📚 D.1 Gravitational Fields | D.1 引力场
Gravitational fields are one of the fundamental field concepts in the IB Physics curriculum. They describe how masses attract each other at a distance, and they underpin everything from falling apples to orbiting satellites. In this article, we will cover Newton's law of universal gravitation, gravitational field strength, gravitational potential and potential energy (HL only), equipotential surfaces, escape speed, orbital motion, and Kepler’s laws. The aim is to provide a clear, self-contained revision resource for both SL and HL students, with paired English–Chinese explanations for every key point.
引力场是IB物理课程中最基本的场概念之一。它描述质量之间如何发生超距吸引，并且支撑着从落下的苹果到轨道卫星的一切现象。本文将覆盖牛顿万有引力定律、引力场强度、引力势和引力势能（仅HL）、等势面、逃逸速度、轨道运动和开普勒定律。目标是提供一份清晰、自足的复习资料，适用于SL和HL学生，每一个要点都配有英中对照解释。
1. Newton's Law of Universal Gravitation | 牛顿万有引力定律
Any two point masses in the universe attract each other 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. This is summarised by Newton's law of universal gravitation:
宇宙中任何两个质点都会相互吸引，引力的大小与两质量的乘积成正比，与它们中心之间距离的平方成反比。这由牛顿万有引力定律概括：
F = G m₁ m₂ / r²
, F is the magnitude of the gravitational force, m₁ and m₂ are the masses, r is the centre‑to‑ ⁻¹¹ N m² kg⁻². The force acts along the line joining the centres and is always attractive.
这里，F是引力的大小，m₁和m₂是两个质量，r是中心距，G是引力常量，数值为6.67×10⁻¹¹ N m² kg⁻²。力沿着两中心连线方向，并且总是吸引力。
2. Gravitational Field Strength g | 引力场强度 g
point mass M (or a spherical mass), the field strength at a distance r from its centre is:
某一点的引力场强度定义为放置在该点的检验质量所受到的万有引力与质量的比值：g = F / m。对于一个点质量M（或球形质量），在距离其中心r处的场强大小为：
g = G M / r²
This is a vector quantity whose direction is toward the centre of the mass creating the field. On the surface of the Earth, g has an approximate value of 9.81 N kg⁻¹, pointing downward.
这是一个矢量，方向指向产生场的质量中心。在地球表面，g的近似值为9.81 N kg⁻¹，方向竖直向下。
3. Gravitational Field Lines and Uniform Fields | 引力场线与均匀场
Gravitational fields are represented by field lines (lines of force). For an isolated point mass, the lines are radial and point inward, indicating an attractive field. The spacing of the lines shows the field strength: closer lines mean a stronger field. Near the surface of a planet, the field can be treated as uniform, meaning the field lines are parallel and equally spaced, and g is constant in both magnitude and direction.
引力场用场线（力线）表示。对于孤立的点质量，场线呈辐射状并指向内，表明场是吸引的。场线的疏密显示场强：线越密场越强。在行星表面附近，场可视为均匀场，即场线平行且等间距，g的大小和方向都恒定。
4. Gravitational Potential (HL only) | 引力势（仅HL）
Gravitational potential V at a point in a gravitational field is defined as the work done per unit mass to bring a small test mass from infinity to that point. Unlike field strength, potential is a scalar. For a point mass M, the potential at a distance r is given by:
引力势V是引力场中某点处的一个标量，定义为将单位检验质量从无穷远处移至该点外力所做的功。对于点质量M，在距离r处的势为：
V = – G M / r
The negative sign indicates that work is done by the field (gravity is an attractive force) and the potential is taken as zero at infinity. Gravitational potential is measured in J kg⁻¹.
负号表明引力场做正功（引力是吸引力），并且规定无穷远处势为零。引力势的单位是J kg⁻¹。
5. Gravitational Potential Energy (HL) | 引力势能（HL）
The gravitational potential energy Ep of a system of two point masses M and m separated by a distance r is given by:
两个点质量M和m相距r所组成的系统的引力势能Ep为：
Ep = – G M m / r
action. The change in potential energy when a mass moves between two points is ΔEp = m ΔV.
如果涉及多个质量，总势能为所有质量对之和。这个量代表了引力相互作用中储存的能量。当质量在两点间移动时，势能的变化为ΔEp = m ΔV。
6. Equipotential Surfaces (HL) | 等势面（HL）
An equipotential surface is a surface on which the gravitational potential is constant. No work is done by the field when a mass moves along an equipotential surface because there is no change in potential energy. For a point mass, the equipotentials are spheres centred on the mass. Field lines are always perpendicular to equipotential surfaces. In a uniform field, equipotentials are planes perpendicular to the field lines.
等势面是引力势处处相等的面。质量沿着等势面移动时，由于势能不变，引力场不做功。对点质量而言，等势面是以该质量为中心的球面。场线总是垂直于等势面。在均匀场中，等势面是垂直于场线的平面。
7. Escape Speed (HL) | 逃逸速度（HL）
The escape speed is the minimum speed an object must have at the surface of a planet (or any massive body) to completely escape its gravitational field without any further propulsion. It is derived by equating the kinetic energy at launch (½ m vₑ²) to the magnitude of the gravitational potential energy (–G M m / R, taking zero at infinity). The result is:
逃逸速度是一个物体在行星（或任何大质量天体）表面必须具有的最小初速度，使其能在没有后续推力的情况下彻底摆脱引力场。它由发射时的动能(½ m vₑ²)等于引力势能的大小（以无穷远处为零）推导得出：
vₑ = √2 G M / R
Here, M is the planet' ape speed does not depend on the mass of the escaping object. For Earth, vₑ ≈ 11.2 km s⁻¹.
其中M是行星质量，R是其半径。逃逸速度与逃逸物体的质量无关。对于地球，vₑ ≈ 11.2 km s⁻¹。
8. Orbital Motion and Kepler's Laws (HL) | 轨道运动与开普勒定律（HL）
l force is provided by gravity: m v² / r = G M m / r². This leads to the orbital speed v = √G M / r. Our understanding of orbits is summarised by Kepler’s three laws:
轨道上的物体，如绕行星的卫星或绕太阳的行星，在有心引力作用下运动。对于圆轨道，向心力由引力提供：m v² / r = G M m / r²，由此可得轨道速率v = √G M / r。我们对轨道的理解可总结为开普勒三定律：
- Kepler’s First Law: The orbit of each planet is an ellipse with the Sun at one focus.
- 开普勒第一定律：行星轨道为椭圆，太阳位于一个焦点上。
- Kepler’s Second Law: A line drawn from the Sun to a planet sweeps out equal areas in equal times.
- 开普勒第二定律：从太阳到行星的连线在相等时间内扫过相等的面积。
- Kepler’s Third Law: The square of the orbital period T is proportional to the cube of the semi‑major axis a: T² ∝ a³. For circular orbits of radius r, this becomes T² = (4π² / G M) r³.
- 开普勒第三定律：轨道周期T的平方正比于半长轴a的立方：T² ∝ a³。对于圆轨道，半径r满足T² = (4π² / G M) r³。
9. Energy in Orbital Motion (HL) | 轨道运动中的能量（HL）
The total mechanical energy E of a satellite of mass m in a circular orbit of radius r around a planet of mass M is the sum of its kinetic energy K and potential energy U:
质量为m的卫星在半径为r的圆轨道上绕质量为M的行星运行，其总机械能E是动能K与势能U之和：
E = K + U = ½ m v² – G M m / r
Using v² = G M / r, we find K = G M m / (2r). Therefore, the total energy is:
利用v² = G M / r，可得K = G M m / (2r)。因此总能量为：
E = – G M m / (2r)
This shows that the total energy is negative for a bound orbit and is exactly half the potential energy magnitude. If the satellite's total energy becomes zero or positive, it is no longer in a closed orbit (i.e. it will escape or follow a parabolic/hyperbolic path).
这表明束缚轨道的总能量为负，并且大小正好是势能绝对值的一半。若卫星的总能量变为零或正值，它将不再处于闭合轨道（即它将逃逸或沿抛物线/双曲线路径运动）。
10. Key Applications and Problem‑Solving Tips | 关键应用与解题技巧
t by clarifying whether the field is uniform (near surface: g = constant) or radial (point‑mass approximation). For HL, frequently used formulas include V = –GM/r, Ep = – y. Remember that gravitational field strength g is the negative gradient of potential: g = – xpress T² in terms of r³ and the mass of the central body; this relationship is fundamental for masses derived from orbital data.
解决引力场问题时，务必先明确场是均匀场（近地表：g恒定）还是辐射场（点质量近似）。对于HL学生，常用公式包括V = –GM/r、Ep = –GMm/r，以及逃逸或束缚能的条件。记住引力场强度g是势的负梯度：沿径向方向g = –ΔV/Δr。要特别注意符号，尤其是在计算反抗引力场做功时。对于开普勒第三定律，将T²表示成r³与中心天体质量的函数；这一关系是通过轨道数据推导天体质量的基础。
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