📚 Fields at Work | 场力做功
In IB Physics Topic 10.2, we explore how forces do work in gravitational and electric fields, leading to concepts of potential energy, potential, and the deep connection between field strength and potential gradient. From the escape speed of a rocket to the motion of charged particles between plates, these ideas unify our understanding of conservative fields.
在IB物理第10.2节中，我们探讨引力场和电场中的力如何做功，由此引出势能、势以及场强与势梯度之间的深层联系。从火箭的逃逸速度到带电粒子在平行板间的运动，这些概念统一了我们对保守场的理解。
1. Gravitational Potential Energy | 引力势能
When a mass m is moved in a gravitational field, work is done against the gravitational force. Since gravity is a conservative force, the work done depends only on the initial and final positions. For a radial field around a point mass or spherical body of mass M, the gravitational potential energy is given by Ug = -GMm/r, where r is the distance from the centre of M. The negative sign indicates that work must be done to separate the masses to infinity, where Ug is defined as zero.
当质量m在引力场中移动时，需克服引力做功。由于引力是保守力，做功仅取决于起点和终点位置。在质点或球形天体M的径向场中，引力势能为Ug = -GMm/r，其中r为距M中心的距离。负号表示必须做功才能将两个质量分离至无穷远，并规定无穷远处Ug为零。
2. Gravitational Potential | 引力势
Gravitational potential Vg is the work done per unit mass to bring a small test mass from infinity to a point in the field. It is a scalar field defined by Vg = -GM/r. The unit is J kg-1. Since only differences in potential matter, we often calculate ΔVg between two points. For a uniform field near a planet's surface, we can approximate ΔVg = g Δh, where g is the field strength and Δh the height change.
引力势Vg是将单位质量试探质量从无穷远移至场中某点所做的功。它是一个标量场，定义为Vg = -GM/r，单位为J kg-1。由于只有势的差值有意义，我们通常计算两点间的ΔVg。在行星表面附近的匀强场中，可近似为ΔVg = g Δh，其中g为场强，Δh为高度变化。
3. Potential Gradient and Field Strength | 势梯度与场强
There is a crucial relationship between field strength and the rate of change of potential with distance. In one dimension, the gravitational field strength g is the negative gradient of the gravitational potential: g = -dVg/dr. The minus sign indicates that the field points in the direction of decreasing potential. For a radial field, differentiating Vg = -GM/r yields g = GM/r2, consistent with Newton's law.
场强与势随距离变化率之间存在重要关系。在一维情况下，引力场强g等于引力势的负梯度：g = -dVg/dr。负号表示场的方向指向势减小的方向。对于径向场，对Vg = -GM/r求导得g = GM/r2，与牛顿定律一致。
This relationship also holds for electric fields: E = -dVe/dr. Where the field is uniform, this reduces to E = ΔV/d, with d being the separation between equipotential surfaces. A steeper potential gradient means a stronger field; closely spaced equipotentials indicate a region of high field strength.
此关系同样适用于电场：E = -dVe/dr。在匀强场中，可简化为E = ΔV/d，其中d为等势面间的距离。势梯度的陡峭程度反映场强大小；等势面越密集，场强越大。
4. Equipotential Surfaces | 等势面
An equipotential surface is a surface on which the potential is constant. In gravitational and electric fields, no work is done when moving a mass or charge along an equipotential surface, because the force is perpendicular to the surface at every point. For a point mass or point charge, the equipotentials are concentric spheres. In a uniform gravitational field, they are horizontal planes; in a uniform electric field between parallel plates, they are planes parallel to the plates.
等势面是势保持恒定的曲面。在引力和电场中，沿等势面移动质量或电荷不做功，因为力处处垂直于该曲面。对于质点和点电荷，等势面为同心球面。在均匀引力场中，等势面为水平面；在平行板间的匀强电场中，等势面为平行于板的平面。
| Gravitational equipotentials | Spheres around point mass; horizontal planes near Earth's surface. |
| Electric equipotentials | Spheres around point charge; planes parallel to charged plates. |
| Work done | Zero for motion along an equipotential; W = m ΔVg or q ΔVe when crossing equipotentials. |
等势面性质：沿等势面移动不做功，跨越等势面时做功 W = m ΔVg 或 q ΔVe。
5. Escape Speed | 逃逸速度
The escape speed is the minimum speed needed for an object to completely leave a gravitational field without further propulsion. Using energy conservation, the total energy at launch must be at least zero (taking Ug = 0 at infinity). Hence, ½mvesc2 + (-GMm/R) = 0, where R is the radius of the planet. Solving gives vesc = √2GM/R. Notice that the escape speed does not depend on the mass of the escaping object.
逃逸速度是物体无需额外推力即可完全脱离引力场的最小速度。根据能量守恒，发射时的总机械能至少为零（规定无穷远处Ug = 0）。因此有½mvesc2 + (-GMm/R) = 0，解得vesc = √2GM/R。注意逃逸速度与逃逸物体的质量无关。
For Earth, substituting M = 5.97×1024 kg, R = 6.37×106 m yields vesc ≈ 11.2 km s-1. This concept also appears when discussing black holes: if a body is so dense that vesc exceeds the speed of light, even light cannot escape.
代入地球质量M = 5.97×1024 kg，半径R = 6.37×106 m，得vesc ≈ 11.2 km s-1。该概念也用于黑洞讨论：若天体密度极大使得vesc超过光速，则连光也无法逃逸。
6. Orbital Motion and Energy | 轨道运动与能量
For a satellite in a stable circular orbit, the centripetal force is provided by gravity: GMm/r2 = mv2/r, giving vorb = √GM/r. The kinetic energy is K = ½mv2 m/(2r), while the potential energy is U = -GMm/r. The total mechanical energy is therefore Etotal = K + U = -GMm/(2r). The negative total energy signifies a bound orbit.
对于稳定圆形轨道上的卫星，向心力由引力提供：GMm/r2 = mv2/r，得vorb = √GM/r。动能为K = ½mv2 = GMm/(2r)，势能为U = -GMm/r，总机械能Etotal = -GMm/(2r)。负的总能量表明轨道是束缚的。
If a satellite is given extra energy, it moves to a higher orbit with a larger radius but lower orbital speed, and its total energy becomes less negative (closer to zero). To move from a lower orbit to a higher orbit, external work must be done, increasing both the potential energy and the total energy, though the kinetic energy decreases.
若卫星获得额外能量，会进入半径更大、但轨道速度更小的较高轨道，总能量负得较少（更接近零）。从低轨道移至高轨道需要外力做功，势能和总能量均增加，但动能减小。
7. Electric Potential Energy and Potential | 电势能与电势
Analogous to gravity, a point charge Q creates an electric field in which a test charge q possesses electric potential energy Ue = (1/(4πε₀)) Qq/r, with the zero of energy taken at infinite separation. The electric potential Ve is the work done per unit charge to bring a small positive test charge from infinity: Ve = (1/(4πε₀)) Q/r, measured in joules per coulomb (J C-1) or volts (V). For a positive Q, the potential is positive and decreases with distance; for a negative Q, it is negative and increases with distance.
与引力场类似，点电荷Q产生电场，试探电荷q具有电势能Ue = (1/(4πε₀)) Qq/r，规定无限远处能量为零。电势Ve是将单位正试探电荷从无穷远移至该点所做的功：Ve = (1/(4πε₀)) Q/r，单位为焦耳每库仑(J C-1) 或伏特(V)。对于正Q，电势为正且随距离减小；对于负Q，电势为负且随距离增大。
Just as with gravity, the electric field is the negative gradient of potential: E = -dVe/dr. In a radial field, differentiating Ve gives E = (1/(4πε₀)) Q/r2, consistent with Coulomb's law. In a uniform field, this becomes E = ΔV/d.
与引力相同，电场是电势的负梯度：E = -dVe/dr。在径向场中对Ve求导得E = (1/(4πε₀)) Q/r2，与库仑定律一致。在匀强电场中则简化为E = ΔV/d。
8. Potential Difference and Work Done | 电势差与做功
The potential difference (p.d.) between two points A and B is defined as ΔV = VB - VA. The work done by the electric field when a charge q moves through a potential difference is W = q ΔV. If the work is done by an external agent against the field, the sign reverses. For example, moving a positive charge from low to high potential requires external work; moving it from high to low potential releases energy.
两点A与B之间的电势差定义为ΔV = VB - VA。当电荷q通过电势差时，电场力做功为W = q ΔV。若为外力反抗电场做功，则符号相反。例如，将正电荷从低电势移至高电势需外力做功；从高电势移至低电势则释放能量。
The electronvolt (eV) is a convenient unit of energy defined as the energy gained by an electron when it moves through a potential difference of 1 V: 1 eV = 1.60 × 10-19 J. This unit is used extensively in particle physics and quantum topics.
电子伏特(eV)是便利的能量单位，定义为一个电子经过1 V电势差所获得的能量：1 eV = 1.60 × 10-19 J。该单位广泛用于粒子物理和量子物理中。
9. Uniform Electric Fields and Potential | 匀强电场与电势
A uniform electric field can be produced by two parallel conducting plates connected to a voltage source. The field strength is constant in magnitude and direction between the plates (ignoring edge effects) and is given by E = ΔV / d, where d is the plate separation. The equipotential surfaces are equally spaced planes parallel to the plates. Every movement perpendicular to the plates corresponds to a linear change in potential.
匀强电场可由连接电源的两块平行导电板产生。忽略边缘效应，板间场强的大小和方向恒定，且满足E = ΔV / d，其中d为板间距离。等势面是间距相等的平行于板的平面。垂直于板的每段移动对应电势的线性变化。
A charged particle in a uniform field experiences a constant force F = qE, resulting in constant acceleration. The kinetic energy gained when moving from one plate to the other equals ½mv2 = q ΔV, which is simply the work–energy theorem for the electric force. This principle is used in particle accelerators and in determining the speed of electrons in cathode-ray tubes.
匀强电场中的带电粒子受到恒力F = qE，产生恒定加速度。从一板移至另一板所获得的动能等于½mv2 = q ΔV，即为电场力的动能定理。该原理应用于粒子加速器以及阴极射线管中电子速度的确定。
10. Conservation of Energy in Fields | 场中的能量守恒
Both gravitational and electrostatic forces are conservative; therefore, the total mechanical energy (kinetic + potential) of a body moving under these forces remains constant, provided no external work is done. For a satellite, ½mv2 - GMm/r = constant. For a charged particle in an electric potential, ½mv2 + qV = constant. This conservation principle enables us to calculate speeds at different positions without knowing the detailed path.
引力和静电力均为保守力；因此，若无外力做功，物体在这些力作用下运动时总机械能（动能+势能）守恒。对人造卫星：½mv2 - GMm/r = 常数。对电场中的带电粒子：½mv2 + qV = 常数。利用该守恒原理，无需了解具体路径即可计算不同位置的速度。
Problems involving escape velocity, motion in electric potentials, or transfer orbits can all be tackled using energy methods. The concept of potential also simplifies the calculation of work done: instead of integrating force over a path, we simply evaluate m ΔVg or q ΔV.
涉及逃逸速度、电势中的运动或轨道转移的问题均可借助能量方法解决。势的概念还简化了做功的计算：无需沿路径积分力，只需计算m ΔVg 或 q ΔV 即可。
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