📚 Fields (HL) – Gravitational, Electric and Magnetic Fields | 场（HL）——引力场、电场与磁场
Fields provide a powerful way to describe how objects interact without direct contact. In IB Physics HL, we study gravitational, electric, and magnetic fields, all of which are vector fields that exert forces over a distance. This article covers the essential concepts, equations, and applications needed to master the Fields topic at Higher Level, from Newton's law of gravitation to Lorentz force and orbital mechanics.
场是描述物体间非接触相互作用的强大工具。在IB物理高级课程中，我们研究引力场、电场和磁场，它们都是能超距施加作用力的矢量场。本文涵盖掌握高级水平“场”这一主题所需的核心概念、方程和应用，从牛顿引力定律到洛伦兹力以及轨道力学。
1. What is a Field? | 什么是场？
A field is a region of space where a force is experienced by a test object with the appropriate property – mass for gravity, charge for electricity, and moving charge or current for magnetism. Fields are represented by field lines that show direction and relative strength, and they can be scalar fields (potential) or vector fields (force, field strength).
场是空间中一个区域，在那里具有相应属性的检验物体（引力对应质量、电场对应电荷、磁场对应运动电荷或电流）会受到力。场用场线来表示方向和相对强弱，并且可以是标量场（势）或矢量场（力、场强度）。
2. Gravitational Fields: Force and Field Strength | 引力场：力与场强
The gravitational force between two point masses is given by Newton's law of gravitation: F = G M m / r², where G = 6.67 × 10⁻¹¹ N m² kg⁻². The gravitational field strength g at a point is the force per unit mass experienced by a small test mass: g = F/m. For a point mass M, this becomes g = G M / r², directed radially inward.
两个质点之间的引力由牛顿引力定律给出：F = G M m / r²，其中 G = 6.67 × 10⁻¹¹ N m² kg⁻²。某点的引力场强度 g 是小检验质量每单位质量受到的力：g = F/m。对于点质量 M，这写作 g = G M / r²，方向沿径向向内。
3. Gravitational Potential | 引力势
Gravitational potential Vg at a point is the work done per unit mass in bringing a test mass from infinity to that point. For a point mass M, Vg = -G M / r. The negative sign indicates that work is done by the field when a mass moves closer. Potential energy is U = m Vg. The potential gradient gives the field strength: g = -dV_g/dr.
某点的引力势 Vg 是将单位质量的检验质量从无穷远移到该点外力所做的功。点质量 M 的引力势为 Vg = -G M / r。负号表示质量靠近时场做正功。势能为 U = m Vg。势的梯度给出场强：g = -dV_g/dr。
4. Electric Fields: Force and Field Strength | 电场：力与场强
Coulomb's law describes the force between two point charges: F = k Q q / r², where k = 8.99 × 10⁹ N m² C⁻². The electric field strength E is the force per unit positive charge: E = F/q. For a point charge Q, E = k Q / r², pointing away from a positive charge and toward a negative charge.
库仑定律描述两点电荷之间的力：F = k Q q / r²，其中 k = 8.99 × 10⁹ N m² C⁻²。电场强度 E 是每单位正电荷受到的力：E = F/q。对于点电荷 Q，E = k Q / r²，方向远离正电荷、指向负电荷。
5. Electric Potential | 电势
Electric potential Ve is the work done per unit charge in bringing a test charge from infinity to that point. For a point charge Q, Ve = k Q / r. The relationship between field and potential is E = -dV_e/dr. The potential energy of a charge q in a potential Ve is U = q Ve.
电势 Ve 是将单位正电荷从无穷远移到该点外力所做的功。点电荷 Q 的电势为 Ve = k Q / r。场与势的关系是 E = -dV_e/dr。电荷 q 在电势 Ve 中的电势能为 U = q Ve。
6. Magnetic Fields: Force on a Moving Charge | 磁场：运动电荷受力
A magnetic field B exerts a force on a moving charge according to F = q v B sin θ, where θ is the angle between velocity and field. In vector form, F = q (v × B). The direction is given by the right‑hand rule for positive charges. The force is always perpendicular to both v and B, so a magnetic field does no work on a moving charge.
磁场 B 对运动电荷施加的力遵循 F = q v B sin θ，其中 θ 是速度与磁场方向的夹角。矢量形式为 F = q (v × B)。方向用正电荷的右手定则判定。力始终垂直于 v 和 B，因此磁场不对运动电荷做功。
7. Magnetic Fields: Force on a Current‑Carrying Conductor | 磁场：载流导体受力
A current‑carrying wire of length L in a uniform magnetic field experiences a force F = B I L sin θ, where I is the current and θ the angle between the wire and the field. The direction is given by Fleming's left‑hand rule. This principle is used in electric motors and moving‑coil galvanometers.
长度为 L 的载流直导线在匀强磁场中会受到力 F = B I L sin θ，其中 I 为电流，θ 为导线与磁场夹角。方向用左手定则判断。这一原理应用于电动机和动圈式电流计。
8. Combined Fields – Lorentz Force | 复合场——洛伦兹力
When a charged particle moves through both electric and magnetic fields, the total electromagnetic force is the Lorentz force: F = q E + q (v × B). This can be used to select particles of a specific velocity in a velocity selector, where electric and magnetic forces balance: q E = q v B, giving v = E / B.
当带电粒子同时在电场和磁场中运动时，总电磁力为洛伦兹力：F = q E + q (v × B)。这一关系可用于速度选择器中选出特定速度的粒子，此时电场力与磁场力平衡：q E = q v B，得出 v = E / B。
9. Equipotential Surfaces and Field Lines | 等势面与场线
Equipotential surfaces are surfaces where the potential is constant. No work is done moving a test mass or charge along an equipotential. Field lines are always perpendicular to equipotential surfaces. In a radial field, equipotentials are spheres; in a uniform field, they are parallel planes. The spacing of equipotentials indicates field strength: closer spacing means stronger field.
等势面是势处处相等的曲面。检验质量或电荷沿等势面移动时不做功。场线总是垂直于等势面。在径向场中，等势面是球面；在均匀场中，等势面是平行平面。等势面的间距反映场强：间距越小场越强。
10. Motion in Uniform Fields | 均匀场中的运动
In a uniform electric field, a charged particle follows a parabolic trajectory, analogous to projectile motion under gravity. The acceleration is constant: a = q E / m. In a uniform magnetic field, a charged particle moves in a circular path if its velocity is perpendicular to B, with radius r = m v / (q B). The period is T = 2π m / (q B), independent of speed.
在匀强电场中，带电粒子轨迹为抛物线，类似于重力场中的抛体运动。加速度恒定：a = q E / m。在匀强磁场中，若速度垂直于 B，带电粒子做匀速圆周运动，半径 r = m v / (q B)，周期 T = 2π m / (q B)，与速率无关。
11. Orbital Motion under Gravity | 引力下的轨道运动
For a satellite in a circular orbit around a planet, the centripetal force is provided by gravity: G M m / r² = m v² / r, giving orbital speed v = √G M / r. Kepler's third law relates period and radius: T² ∝ r³. The total energy of an orbiting satellite is E = -G M m / (2r), and escape speed is vesc = √2G M / r.
对于绕行星做圆周轨道运动的卫星，万有引力提供向心力：G M m / r² = m v² / r，可得轨道速度 v = √G M / r。开普勒第三定律联系周期与半径：T² ∝ r³。轨道卫星的总机械能为 E = -G M m / (2r)，逃逸速度为 vesc = √2G M / r。
12. Comparing Gravitational and Electric Fields | 引力场与电场的比较
Both gravitational and electric fields follow inverse‑square laws and have associated potentials that vary as 1/r. However, gravity is always attractive, while electric forces can be attractive or repulsive. The gravitational constant G is much smaller than the Coulomb constant k, making gravity extremely weak compared to electromagnetic forces. Magnetic fields are different in that they only act on moving charges and are non‑conservative in the sense that they do no work.
引力场和电场都遵循平方反比定律，相应的势均按 1/r 变化。然而，引力总是吸引的，而电力可以是吸引或排斥。引力常量 G 远小于库仑常量 k，使得引力远弱于电磁力。磁场则不同，它只作用于运动电荷，并且因其不做功而是非保守力。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图