📚 A-Level WJEC Physics: Gravitational Fields Key Points Revision | A-Level WJEC 物理：万有引力 考点精讲
Gravitational fields are a central topic in the WJEC A-Level Physics specification, linking concepts of force, energy, and motion on both planetary and cosmic scales. Understanding the nature of these fields, the associated field strength and potential, and how they govern orbital mechanics is essential for mastering the unit. This revision guide distills the key points into clear explanations, essential equations, and practical examples tailored to the WJEC exam requirements.
引力场是WJEC A-Level物理大纲中的核心课题，它把力、能量和运动的规律从地球表面拓展到行星乃至宇宙尺度。理解引力场的性质、相关的场强与势，以及它们如何支配轨道力学，是学好本单元的关键。这篇考点精讲将核心知识提炼为清晰的解释、必备方程和紧扣WJEC考试要求的典型实例。
1. What is a Gravitational Field? | 什么是引力场？
A gravitational field is a region of space around a mass in which any other mass experiences an attractive force. The field is a vector field, meaning it has both magnitude and direction at every point. The direction of the field is the direction of the force that would act on a small test mass placed at that point. Field lines are used to represent gravitational fields: they point towards the mass creating the field and never cross each other. The density of field lines indicates the field strength.
引力场是质量周围空间中存在的区域，任何其他质量进入该区域都会受到引力作用。引力场是矢量场，意味着空间每一点既有大小也有方向。场的方向就是把一个很小的检验质量放在该点时所受力的方向。我们常用引力线来表示引力场：它们指向产生场的质量，永不相交。引力线的疏密反映了场强的大小。
2. Newton’s Law of Universal Gravitation | 牛顿万有引力定律
Every point mass attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. In equation form, this law is written as:
任何两个质点之间都存在引力，该力的大小与两质点的质量乘积成正比，与它们中心之间距离的平方成反比。方程表示为：
F = − G M m / r²
Here, F is the gravitational force acting along the line joining the centres, G is the universal gravitational constant (6.67 × 10⁻¹¹ N m² kg⁻²), M and m are the masses, and r is the separation distance. The negative sign indicates that the force is always attractive—it pulls the masses together. For examinations, you must be able to apply this law to both point masses and spherical objects (where the mass may be treated as if it is concentrated at the centre, provided you are outside the sphere).
式中，F 是沿两中心连线方向的引力，G 是万有引力常数（6.67 × 10⁻¹¹ N m² kg⁻²），M 和 m 是两个质量，r 是两者间的距离。负号表示力总是吸引的，即把质量拉向彼此。考试中，你必须能将该定律应用于点质量和球对称物体（只要在物体外部，就可将质量视为集中在球心）。
3. Gravitational Field Strength, g | 引力场强度 g
Gravitational field strength at a point is defined as the gravitational force exerted per unit mass on a small test mass placed at that point. It is a vector quantity, symbol g, with SI units of N kg⁻¹ (equivalent to m s⁻²). For a point mass or outside a uniform sphere, using Newton’s law, the magnitude of the field strength is:
引力场中某点的场强定义为放于该点的小检验质量单位质量所受到的引力。它是一个矢量，符号为 g，SI 单位是 N kg⁻¹（等价于 m s⁻²）。对于点质量或在均匀球体外，根据牛顿定律，场强的大小为：
g = G M / r²
The direction of g is towards the centre of the mass M. Because the force per unit mass points radially inward, we often write the vector form as g = −(G M / r²) r̂, where r̂ is a unit vector pointing radially outward from the source mass. This equation tells us that field strength follows an inverse-square law and is independent of the test mass.
g 的方向指向产生场的质量 M 的中心。由于单位质量所受的力沿径向向内，我们经常把矢量形式写作 g = −(G M / r²) r̂，其中 r̂ 是从源质量径向向外的单位矢量。这个公式表明场强遵循平方反比规律，且与检验质量无关。
4. Radial Gravitational Fields | 径向引力场
A radial gravitational field is the field surrounding a point mass or a spherical mass. The field lines are radially directed towards the centre of the mass, and the magnitude of g decreases as 1/r². For a planet of mass M and radius R, the field strength at its surface is g_surface = G M / R². As you move away from the planet, g decreases according to g ∝ 1/r². This radial dependence is crucial for calculations involving height above the Earth’s surface: if r = R + h, then g = G M / (R + h)².
径向引力场是包围点质量或球形质量的场。场线沿径向指向质量中心，场强大小按 1/r² 减小。对于质量为 M、半径为 R 的行星，其表面的场强为 g_surface = G M / R²。随着距离行星越来越远，g 按照 g ∝ 1/r² 减小。这种径向依赖关系对于涉及地面上空高度的计算至关重要：若 r = R + h，则 g = G M / (R + h)²。
Typical graphs of g against r show a curve that decreases steeply near the surface and levels out at larger distances. Inside a uniform solid sphere, the field strength falls linearly to zero at the centre, but WJEC usually focuses on the external field.
典型的 g 对 r 的图像显示，靠近表面时曲线急剧下降，远处变缓。在均匀实心球体内部，场强线性减小到球心处为零，但 WJEC 通常只考察外部场。
5. Uniform Gravitational Fields | 均匀引力场
Over small distances near the Earth’s surface, the curvature of the field lines is negligible and g is approximately constant, both in magnitude and direction. This gives a uniform gravitational field, where field lines are parallel and equally spaced. The value of g near the Earth’s surface is taken as 9.81 N kg⁻¹. In a uniform field, the change in gravitational potential energy when a mass m is raised through a height h is simply ΔE_p = m g Δh. The corresponding change in gravitational potential (potential per unit mass) is ΔV = g Δh. Uniform field approximations are used whenever the vertical displacement is much smaller than the Earth’s radius.
在地球表面附近很小的距离范围内，场线的弯曲可忽略不计，g 的大小和方向几乎恒定。这便形成了一个均匀引力场，场线平行且等间距。地球表面附近的 g 近似取值为 9.81 N kg⁻¹。在均匀场中，当质量为 m 的物体被举升高度 h 时，引力势能的变化量为 ΔE_p = m g Δh。相应的引力势（单位质量的势能）变化为 ΔV = g Δh。只要竖直位移远小于地球半径，就可以使用均匀场近似。
6. Gravitational Potential, V | 引力势 V
Gravitational potential at a point is defined as the work done per unit mass to bring a small test mass from infinity to that point. Since the gravitational force is attractive, external work must be done to move a mass away from the source, and the field does work when the mass moves closer. By convention, the potential at infinity is zero. The potential at a distance r from a point mass M is given by:
引力势定义为将单位检验质量从无穷远处移到该点外力所做的功。因为引力是吸引的，将质量移离源质量时外力必须做正功，而质量靠近时场力做正功。按约定，无穷远处的势为零。距离点质量 M 为 r 处的引力势为：
V = − G M / r
Gravitational potential is a scalar quantity, measured in J kg⁻¹. The negative sign indicates that the potential is less than zero at any finite distance—you would need to supply energy to remove a mass completely from the field. The potential becomes more negative as you move closer to the mass. For a spherical mass, this formula is valid for r ≥ R.
引力势是标量，单位为 J kg⁻¹。负号表示在有限距离上的势小于零——要把质量完全移出引力场，需要提供能量。越靠近质量，势值越负。对于球形质量，此式在 r ≥ R 范围内均成立。
7. Gravitational Potential Energy, U | 引力势能 U
The gravitational potential energy of a system of two point masses is the work done to assemble the system from infinite separation. For two masses M and m separated by a distance r, the potential energy is:
两个点质量组成的系统的引力势能是将它们从无限远处聚集到一起所做的功。对于相距 r 的质量 M 和 m，势能为：
U = − G M m / r
Like potential, potential energy is also negative, reflecting the bound nature of the system. The change in potential energy determines the work done by or against the field. For a satellite moving between orbits, ΔU = U_final − U_initial. If more than two masses are involved, the total potential energy is the sum of the pair-wise contributions, taking care not to double count.
与引力势类似，势能也是负值，这反映了系统的束缚特性。势能的变化决定了场力做正功还是外力做正功。对于在不同轨道之间移动的卫星，ΔU = U_末态 − U_初态。如果涉及两个以上的质量，总势能是所有两两贡献之和，注意不要重复计算。
8. Escape Velocity | 逃逸速度
Escape velocity is the minimum initial speed an object must have at the surface of a planet (or other celestial body) in order to escape the gravitational field completely, reaching infinity with zero final speed. By applying the principle of conservation of energy, the total mechanical energy at launch must be at least zero for escape. At the planet’s surface:
逃逸速度是物体在行星（或其他天体）表面必须具备的最小初速度，使其能够完全脱离引力场到达无穷远处且最终速度为零。根据能量守恒原理，发射时总机械能必须至少为零才能逃逸。在行星表面：
½ m v_esc² + ( − G M m / R ) = 0
Solving for v_esc gives:
解出 v_esc 得到：
v_esc = √( 2 G M / R )
This result shows that escape velocity depends only on the mass and radius of the planet, not on the mass of the escaping object. For Earth, using M = 5.97 × 10²⁴ kg and R = 6.37 × 10⁶ m, the escape velocity is about 11.2 km s⁻¹. This concept is frequently tested in WJEC papers, often combined with questions on energy or orbital mechanics.
该结果表明，逃逸速度只取决于行星的质量和半径，与逃逸物体的质量无关。对于地球，代入 M = 5.97 × 10²⁴ kg 和 R = 6.37 × 10⁶ m，逃逸速度约为 11.2 km s⁻¹。这个概念在 WJEC 试卷中频繁出现，常与能量或轨道运动问题结合考查。
9. Orbital Motion of Satellites | 卫星的轨道运动
For a satellite in a stable circular orbit around a planet, the gravitational force provides the necessary centripetal force. Setting F_gravity = F_centripetal gives:
对于在环绕行星的稳定圆形轨道上运行的卫星，引力提供所需的向心力。令引力等于向心力：
G M m / r² = m v² / r
This simplifies to an expression for the orbital speed:
化简后得到轨道速度：
v = √( G M / r )
The orbital period T is the time taken to complete one revolution: T = 2π r / v. Substituting for v yields Kepler’s third law for circular orbits:
轨道周期 T 是完成一整圈所需的时间：T = 2π r / v。将 v 代入即得到圆轨道的开普勒第三定律：
T² = (4π² / G M) r³
Higher orbits correspond to lower orbital speeds and longer periods—an inverse relationship for speed and a direct proportionality between T² and r³. These equations assume that the satellite’s mass is negligible compared to the central mass. They are essential for analysing planetary motion, moons, and artificial satellites.
轨道越高，轨道速度越低，周期越长——速度与半径存在反比关系，而 T² 与 r³ 成正比。这些方程都假定卫星质量与中心天体质量相比可忽略不计。它们对于分析行星运动、天然卫星和人造卫星至关重要。
10. Kepler’s Laws of Planetary Motion | 开普勒行星运动定律
Johannes Kepler formulated three empirical laws that describe planetary motion, all of which can be derived from Newton’s law of gravitation:
约翰内斯·开普勒总结出描述行星运动的三条经验定律，这些定律均可从牛顿万有引力定律推导得出：
First law: Planets move in elliptical orbits with the Sun at one focus. For most WJEC problems, orbits are treated as circular, which is a special case of an ellipse with eccentricity zero.
第一定律：行星沿椭圆轨道运行，太阳位于椭圆的一个焦点。在 WJEC 考试的大多数问题中，轨道被简化为圆形，这是偏心率为零的特殊椭圆轨道。
Second law: A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This implies that a planet moves faster when closer to the Sun and slower when farther away.
第二定律：连接行星与太阳的线段在相等时间内扫过相等的面积。这意味着行星靠近太阳时运动较快，远离时运动较慢。
Third law: The square of the orbital period of a planet is directly proportional to the cube of the semi‑major axis of its orbit: T² ∝ a³. For circular orbits, a = r, and the constant of proportionality is 4π² / G M, where M is the mass of the central star.
第三定律：行星轨道周期的平方与其轨道半长轴的立方成正比：T² ∝ a³。对于圆形轨道 a = r，比例常数为 4π² / G M，其中 M 是中心恒星的质量。
11. Geostationary Orbits | 地球同步轨道
A geostationary satellite orbits the Earth above the equator with a period of exactly 24 hours (equal to the Earth’s rotational period) and in the same direction as Earth’s rotation. As a result, it appears stationary to an observer on the ground. Using T = 24 h = 86400 s and the relation T² = (4π² / G M) r³, the orbital radius can be calculated. For Earth, this radius is approximately 4.23 × 10⁷ m, which corresponds to an altitude of about 3.58 × 10⁷ m above the surface. Geostationary orbits are used for communication, weather, and broadcasting satellites because ground-based antennas can remain fixed in one direction.
地球同步卫星在赤道上空环绕地球运行，周期恰好为 24 小时（等于地球自转周期），且运行方向与地球自转方向相同。因此，它相对于地面观察者看起来静止不动。利用 T = 24 h = 86400 s 和 T² = (4π² / G M) r³ 可以计算出轨道半径。对于地球，该半径约为 4.23 × 10⁷ m，对应地面之上约 3.58 × 10⁷ m 的高度。地球同步轨道用于通信、气象和广播卫星，因为地面天线可以保持指向固定方向。
WJEC questions may ask you to derive the radius, show that the height is independent of the satellite’s mass, or explain why the orbit must be equatorial for a truly geostationary satellite.
WJEC 考题可能会要求你推导轨道半径，证明高度与卫星质量无关，或解释为什么真正的同步轨道必须在赤道平面内。
12. Relationship Between Field Strength and Potential | 场强与势的关系
In a general gravitational field, the field strength g is the negative gradient of the gravitational potential V. For a one-dimensional radial field, this is expressed as:
在一般的引力场中，场强 g 是引力势 V 的负梯度。对于一维径向场，这一关系可表达为：
g = − dV / dr
Applying this to the potential function V = −G M / r, we find dV/dr = G M / r², and therefore g = −G M / r², consistent with the inverse-square law and the inward direction. In a uniform field, where V varies linearly with displacement, g is constant and the relationship reduces to g = −ΔV / Δh. Understanding this link helps to interpret potential–distance graphs and solve problems where you are given a V(r) graph and asked to find g at a point by measuring the slope.
将此关系应用于势函数 V = −G M / r，可得 dV/dr = G M / r²，因此 g = −G M / r²，这与平方反比定律及向内方向一致。在均匀场中，V 随位移线性变化，g 为常数，关系简化为 g = −ΔV / Δh。理解这一联系有助于解释势–距离图像，并在给出 V(r) 图像后通过测量斜率求取某点的 g 值。
This gradient relationship is a unifying concept in field theory, linking scalar potential with vector field strength, and is highly examinable in WJEC papers.
这个梯度关系是场论中的一个统合概念，把标量势与矢量场强联系在一起，在 WJEC 试卷中常被考查。
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