Gravitational Fields | 引力场

📚 Gravitational Fields | 引力场

Gravitational fields describe how masses interact across empty space. This article summarises key definitions, equations and applications required for CIE A-Level Physics, from Newton’s law to satellite orbits and escape velocity.

引力场描述质量如何在真空中相互作用。本文总结了 CIE A-Level 物理中所需的关键定义、公式和应用，从牛顿万有引力定律到卫星轨道和逃逸速度。

1. Gravitational Field and Newton’s Law of Gravitation | 引力场与牛顿万有引力定律

A gravitational field is a region of space around a mass in which another mass experiences a gravitational force. The field is a vector field because the force has both magnitude and direction.

引力场是质量周围空间中另一质量会受到引力作用的区域。由于引力既有大小又有方向，引力场是矢量场。

Newton’s law of gravitation states that every two point masses attract each other with a force directly proportional to the product of their masses and inversely proportional to the square of their separation.

牛顿万有引力定律指出，任意两个质点都相互吸引，引力的大小与两质量的乘积成正比，与它们之间距离的平方成反比。

F = Gm₁m₂/r²

G is the universal gravitational constant, 6.67 × 10⁻¹¹ N m² kg⁻². The negative sign often used to show attraction is omitted in scalar form.

G 是万有引力常量，约为 6.67 × 10⁻¹¹ N m² kg⁻²。标量形式中通常省略表示吸引的负号。

2. Gravitational Field Strength g | 引力场强度 g

Gravitational field strength at a point is defined as the gravitational force per unit mass acting on a small test mass placed at that point.

引力场强度定义为放在该点的小检验质量所受的引力与质量的比值。

g = F/m

For a point mass or a spherical body of mass M, the field strength at distance r from its centre is g = GM/r². This is uniform only in a small region near the surface of a large body.

对于点质量或质量为 M 的球体，距其中心 r 处的引力场强度为 g = GM/r²。只有在大天体表面附近的小区域内，场强才近似均匀。

The unit of gravitational field strength is N kg⁻¹, which is equivalent to m s⁻².

引力场强度的单位是 N kg⁻¹，与 m s⁻² 等价。

3. Gravitational Field Lines | 引力场线

Field lines help visualise a gravitational field. The direction of the field at any point is the tangent to the line; the spacing indicates field strength.

场线有助于直观表示引力场。某点的场方向沿该处场线切线方向，场线间距表示场强大小。

Around a point mass, the lines are radial and point inward, showing that gravitational force is always attractive. Near Earth’s surface, the lines are approximately parallel and evenly spaced, representing a uniform field.

点质量周围的场线呈放射状并指向内部，表明引力总是吸引力。在地球表面附近，场线近似平行且等距，表示均匀场。

In a uniform field, a mass experiences the same force in magnitude and direction at every point, which simplifies calculations near the surface of a planet.

在均匀场中，质量在各点所受的引力大小和方向都相同，这简化了行星表面附近的计算。

4. Gravitational Potential | 引力势

Gravitational potential at a point is the work done per unit mass in bringing a small test mass from infinity to that point without changing its kinetic energy.

引力势是指在不改变检验质量动能的前提下，将单位检验质量从无穷远移到该点所做的功。

V = -GM/r

The potential is negative because gravitational forces are attractive, so work is done by the field when a mass moves from infinity to a point. Potential is a scalar quantity.

引力势为负值，因为引力是吸引力，质量从无穷远移到某点时引力场做正功。引力势是标量。

The unit of gravitational potential is J kg⁻¹. Potential at infinity is conventionally taken as zero.

引力势的单位是 J kg⁻¹。通常取无穷远处的引力势为零。

5. Gravitational Potential Energy | 引力势能

The gravitational potential energy of a system of two point masses is the work done in assembling them from infinite separation to a separation r.

两个点质量组成的系统的引力势能，是将它们从相距无穷远移动到相距 r 时所做的功。

Eₚ = -GMm/r

This energy is negative, showing that the masses are bound. To separate them to infinity, an external agent must supply energy equal to GMm/r.

该势能为负，表明两个质量相互束缚。要将它们分离到无穷远，外界需要提供等于 GMm/r 的能量。

For a mass m in the field of a much larger mass M, we often treat the potential energy as belonging to mass m in the external field.

对于处于远大于自身的质量 M 的场中的质量 m，我们通常将势能视为质量 m 在外场中的势能。

6. Relation Between Field Strength and Potential | 场强与势的关系

In a radial field, gravitational field strength is the negative gradient of gravitational potential with respect to distance.

在径向场中，引力场强度等于引力势随距离变化率的负值。

g = -dV/dr

For a point mass, differentiating V = -GM/r gives g = GM/r², which matches the inverse-square law. This relation is useful for analysing non-uniform fields.

对 V = -GM/r 求导可得到 g = GM/r²，与平方反比定律一致。这一关系对分析非均匀场很有用。

The negative sign means the field points in the direction of decreasing potential, toward the mass.

负号表明场指向势减小的方向，即指向质量中心。

7. Orbital Motion of Satellites | 卫星轨道运动

For a satellite in a circular orbit, the gravitational force provides the centripetal force required for circular motion.

对于圆轨道上的卫星，万有引力提供圆周运动所需的向心力。

GMm/r² = mv²/r

Rearranging gives the orbital speed v = √(GM/r). The speed decreases as the orbital radius increases.

整理可得轨道速度 v = √(GM/r)。轨道半径越大，轨道速度越小。

The orbital period T is found from v = 2πr/T, giving Kepler’s third law: T² = 4π²r³/(GM).

轨道周期 T 由 v = 2πr/T 求得，得到开普勒第三定律：T² = 4π²r³/(GM)。

8. Geostationary Orbits | 地球同步轨道

A geostationary satellite has a period of 24 hours and orbits in the equatorial plane in the same direction as Earth’s rotation, so it remains fixed above one point on the equator.

地球同步卫星的周期为 24 小时，在赤道平面内沿地球自转方向运行，因此始终停留在赤道某一点上方。

Using T² = 4π²r³/(GM) with T = 24 h gives an orbital radius of about 4.2 × 10⁷ m from Earth’s centre, equivalent to an altitude of about 3.6 × 10⁷ m.

将 T = 24 h 代入 T² = 4π²r³/(GM)，可算出轨道半径约为 4.2 × 10⁷ m，即距地面高度约 3.6 × 10⁷ m。

Geostationary orbits are used for communication and weather satellites because the satellite appears stationary relative to the ground.

地球同步轨道用于通信和气象卫星，因为卫星相对地面看起来静止不动。

9. Escape Velocity | 逃逸速度

Escape velocity is the minimum speed an object must have at the surface of a planet to escape its gravitational field completely, reaching infinity with zero final kinetic energy.

逃逸速度是物体在天体表面要完全摆脱其引力场、到达无穷远且最终动能为零所需的最小速度。

v = √(2GM/R)

It is independent of the mass of the escaping object. For Earth, escape velocity is approximately 11.2 km s⁻¹.

逃逸速度与逃逸物体的质量无关。地球的逃逸速度约为 11.2 km s⁻¹。

The derivation equates the initial kinetic energy ½mv² to the gravitational potential energy GMm/R.

推导时令初始动能 ½mv² 等于引力势能 GMm/R。

10. Energy of an Orbiting Satellite | 轨道卫星的能量

For a satellite in a circular orbit, kinetic energy and gravitational potential energy are related by E_k = ½mv² = GMm/(2r) and Eₚ = -GMm/r.

对于圆轨道卫星，动能与引力势能的关系为 E_k = ½mv² = GMm/(2r)，Eₚ = -GMm/r。

E_total = E_k + Eₚ = -GMm/(2r)

The total energy is negative, indicating a closed or bound orbit. If the satellite loses energy, it moves to a lower orbit and its speed actually increases.

总能量为负，表明轨道是闭合或束缚的。如果卫星损失能量，它会进入更低轨道，而速度实际上会增大。

To move to a higher orbit, the satellite must gain energy; this energy may come from rocket thrusters.

若卫星要进入更高轨道，必须获得能量；这些能量可来自火箭推进器。

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Gravitational Fields | 引力场