, and both can be represented by field lines whose spacing indicates strength. In CIE terminology the test object is a small point mass for a gravitational field and a small positive test charge for an electric field.
场是空间中的一个区域，位于其中的合适试探物体会受到非接触力。任何有质量的物体周围都存在引力场；任何带电的物体周围都存在电场。两种场都能在真空中作用，都随距离增大而减弱，都可以用场线表示，场线的疏密表示场的强弱。按 CIE 的表述，引力场的试探物体是一个小质点，电场的试探物体是一个小的正试探电荷。
The single most important structural difference is the number of signs available. Mass has only one kind, so gravitational forces are always attractive and gravitational field lines always point inwards towards the source. Charge has two kinds - positive and negative - so electric forces may attract or repel, and electric field lines point outwards from positive charge and inwards towards negative charge. Almost every other difference in this article follows from that one fact.
最根本的结构性差异在于"符号"的数量。质量只有一种，因此引力永远是吸引力，引力场线总是指向场源内部。电荷有两种，正电荷与负电荷，因此电场力可以是引力也可以是斥力，电场线从正电荷发出、指向负电荷。本文后面的几乎所有差异，都可以由这一条推出。
2. Inverse-Square Force Laws | 平方反比力律
For two point masses separated by a distance r, Newton's law of gravitation gives the force magnitude. For two point charges in a vacuum, Coulomb's law gives the electrostatic force magnitude. Written together, the parallel structure is obvious.
对于相距 r 的两个质点，牛顿万有引力定律给出引力大小；对于真空中的两个点电荷，库仑定律给出静电力大小。把两式并列写出，其平行结构一目了然。
= G m₁ m₂ / r² and F = Q q / (4π ε₀ r²) = k Q q / r²
are central, radial, conservative and infinite in range. Neither law contains a velocity term, and both are strictly valid only for point sources or spherically symmetric bodies, for which the source behaves as if all its mass or charge were concentrated at the centre.
其中 G = 6.67 × 10⁻¹¹ N m² kg⁻²，k = 1 / (4π ε₀) = 8.99 × 10⁹ N m² C⁻²。两个力都是有心力、沿径向、属于保守力，作用范围都是无限的。两个定律都不含速度项，且严格来说只适用于点场源或球对称场源；对球对称物体，可视为全部质量或电荷集中在球心。
ormous physical difference. The gravitational constant is tiny, so gravity is the weakest known fundamental interaction; the electrostatic constant is enormous, so electric forces between charged particles dominate completely. For a proton and an electron, the ratio Fₑ / Fg is why the electric force governs atomic structure while gravity governs planetary orbits.
这两个常数的数值揭示了巨大的物理差异。万有引力常量极小，因此引力是已知最基本的相互作用中最弱的一种；静电力常量极大，因此带电粒子之间的电场力完全占主导。对质子与电子而言，Fₑ / Fg 约为 2.3 × 10³⁹，这正是电场力支配原子结构、而引力支配行星轨道的原因。
3. Field Strength: g and E | 场强：g 与 E
ric field strength E at a point is the electric force per unit positive charge on a small test charge placed there: E = F / Q, measured in N C⁻¹. Both are vectors, and for point sources both are given by the same inverse-square form.
某点的引力场强 g 是放在该点的小试探质量所受引力与质量的比值：g = F / m，单位为 N kg⁻¹。某点的电场强度 E 是放在该点的小正试探电荷所受电场力与电荷量的比值：E = F / Q，单位为 N C⁻¹。两者都是矢量，对点场源而言，两者都写成相同的平方反比形式。
g = G M / r² and E = Q / (4π ε₀ r²) = k Q / r²
Note carefully that g is defined with a test mass, which is always attracted, so the direction of g is uniquely defined as towards the source. E is defined with a positive test charge, so its direction is the direction of the force on a positive charge: away from a positive source, towards a negative source. The units N kg⁻¹ and N C⁻¹ are equivalent to m s⁻² and V m⁻¹ respectively, and examiners expect you to move fluently between these forms.
要注意，g 用试探质量定义，而质量总被吸引，所以 g 的方向唯一地指向场源。E 用正试探电荷定义，因此它的方向就是正电荷受力的方向：从正场源向外，指向负场源。单位 N kg⁻¹ 与 m s⁻² 等价，N C⁻¹ 与 V m⁻¹ 等价，考试要求你能在这些形式之间自如转换。
4. Direction, Sign and Superposition | 方向、符号与叠加
Gravitational field strength is always directed towards the mass creating it, and because there is no negative mass, there is never a repulsive gravitational field to cancel it. Electric field strength, by contrast, can be treated algebraically with a sign convention along an axis: a field pointing in the positive x-direction is positive, and one pointing the other way is negative. This is why electric potential and field-strength graphs for a pair of charges can cross zero, while gravitational graphs usually cannot.
引力场强总是指向产生它的质量，而且由于不存在负质量，永远不存在能把引力抵消的"斥力场"。电场强度则可以在一条轴上用正负号作代数处理：沿正 x 方向的场为正，反向为负。这正是"一对电荷"的电场强度和电势图像可以穿过零点、而引力图像通常不能的原因。
Both fields obey the principle of superposition: the resultant field strength at a point is the vector sum of the field strengths produced by each source separately. In one dimension this reduces to an algebraic sum with signs. Between two masses the fields oppose and give a single neutral point closer to the smaller mass; between a positive and a negative charge the fields reinforce, while between two like charges there is a neutral point between them.
两种场都服从叠加原理：某点的合场强等于各场源单独产生的场强的矢量和。在一维情形下，这退化为带符号的代数和。两个质量之间的场方向相反，存在一个零点，且零点离质量较小的一方更近；一正一负电荷之间两场互相加强，而两个同种电荷之间存在一个零点。
5. Field Lines and Uniform Fields | 场线与匀强场
Field lines are a visual tool with strict rules: their direction gives the direction of the field, their spacing indicates strength (closer lines mean a stronger field), they never cross, and they are always perpendicular to the surface of a conductor in electrostatic equilibrium. Radial fields are drawn with evenly spaced angular lines whose separation increases with distance; uniform fields are drawn as parallel, equally spaced straight lines.
场线是一种有严格规则的图示工具：其方向表示场的方向，其疏密表示场的强弱（越密越强），场线永不相交，且在静电平衡时总与导体表面垂直。径向场画成角向均匀分布、随距离增大而变疏的直线；匀强场则画成平行且等间距的直线。
A uniform gravitational field exists locally near a planet' ssentially constant over laboratory distances. A uniform electric field is produced between two parallel charged plates, and its strength is given by the potential difference divided by the plate separation. In both cases the force on a test object is constant, so the motion is described by the equations of uniform acceleration.
在行星表面附近局部存在匀强引力场，其中 g ≈ 9.81 N kg⁻¹，在实验室尺度内基本恒定。匀强电场产生于两块平行带电板之间，其场强等于两板间的电势差除以板间距。两种情形中试探物体所受的力都是恒力，因此运动可用匀加速运动公式描述。
E = V / d
6. Potential and Potential Energy | 势与势能
Gravitational potential φ at a point is the work done per unit mass in bringing a small test mass from infinity to that point. Because the gravitational force is always attractive, the field does positive work as a mass falls in from infinity, so φ is always negative, reaching zero only at infinity. Around a point mass of mass M, and for a mass m placed there, the quantities are as follows.
某点的引力势 φ 是把单位质量的小试探质量从无穷远处移到该点时外界所做的功。由于引力总是吸引力，质量从无穷远处移入时场做正功，因此 φ 恒为负值，只有在无穷远处才为零。对质量为 M 的点场源，以及放在该处的质量 m，有如下关系。
φ = - G M / r and Eₚ = m φ = - G M m / r
Electric potential V at a point is the work done per unit positive charge in bringing a small positive test charge from infinity to that point. Because the electric force can attract or repel, V can be positive (around a positive charge) or negative (around a negative charge), and the potential energy of a charge q is qV, which may be positive or negative depending on the signs of q and the source charge.
某点的电势 V 是把单位正电荷的小试探电荷从无穷远处移到该点时外界所做的功。由于电场力可以是引力也可以是斥力，V 可正（正电荷周围）也可负（负电荷周围），电荷 q 的电势能为 qV，其正负取决于 q 与场源电荷的符号组合。
V = Q / (4π ε₀ r) = k Q / r and Eₚ = q V
A key shared idea is that both potentials are scalar and both are defined relative to infinity. Because they are scalars, total potentials add algebraically without vector components, which makes potential much easier to use than field strength in multi-source problems.
两者共有的一点是：势都是标量，且都以无穷远处为零势参考点。由于是标量，总势可以直接代数相加，不需要分解矢量分量，因此在多场源问题中用势比用场强方便得多。
7. Potential Gradient and Field Strength | 势梯度与场强
Field strength and potential are linked in exactly the same way for both fields: the field strength at a point equals the negative of the potential gradient at that point, i.e. the negative of the slope of the potential-distance graph. The minus sign simply states that the field points in the direction of decreasing potential, which for gravity means towards the mass and for a positive charge means away from it.
两种场中，场强与势的联系方式完全相同：某点的场强等于该点处势梯度的负值，也就是"势—距离"图像斜率的负值。负号只表示场指向电势降低的方向：对引力场即指向质量，对正电荷即背离正电荷。
g = - Δφ / Δr and E = - ΔV / Δr
the potential difference divided by the distance, which recovers g = ΔV / Δh near a planet' s r⁻². Plotting φ or V against r and differentiating graphically is a standard exam skill.
在匀强场中梯度恒定，故场强恒定，等于势差除以距离，由此得到行星表面附近的 g = ΔV / Δh 以及平行板间的 E = V / d。在径向场中，r 越小梯度越陡，故场强按 r⁻² 增大。画出 φ 或 V 随 r 变化的图像并作图解微分，是标准的考试技能。
8. Shielding and Screening | 屏蔽与静电屏蔽
The most striking practical difference concerns shielding. Electric fields can be screened. In a conductor, free electrons redistribute until the field inside is zero, so a hollow conductor forms a Faraday cage that protects its interior from external electric fields. This is why sensitive electronics are housed in metal cases, why coaxial cable has a braided screen, and why a car is a relatively safe place during a lightning storm.
两者在实用性上最显著的差异是屏蔽。电场可以被屏蔽。在导体中自由电子重新分布，直到导体内部场强为零，于是空心导体形成法拉第笼，使内部不受外电场影响。这就是精密电子设备装在金属外壳内、同轴电缆带编织屏蔽层、雷暴时汽车相对安全的原因。
There is no gravitational analogue. Because there is no negative mass, no arrangement of matter can cancel a gravitational field in a region of space; the only way to reduce g is to move far from all masses. A gravitational shield does not exist even in principle, which is why gravity is the force that ultimately shapes the large-scale structure of the universe.
引力没有对应的机制。由于不存在负质量，任何物质的排布都无法在空间中把引力场抵消；要减小 g，唯一的方法是远离所有质量。引力屏蔽层即使在原理上也不存在，这正是引力最终塑造宇宙大尺度结构的原因。
9. Motion of Particles: Orbits and Projectiles | 粒子运动：轨道与抛射
For an object in a circular orbit of radius r around a much more massive body, gravity provides the centripetal force, giving orbital speed and the period relation below. The speed is independent of the orbiting mass, and Kepler's third law T² ∝ r³ follows directly. Gravitational motion is therefore always an attractive, bound or unbound orbit problem.
对以半径 r 绕远比它重的天体做圆轨道运动的物体，引力提供向心力，于是有下面的轨道速度和周期关系。轨道速度与环绕物体的质量无关，开普勒第三定律 T² ∝ r³ 由此直接得出。因此引力运动总是"吸引型"的束缚或非束缚轨道问题。
v = √G M / r and T² = 4π² r³ / (G M)
celeration a = qE / m, since gravity is usually negligible for electrons and ions. In a radial electric field around a point charge, a particle of opposite sign performs an orbit mathematically identical to a planetary orbit, whereas a particle of the same sign is deflected away and cannot be bound at all.
匀强电场中的带电粒子表现得像抛体：若垂直进入场区，它沿着抛物线运动，加速度 a = qE / m，因为对电子和离子而言引力通常可忽略。在点电荷的径向电场中，异号粒子的轨道在数学上与行星轨道完全相同；而同号粒子被推开，根本无法被束缚。
10. Escape Velocity, Ionisation and Energy Scales | 逃逸速度、电离与能量尺度
Escape speed from a spherical mass is found by setting the total mechanical energy to zero, and the same energy argument determines the energy needed to remove a charge to infinity. Setting the sum of kinetic and potential energy to zero gives the shared algebraic form, which is why the two topics are examined together.
从球形天体逃逸的速度，可由总机械能等于零求得；同样的能量论证也决定了把电荷移到无穷远所需的能量。令动能与势能之和为零，可得两者共有的代数形式，这也是这两个知识点常被放在一起考查的原因。
v(escape) = √2 G M / r and E(ionisation) = k Q q / r
The numerical scale, however, is utterly different. Earth' tomic ionisation is quoted in electronvolts (13.6 eV) rather than joules. Typical A-level data: e = 1.60 × 10⁻¹⁹ C, mₑ = 9.11 × 10⁻³¹ kg, mₚ = 1.67 × 10⁻²⁷ kg, ε₀ = 8.85 × 10⁻¹² F m⁻¹.
然而两者的数值尺度截然不同。地球的逃逸速度约为 11.2 km s⁻¹，而把氢原子从基态电离所需的能量仅约 2.18 × 10⁻¹⁸ J，因此原子电离能通常用电子伏特（13.6 eV）而不用焦耳表示。A-level 常见数据：e = 1.60 × 10⁻¹⁹ C，mₑ = 9.11 × 10⁻³¹ kg，mₚ = 1.67 × 10⁻²⁷ kg，ε₀ = 8.85 × 10⁻¹² F m⁻¹。
11. Full Comparison Table | 完整对比表
The table below summarises the points made so far and is worth memorising as a revision checklist before any fields question.
下表汇总了前文的要点，建议作为应对任何场学题目前的复习清单加以记忆。
| Feature | 项目 | Gravitational field | 引力场 | Electric field | 电场 |
| Source | 场源 | Mass | 质量 | Charge | 电荷 |
| Force law | 力律 | F = G M m / r² | F = Q q / (4π ε₀ r²) |
| Direction | 方向 | Always attractive | 总是吸引 | Attractive or repulsive | 可吸引可排斥 |
| Field strength | 场强 | g
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