📚 Electric and Magnetic Fields | 电场与磁场
Electric and magnetic fields are cornerstones of A-Level physics. They explain forces between charges, the motion of particles in accelerators, and the principles behind countless technologies. Starting from Coulomb's law, you will build an understanding of electric field strength, potential, and particle dynamics, then explore magnetic forces on currents and moving charges, culminating in devices such as the mass spectrometer, cyclotron, and Hall probe.
电场与磁场是 A-Level 物理的基石，它们解释了电荷间的相互作用、粒子在加速器中的运动以及无数技术背后的原理。从库仑定律开始，你将逐步掌握电场强度、电势和粒子动力学，继而探索磁场对电流和运动电荷的作用，并最终理解质谱仪、回旋加速器和霍尔探针等设备。
1. Electric Charge and Coulomb's Law | 电荷与库仑定律
Electric charge is a scalar property of matter that appears as positive or negative. Like charges repel and opposite charges attract. The force between two stationary point charges is given by Coulomb's law, which states that the force is directly proportional to the product of the charges and inversely proportional to the square of their separation.
电荷是物质的一种标量属性，分为正电荷和负电荷。同性相斥，异性相吸。两个静止点电荷之间的力由库仑定律描述：力的大小与电荷量的乘积成正比，与它们之间距离的平方成反比。
F = k Q₁ Q₂ / r²
n k = 1/(4πε₀), where ε₀ is the permittivity of free space, ε₀ ≈ 8.85 × 10⁻¹² F·m⁻¹. Coulomb' strictly to point charges in a vacuum; in a material medium, ε₀ is replaced by ε = ε_r ε₀.
这里 F 是力，Q₁ 和 Q₂ 是电荷量，r 是距离，k 是库仑常数，k ≈ 8.99 × 10⁹ N·m²·C⁻²。该常数也可表示为 k = 1/(4πε₀)，其中 ε₀ 是真空介电常数，ε₀ ≈ 8.85 × 10⁻¹² F·m⁻¹。库仑定律严格适用于真空中的点电荷；在介质中，ε₀ 应替换为 ε = ε_r ε₀。
Because force is a vector, when more than two charges are present you must add the individual forces vectorially. This procedure is called superposition. A useful numerical check is that charges of +2.0 μC and -3.0 μC separated by 5.0 cm in air exert a force of about 21.6 N, and the attraction is indicated by the opposite signs of the charges.
由于力是矢量，当存在两个以上电荷时，必须把各个力按矢量相加，这一过程称为叠加原理。一个有用的数值检验是：空气中相距 5.0 cm 的 +2.0 μC 与 -3.0 μC 电荷之间的作用力约为 21.6 N，而电荷符号相反表示彼此吸引。
2. Electric Field Strength | 电场强度
int, so E = F/Q. It is a vector quantity, and its direction is the direction of the force on a positive test charge. The SI unit is newton per coulomb, N·C⁻¹, which is identical to volt per metre, V·m⁻¹.
电场是电荷在其中受到电场力的区域。电场强度 E 定义为某点处单位正电荷所受的力，因此 E = F/Q。它是矢量，方向与正试探电荷所受力的方向相同。国际单位是牛顿每库仑，N·C⁻¹，与伏特每米 V·m⁻¹ 等价。
E = F / Q
he field of a positive point charge points radially outwards, while that of a negative charge points radially inwards. Where several charges are present, the resultant field is the vector sum of the individual fields.
对于点电荷 Q，距离 r 处的场强为 E = kQ/r²，或 E = Q/(4πε₀r²)。正点电荷的电场沿径向向外，负点电荷的电场沿径向向内。当存在多个电荷时，合场强等于各个场强的矢量和。
ield is represented by equally spaced parallel lines. In a uniform field between parallel plates, the magnitude is also given by E = V/d, where V is the potential difference and d is the plate separation.
电场线是一种直观工具：其方向表示电场方向，疏密表示场强大小，且电场线永不相交。匀强电场用等间距的平行线表示。在平行板之间的匀强电场中，场强大小也可由 E = V/d 给出，其中 V 是电势差，d 是板间距。
3. Electric Potential and Potential Difference | 电势与电势差
d its unit is the volt, where 1 V = 1 J·C⁻¹. For a point charge Q, the potential at distance r is V = kQ/r. Potential may be positive or negative, but potential difference is what drives charge flow.
某点的电势 V 是把单位正电荷从无穷远处移到该点所做的功。它是标量，单位是伏特，1 V = 1 J·C⁻¹。对于点电荷 Q，距离 r 处的电势为 V = kQ/r。电势可以为正也可以为负，但驱动电荷流动的是电势差。
V = W / Q Ep = QV
n them, V = W/Q. A charge Q moved through a potential difference V gains or loses energy Ep re always perpendicular to field lines, and no work is done when a charge moves along an equipotential surface.
两点之间的电势差是把单位电荷在两点间移动时所做的功，V = W/Q。电荷 Q 通过电势差 V 时会获得或失去能量 Ep = QV。正电荷自然地由高电势移向低电势，负电荷则相反。等势面总是与电场线垂直，电荷沿等势面移动时电场力不做功。
, where d is the distance measured along the field direction. This relationship is central to problems involving parallel plates and charged particle beams.
电场方向指向电势降低的方向。在匀强电场中，场强大小等于电势梯度：E = V/d，其中 d 是沿电场方向测量的距离。这一关系是解决平行板和带电粒子束问题的核心。
4. Uniform Electric Fields and Field Lines | 匀强电场与电场线
A uniform electric field can be produced between two parallel conducting plates connected to a d.c. supply. The field lines are straight, parallel, and equally spaced, running from the positive plate to the negative plate. Near the edges the field becomes non-uniform, so ideal parallel-plate calculations assume the central region.
匀强电场可由两块接直流电源的平行金属板产生。电场线是直线，彼此平行且等间距，从正极板指向负极板。在边缘附近电场变得不均匀，因此理想平行板计算通常只考虑中央区域。
E = V / d
rce is independent of position, so the acceleration is constant and the motion can be analysed using the equations of uniformly accelerated motion. The potential difference between the plates and the separation d determine the field strength: doubling V doubles E, while doubling d halves E.
在该匀强电场中，电荷 Q 受到恒定的力 F = QE。该力与位置无关，因此加速度恒定，运动可用匀加速运动方程分析。两板间的电势差与间距 d 决定场强：V 加倍则 E 加倍，d 加倍则 E 减半。
Field lines and equipotentials provide a complete picture. Field lines are perpendicular to the plates and to every equipotential surface. The spacing of equipotentials reflects the field strength: close equipotentials mean a strong field. This is why a charged particle moving parallel to the plates is deflected, while one moving along an equipotential requires no work.
电场线与等势面提供了完整图景。电场线垂直于极板，也垂直于每一个等势面。等势面的疏密反映场强：等势面越密，场强越大。这解释了为什么平行于极板运动的带电粒子会发生偏转，而沿等势面运动的电荷不需要做功。
5. Motion of Charged Particles in Electric Fields | 带电粒子在电场中的运动
vided the particle starts from rest. For charge Q and mass m, the speed gained is found from QV = ½mv², giving v = √2QV/m. This principle is used in electron guns and particle accelerators.
当带电粒子通过电势差 V 被加速时，若粒子由静止开始，则损失的电势能等于获得的动能。对于电荷量 Q 和质量 m，由 QV = ½mv² 可得速度 v = √2QV/m。电子枪和粒子加速器都应用了这一原理。
QV = ½ m v² v = √2QV/m
acceleration is a = QE/m = QV/(md). If the initial horizontal speed is v and the plate length is L, the time spent between the plates is t = L/v. The vertical deflection is y = ½at² = QVL²/(2mdv²).
在横向方向上，垂直进入匀强电场的带电粒子行为类似抛体。加速度为 a = QE/m = QV/(md)。若初始水平速度为 v，板长为 L，则在板间运动时间 t = L/v。竖直偏转为 y = ½at² = QVL²/(2mdv²)。
After leaving the plates, the particle travels in a straight line along its new velocity vector. The direction of deflection reveals the sign of the charge: a positive charge is deflected towards the negative plate, and a negative charge towards the positive plate. In the cathode-ray oscilloscope, this deflection system controls the bright spot on the screen.
离开极板后，粒子沿新的速度矢量做直线运动。偏转方向可以判断电荷的正负：正电荷偏向负极板，负电荷偏向正极板。在阴极射线示波器中，正是这一偏转系统控制屏幕上的亮点位置。
6. Magnetic Fields and Magnetic Flux Density | 磁场与磁感应强度
sity B measures the strength of the field and is defined by the force on a current-carrying conductor. Its unit is the tesla, T, where 1 T = 1 N·A⁻¹·m⁻¹.
磁场是对运动电荷、载流导体或磁性材料施加磁力的区域。磁感应强度 B 衡量磁场强弱，由载流导体所受的力来定义。其单位是特斯拉 T，1 T = 1 N·A⁻¹·m⁻¹。
F = B I L sin θ B = F / (I L sin θ)
Magnetic field lines form closed loops: they emerge from the north pole of a magnet and enter the south pole outside the magnet, and continue inside the magnet from south to north. Unlike electric field lines, they have no starting or ending points. Closer field lines indicate a stronger field, and a uniform magnetic field is shown by equally spaced parallel lines.
磁感线形成闭合回路：在磁体外部从北极出发进入南极，在磁体内部从南极回到北极。与电场线不同，磁感线没有起点和终点。磁感线越密表示磁场越强，匀强磁场用等间距的平行线表示。
A current-carrying wire also produces a magnetic field. The field around a long straight wire forms concentric circles, and the direction is given by the right-hand grip rule: point the thumb along the conventional current, and the fingers curl in the direction of the field. A solenoid produces a field similar to that of a bar magnet, and its strength increases with current and number of turns per unit length.
载流导线也会产生磁场。长直导线周围的磁场形成同心圆，方向由右手螺旋定则确定：拇指指向常规电流方向，四指弯曲方向即为磁场方向。螺线管产生的磁场与条形磁铁相似，其强度随电流和单位长度匝数的增加而增大。
7. Magnetic Force on a Current-Carrying Conductor | 磁场对通电导体的作用力
ield. The force is maximum when the current is perpendicular to the field and zero when the current is parallel to the field. The magnitude is given by F = BIL sin θ.
当载流导体置于磁场中时，只要电流有垂直于磁场的分量，导体就会受到磁力。电流与磁场垂直时力最大，电流与磁场平行时力为零。力的大小由 F = BIL sin θ 给出。
F = B I L sin θ
The direction of the force is found using Fleming's left-hand rule. Hold the thumb, first finger, and second finger mutually perpendicular: the first finger points along the magnetic field from north to south, the second finger points along the conventional current, and the thumb gives the direction of the force, also called the motor effect. The force is always perpendicular to both the current and the magnetic field.
力的方向由弗莱明左手定则确定。让拇指、食指和中指互相垂直：食指指向磁场方向，由 N 极到 S 极；中指指向常规电流方向；拇指即力的方向，也称为电动机效应。该力总是同时垂直于电流和磁场。
A simple application is the electric motor. A rectangular coil carrying current in a magnetic field experiences a pair of forces on opposite sides, producing a turning effect. A split-ring commutator reverses the current every half turn so that the coil continues to rotate in the same direction. Loudspeakers use the same principle: a varying current in a coil in a magnetic field produces a varying force that moves a cone and creates sound.
一个简单应用是电动机。磁场中的矩形载流线圈在相对两边受到一对力，产生转动效应。换向器每半圈反转电流，使线圈持续沿同一方向转动。扬声器使用同一原理：磁场中线圈内的变化电流产生变化的力，推动纸盆运动并产生声音。
8. Magnetic Force on Moving Charges | 磁场对运动电荷的作用力
tween the velocity v and the magnetic flux density B. The force is greatest when the velocity is perpendicular to the field and zero when the charge moves parallel to the field.
磁场中的运动电荷也会受到力。其大小为 F = BQv sin θ，其中 θ 是速度 v 与磁感应强度 B 之间的夹角。当速度垂直于磁场时力最大，当电荷沿磁场方向运动时力为零。
F = B Q v sin θ
For a positive charge, the direction of the force is given by Fleming's left-hand rule, with the second finger pointing along the velocity. For a negative charge such as an electron, the force is in the opposite direction. The magnetic force is always perpendicular to the velocity, so it does no work on the charge and cannot change its speed; it only changes the direction of motion.
对于正电荷，力的方向由弗莱明左手定则确定，中指指向
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