📚 Gravitational Potential | 引力势
Gravitational potential is one of the most important field concepts in CIE A-Level Physics. It leads directly to gravitational potential energy, escape velocity, satellite total energy and the shape of gravitational fields. This article explains the definition, equations, graphs and common exam applications.
引力势是 CIE A-level 物理中最重要的场概念之一。它直接联系引力势能、逃逸速度、卫星总能量和引力场的图像。本文讲解定义、公式、图线以及常见考试应用。
1. Definition of Gravitational Potential | 引力势的定义
In CIE A-Level Physics, gravitational potential φ at a point is defined as the work done per unit mass in bringing a small test mass from infinity to that point. It is a scalar quantity, and its SI unit is J kg⁻¹. The definition uses a small test mass so that the source field is not disturbed.
在 CIE A-level 物理中，某点的引力势 φ 定义为把一个小试验质量从无穷远移到该点过程中每单位质量所做的功。它是标量，SI 单位是 J kg⁻¹。定义中使用小试验质量是为了不扰动源场。
More precisely, the work is the work done by the gravitational field itself. Because the gravitational force is attractive, the field does positive work on the test mass as it moves from infinity towards the source mass. This is why gravitational potential at a finite distance from a point mass is always negative.
更准确地说，这里的功是引力场本身所做的功。由于万有引力是吸引力，试验质量从无穷远移向源质量时，引力场对它做正功。因此，点质量附近有限距离处的引力势总是负值。
Potential is a property of the field at a point, not a property of the test mass. The test mass is only used to probe the field. Gravitational potential is also independent of the path taken from infinity to the point, because gravity is a conservative force.
引力势是场在某点的性质，而不是试验质量的性质。试验质量只是用来探测场的工具。引力势还与从无穷远到该点的路径无关，因为引力是保守力。
φ = W / m
In this equation, W is the work done by the gravitational field in bringing mass m from infinity to the point, and m is the test mass. The value of φ does not depend on m, because W is proportional to m.
在这个公式中，W 是引力场把质量 m 从无穷远移到该点所做的功，m 是试验质量。φ 的数值不依赖于 m，因为 W 与 m 成正比。
2. Why Infinity is the Zero Reference | 为什么选择无穷远为零势点
Zero potential at infinity is a natural convention for inverse-square laws. It means a body at infinity has zero gravitational potential energy relative to the source. As the body moves closer, the field does positive work, so the potential energy must decrease below zero.
对于平方反比定律，无穷远为零势点是一个自然的约定。它意味着无穷远的物体相对源质量具有零引力势能。随着物体靠近，引力场做正功，因此势能必须降到零以下。
were chosen, the formula would require an added constant, and the physics would become unnecessarily complicated.
这个选择还使势的表达式在数学上非常简单：φ = −GM/r。如果选择别的零点，公式就需要额外加一个常数，物理表达会变得不必要地复杂。
Another way to see the convention is to consider the force law. For a point mass, the gravitational force varies as 1/r². The potential is obtained by integrating this force with respect to distance, and the integration constant is chosen so that φ tends to zero as r tends to infinity.
理解这个约定的另一种方法是考虑力的规律。对于点质量，引力按 1/r² 变化。势由这个力对距离积分得到，积分常数取为当 r 趋于无穷大时 φ 趋于零。
In exam answers, you should state clearly that the zero of gravitational potential is taken at infinity. You should also explain that gravitational potential is negative near a mass because the field does positive work when a mass falls in from infinity.
在考试答案中，你应该明确说明引力势的零点取在无穷远。你还应解释，质量附近引力势为负，是因为质量从无穷远落入时引力场做正功。
3. Formula for a Point Mass | 点质量的引力势公式
ere G is the gravitational constant, M is the mass of the source, and r is the distance from the centre of the mass.
对于点质量 M，或均匀球体外部的点，距离球心 r 处的引力势由 φ = −GM/r 给出。其中 G 是引力常量，M 是源质量，r 是到质量中心的距离。
The formula applies outside a uniform sphere because a uniform spherical shell behaves, for points outside it, as if all its mass were concentrated at its centre. This is a standard result in CIE A-Level Physics and is often used in satellite and planetary calculations.
该公式适用于均匀球体外部，因为均匀球壳对壳外点的作用，等效于全部质量集中在球心。这是 CIE A-level 物理中的标准结论，常用于卫星和行星计算。
total = φ₁ + φ₂ = −GM₁/r₁ − GM₂/r₂, where r₁ and r₂ are the distances from the point to each mass.
引力势是标量，因此多个质量产生的势可以直接代数相加。对于两个质量 M₁ 和 M₂，某点的总势为 φ_total = φ₁ + φ₂ = −GM₁/r₁ − GM₂/r₂，其中 r₁ 和 r₂ 是该点到两个质量的距离。
φ = −GM / r
The potential has units of J kg⁻¹, while gravitational field strength has units of N kg⁻¹ or m s⁻².
注意，对于点质量，φ 总是负值，且其大小随 r 增大而减小。在无穷远处，φ = 0。势的单位是 J kg⁻¹，而引力场强度的单位是 N kg⁻¹ 或 m s⁻²。
4. Gravitational Potential Energy and Mass | 引力势能与质量
is the energy associated with the position of mass m in the gravitational field of mass M.
质量 m 在势为 φ 的点的引力势能 U 由 U = mφ 给出。对于点质量 M，这变为 U = −GMm/r。这是质量 m 在质量 M 的引力场中由于位置而具有的能量。
Since φ is negative, U is also negative. As r increases, U becomes less negative and approaches zero at infinity. This means that a mass farther from the source has higher gravitational potential energy than a mass closer to the source, even though both values are negative.
由于 φ 为负，U 也为负。随着 r 增大，U 变得不那么负，并在无穷远处趋于零。这意味着离源质量较远的物体比离源质量较近的物体具有更高的引力势能，尽管两者都是负值。
es away from the source, ΔU is positive; if it falls towards the source, ΔU is negative and the field does positive work.
当质量从 r₁ 移到 r₂ 时，引力势能的变化为 ΔU = mΔφ = m(φ₂ − φ₁)。如果质量远离源质量，ΔU 为正；如果它落向源质量，ΔU 为负，引力场做正功。
inity. Since U = −GMm/r at distance r and U = 0 at infinity, the binding energy is +GMm/r.
质量 m 在距离质量 M 为 r 处的结合能，是把它完全移出引力场即移到无穷远所需的能量。由于在距离 r 处 U = −GMm/r，在无穷远处 U = 0，所以结合能为 +GMm/r。
U = mφ = −GMm / r
f radius R, a point on the surface has r = R, so φ_surface = −GM/R and U_surface = −GMm/R.
计算时一定要用从源质量中心算起的距离，而不是从表面算起。对于半径为 R 的行星，表面上的点有 r = R，因此 φ_surface = −GM/R，U_surface = −GMm/R。
5. Potential Gradient and Field Strength | 势梯度与场强
φ/dr. The minus sign shows that the field points in the direction in which potential decreases most rapidly.
引力场强度 g 与引力势的梯度有关。在一维情况下，g = −dφ/dr。负号表示场指向势下降最快的方向。
For a uniform field, such as the field near the Earth' −Δφ/Δx, where Δx is a small displacement. This is why gravitational field strength can also be expressed in N kg⁻¹, which is equivalent to J kg⁻¹ m⁻¹.
对于均匀场，例如地球表面附近小竖直范围内的场，该关系可写成 g = −Δφ/Δx，其中 Δx 是小位移。这就是引力场强度也可以用 N kg⁻¹ 表示的原因，它等价于 J kg⁻¹ m⁻¹。
a point mass, substituting φ = −GM/r into g = −dφ/dr gives g = GM/r². This recovers Newton' ation in field form. The field strength varies as 1/r², while the potential varies as 1/r.
对于点质量，把 φ = −GM/r 代入 g = −dφ/dr，得到 g = GM/r²。这以场的形式重现了牛顿万有引力定律。场强按 1/r² 变化，而势按 1/r 变化。
A useful graphical interpretation is that the magnitude of the field strength at any point equals the magnitude of the gradient of the φ-r graph at that point. A steeper φ-r graph means a stronger gravitational field.
一个有用的图像解释是：某点的场强大小等于该点 φ-r 图线斜率的绝对值。φ-r 图线越陡，引力场越强。
g = −Δφ / Δr
Students often confuse potential and field strength. Potential is a scalar and is measured in J kg⁻¹; field strength is a vector and is measured in N kg⁻¹. The field strength depends on the gradient of potential, not on the potential itself.
学生常混淆势和场强。势是标量，单位是 J kg⁻¹；场强是矢量，单位是 N kg⁻¹。场强取决于势的梯度，而不是势本身的大小。
6. Graphs of φ and g against r | φ 和 g 随 r 变化的图线
The graph of gravitational potential φ against distance r for a point mass is a negative curve that approaches zero asymptotically as r tends to infinity. It never crosses the r-axis, because φ is negative for all finite r.
对于点质量，引力势 φ 随距离 r 变化的图线是一条负的曲线，当 r 趋于无穷大时渐近地趋于零。它不会穿过 r 轴，因为所有有限 r 处的 φ 都是负值。
The graph of gravitational field strength g against r is a positive curve proportional to 1/r². It falls more rapidly than the potential graph. At small r, both the potential and the field strength have large magnitudes.
引力场强度 g 随 r 变化的图线是一条与 1/r² 成正比的正曲线。它比势的图线下降得更快。在 r 很小时，势和场强的大小都很大。
ith r from zero at the centre to GM/R² at the surface. This linear region is often tested in multiple-choice questions.
对于半径为 R 的均匀实心球体，球外场强为 g = GM/r²。球内场强为 g = GMr/R³，它随 r 从球心处的零线性增加到表面处的 GM/R²。这个线性区域常在选择题中考查。
| Quantity | Outside a point mass or sphere | Graph shape |
|---|---|---|
| Potential φ | −GM/r | Negative curve tending to 0 as r → ∞ |
| Field strength g | GM/r² | Positive curve tending to 0 as r → ∞ |
When sketching these graphs, label the axes correctly, show the correct sign, and mark that the curves approach zero without touching the axis at a finite distance. Do not draw the potential graph crossing zero, and do not draw the field strength graph as negative for an attractive field.
画这些图时，要正确标注坐标轴，显示正确的正负号，并标明曲线在有限距离内不接触坐标轴而是趋于零。不要把势的图线画成穿过零，也不要把吸引场的场强图线画成负值。
7. Potential Inside and Outside a Uniform Sphere | 均匀球体内部与外部的引力势
r is greater than or equal to the radius R of the sphere. This is the same as for a point mass of the same total mass M located at the centre.
在均匀球体外部，引力势为 φ = −GM/r，其中 r 大于或等于球体半径 R。这与同样总质量 M 集中在球心的点质量相同。
gative value of potential outside the sphere and the boundary value used in many calculations.
在球体表面，r = R，所以 φ_surface = −GM/R。这是球体外部势的最不负的值，也是许多计算中使用的边界值。
which is more negative than the surface value. The potential is continuous at the surface.
在均匀实心球体内部，势不是常数，而是由 φ = −GM(3R² − r²)/(2R³) 给出。在球心处，r = 0，所以 φ_centre = −3GM/(2R)，比表面处的值更负。势在表面处是连续的。
Inside a thin uniform spherical shell, however, the gravitational field is zero everywhere. Since the field is the negative gradient of potential, a zero field means the potential is constant inside the shell. Its value equals the potential at the shell's surface, −GM/R.
但在均匀薄球壳内部，引力场处处为零。由于场是势的负梯度，场为零意味着壳内势为常数。其值等于球壳表面的势，即 −GM/R。
These results explain why a person inside a hollow uniform shell would feel no net gravitational force from the shell, even though the potential inside is not zero. Potential and field strength are different quantities and must not be confused.
这些结果解释了为什么在均匀空心球壳内的人不会感受到球壳的合引力，尽管壳内的势并不为零。势和场强是不同的物理量，不能混淆。
8. Potential Difference and Work Done | 势差与做功
The work done by the gravitational field when a mass m moves from point 1 to point 2 is Wfield = −mΔφ = −m(φ₂ − φ₁).
两点之间的势差为 Δφ = φ₂ − φ₁。当质量 m 从点 1 移到点 2 时，引力场做的功为 Wfield = −mΔφ = −m(φ₂ − φ₁)。
ent is W_external = +mΔφ = m(φ₂ − φ₁). This is the negative of the work done by the field.
如果外力缓慢移动该质量，不改变其动能，则外力做的功为 W_external = +mΔφ = m(φ₂ − φ₁)。这是引力场做功的负值。
1/r₁ − 1/r₂). This quantity is positive, because the satellite is being moved away from the planet against the gravitational attraction.
例如，把质量为 m 的卫星从半径 r₁ 移到更大的半径 r₂，所需的外力做功为 W = GMm(1/r₁ − 1/r₂)。这个量为正，因为卫星是逆着引力被移离行星的。
In orbital mechanics, the work done by the gravitational field alone changes the kinetic energy of the satellite. For a circular orbit, however, the gravitational force is always perpendicular to the
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