📚 Electric and Magnetic Fields | 电场与磁场
Electric and magnetic fields are fundamental concepts in physics that describe the interactions between charged particles and currents. Understanding these fields is essential for explaining a vast array of phenomena, from the attraction of static electricity to the operation of electric motors and generators. These invisible forces shape how charges move and how energy is transferred in circuits and devices.
电场与磁场是描述带电粒子与电流之间相互作用的基本物理概念。理解这些场对于解释从静电吸引到电动机和发电机运行等多种现象至关重要。这些无形的力决定了电荷的运动方式以及电路和器件中的能量传递。
1. Electric Charge and Coulomb's Law | 电荷与库仑定律
Electric charge exists in two types, positive and negative. Like charges repel each other, while opposite charges attract. The electrostatic force between two stationary point charges is quantified by Coulomb's law. The magnitude of the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
电荷有正负两种，同性相斥、异性相吸。两个静止点电荷之间的静电力由库仑定律定量描述。力的大小与电荷量的乘积成正比，与距离的平方成反比。
F = k |q₁ q₂| / r²
Coulomb&# C⁻². The force acts along the straight line joining the two charges. If the charges are of opposite sign, the force is attractive; if they have the same sign, it is repulsive. This law is the foundation for understanding electric interactions.
库仑常量 k = 8.99 × 10⁹ N m² C⁻²。力沿两点电荷连线方向作用。若电荷异号，力为引力；同号则为斥力。该定律是理解电相互作用的基础。
2. Electric Field Strength | 电场强度
An electric field is a region of space in which a stationary electric charge experiences a force. The electric field strength E at a point is defined as the force per unit positive charge placed at that point: E = F / q. Its unit is N C⁻¹, which is equivalent to V m⁻¹.
电场是静止电荷会受到力的空间区域。某点的电场强度 E 定义为该点单位正电荷所受的力：E = F / q，单位为 N C⁻¹，等同于 V m⁻¹。
For a point charge Q, the field strength at a distance r is given by E = kQ / r². The direction of the field is radially outward from a positive charge and radially inward toward a negative charge. Electric field lines are used to represent the field direction and strength, with closer lines indicating a stronger field.
对于点电荷 Q，距离 r 处的电场强度为 E = kQ / r²。正电荷的电场方向沿径向向外，负电荷则向内。电场线用于表示场的方向和强度，线越密场越强。
3. Electric Potential and Potential Energy | 电势与电势能
Electric potential V at a point is the electric potential energy per unit charge: V = U / q. It is a scalar quantity measured in volts (J C⁻¹). The potential due to a point charge Q at a distance r is V = kQ / r, taking the potential at infinity to be zero.
某点的电势 V 是单位电荷在该点的电势能：V = U / q，为标量，单位为伏特（J C⁻¹）。点电荷 Q 在距离 r 处的电势为 V = kQ / r，取无穷远处电势为零。
The potential difference ΔV between two points is the work done per unit charge in moving a charge from one point to the other: W = qΔV. In a uniform electric field, the magnitude of the field is related to the potential gradient by E = ΔV / d, where d is the distance between the equipotential surfaces.
两点间的电势差 ΔV 是移动单位电荷所做的功：W = qΔV。在匀强电场中，场强大小与电势梯度的关系为 E = ΔV / d，其中 d 为等势面之间的距离。
4. Uniform Electric Fields | 匀强电场
A uniform electric field has constant magnitude and direction throughout a region. The most common way to produce a uniform electric field is by applying a potential difference across two parallel conducting plates. The field lines are straight, parallel, and equally spaced, and the field strength is E = V / d, where V is the voltage between the plates and d is their separation.
匀强电场在区域各处的大小和方向均相同。最常见的产生方法是给两块平行导电板施加电势差。电场线为平直、平行且等间距，场强大小为 E = V / d，其中 V 为板间电压，d 为板间距。
A charged particle in such a field experiences a constant force F = qE. If this is the only force, the particle will accelerate uniformly, making the motion analogous to a mass in a uniform gravitational field. This simplicity makes uniform electric fields ideal for studying charged particle dynamics.
处于这种场中的带电粒子受到恒力 F = qE。若该力是唯一的力，粒子将做匀加速运动，类似于均匀重力场中的质点。这种简单性使匀强电场成为研究带电粒子动力学的理想模型。
5. Magnetic Fields and Field Lines | 磁场与磁感线
Magnetic fields are produced by moving charges (electric currents) or by intrinsic magnetic moments of elementary particles. They can be represented by magnetic field lines (also called lines of magnetic flux). These lines emerge from the north pole of a magnet, go through space, enter the south pole, and continue inside the magnet to form closed loops. The direction of the field at any point is the direction of the force on a small north pole placed there.
磁场由运动电荷（电流）或基本粒子的内禀磁矩产生。磁场可用磁感线（磁通线）来表示。磁感线从磁体的 N 极出发，穿越空间进入 S 极，再通过磁体内部形成闭合回路。某点的磁场方向为置于该处的小磁针 N 极的受力方向。
Magnetic field strength is quantified by the magnetic flux density B, measured in tesla (T). One tesla is a strong field; the Earth's magnetic field is about 5 × 10⁻⁵ T. Unlike electric field lines, magnetic field lines never start or end — they are continuous loops, reflecting the absence of magnetic monopoles.
磁场强度用磁感应强度 B 度量，单位为特斯拉（T）。1 T 是一个很强的场；地球磁场约为 5 × 10⁻⁵ T。与电场线不同，磁感线无始无终 —— 它们是连续的闭合曲线，这反映了不存在磁单极的事实。
6. Force on a Moving Charge (Lorentz Force) | 运动电荷在磁场中的受力（洛伦兹力）
When a charged particle moves through a magnetic field with velocity v, it experiences a magnetic force given by the Lorentz force law. The magnitude of this force is F = q v B sinθ, where θ is the angle between v and B. The direction of the force is perpendicular to both v and B, as given by the right-hand rule for a positive charge (or left-hand rule for negative).
当带电粒子以速度 v 在磁场中运动时，会受到洛伦兹力。力的大小为 F = q v B sinθ，θ 为 v 与 B 的夹角。力的方向垂直于 v 和 B 所在的平面，对于正电荷用右手定则判断（负电荷相反）。
Because the magnetic force is always perpendicular to the velocity, it does no work on the particle and cannot change its speed. However, it can change the direction of motion. When v is perpendicular to a uniform B, the force provides a constant centripetal acceleration, leading to circular motion of radius r = mv / (qB). The period of revolution is T = 2πm / (qB), independent of speed. This principle is used in cyclotrons.
由于磁力始终垂直于速度，因而对粒子不做功，不能改变其速率。但可以改变运动方向。当 v 垂直于均匀 B 时，磁力提供恒定的向心加速度，粒子做圆周运动，半径 r = mv / (qB)。旋转周期 T = 2πm / (qB) 与速率无关。这一原理应用于回旋加速器。
7. Force on a Current-Carrying Conductor | 电流在磁场中的受力（安培力）
A current is a flow of charge, so a conductor carrying current in a magnetic field experiences a force that is the sum of the magnetic forces on all moving charges. For a straight wire of length L carrying current I in a uniform magnetic field B, the magnitude of the force is F = B I L sinθ, where θ is the angle between the wire and the field. The direction is given by Fleming's left-hand rule: thumb for force, first finger for field, second finger for current.
电流是电荷的流动，因此载流导线在磁场中受到的力是所有运动电荷所受磁力的总和。对于长度为 L、载流为 I 的直导线在均匀磁场 B 中，受力大小为 F = B I L sinθ，其中 θ 为导线与磁场的夹角。方向用左手定则：拇指为力，食指为场，中指为电流。
This effect is the basis for many electromagnetic devices, such as the electric motor and the moving-coil loudspeaker. For two long, parallel wires separated by distance r, the force per unit length between them is F/L = μ₀ I₁ I₂ / (2πr). The force is attractive when the currents are in the same direction and repulsive when they are opposite.
这一效应是许多电磁设备的基础，如电动机和动圈式扬声器。对于相距 r 的两根长平行导线，单位长度上的作用力为 F/L = μ₀ I₁ I₂ / (2πr)。同向电流相吸，反向电流相斥。
8. Magnetic Fields due to Electric Currents | 电流产生的磁场
Electric currents are the primary source of magnetic fields. For a long, straight wire, the magnetic field lines are concentric circles around the wire. The magnitude of B at a perpendicular distance r from the wire is B = &mu
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