Electric and Magnetic Fields: Combined Effects | 电场与磁场的综合效应

📚 Electric and Magnetic Fields: Combined Effects | 电场与磁场的综合效应

In Edexcel International A-Level Physics, especially Unit 4, you must link electric and magnetic fields to explain the motion of charged particles and electromagnetic induction. Combined-field questions often ask you to identify forces, balance conditions and energy changes in devices such as velocity selectors and cyclotrons.

在 Edexcel 国际 A-Level 物理（特别是 Unit 4）中，你需要将电场与磁场联系起来，解释带电粒子的运动以及电磁感应。复合场题目常要求你判断受力、平衡条件以及速度选择器、回旋加速器等装置中的能量变化。

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1. Defining the Fields | 场的基本定义

An electric field is a region where a charged particle experiences an electric force. The electric field strength E is defined as the force per unit positive charge:

电场是带电粒子受到电场力的区域。电场强度 E 定义为单位正电荷所受的力：

E = F / Q

The unit of E is N C⁻¹ or V m⁻¹. Since electric field strength is a vector, it points in the direction a positive test charge would move.

E 的单位是 N C⁻¹，也等价于 V m⁻¹。电场强度是矢量，方向与正试探电荷的受力方向相同。

A magnetic field is a region where moving charges or permanent magnets experience a magnetic force. Magnetic flux density B is measured in tesla (T), where 1 T = 1 N A⁻¹ m⁻¹. For a charged particle moving perpendicular to a uniform magnetic field, the force is F = BQv.

磁场是运动电荷或永磁体受到磁力的区域。磁通量密度 B 的单位是特斯拉（T），1 T = 1 N A⁻¹ m⁻¹。当带电粒子垂直于均匀磁场运动时，所受磁力为 F = BQv。

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2. Forces on Charged Particles | 带电粒子的受力

When a particle carries charge q and moves with velocity v in both an electric field and a magnetic field, the total Lorentz force is the vector sum:

当粒子带电荷 q，并以速度 v 在电场和磁场中运动时，总的洛伦兹力是两者的矢量和：

F = qE + qvB sin θ

Here θ is the angle between v and B. In many exam questions, E and B are perpendicular, so you must add perpendicular force components rather than simple scalars.

其中 θ 是 v 与 B 的夹角。在许多考题中，电场和磁场相互垂直，因此你必须按垂直分量合成力，而不是简单标量相加。

Use Fleming’s left-hand rule for the magnetic force: thuMb = Motion, First finger = Field, seCond finger = Current. For a positive charge, current direction is the same as v; for a negative charge, reverse the current direction.

磁力方向用弗莱明左手定则：拇指表示运动方向，食指表示磁场，中指表示电流。对正电荷，电流方向与 v 相同；对负电荷，电流方向相反。

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3. Motion in a Uniform Electric Field | 均匀电场中的运动

In a uniform electric field, a charge experiences constant acceleration a = F/m = qE/m. If the particle enters perpendicular to E, it follows a parabolic path, just like a projectile in a gravitational field.

在均匀电场中，电荷受到恒定加速度 a = F/m = qE/m。如果粒子垂直于电场进入，它会走抛物线路径，就像重力场中的抛体运动一样。

The displacement components can be found from the constant-acceleration equations. For an electron in a cathode-ray tube, horizontal motion is uniform while vertical motion is uniformly accelerated by the deflecting plates.

位移分量可由匀加速运动方程求出。以阴极射线管中的电子为例，水平方向是匀速运动，而垂直方向在偏转板作用下做匀加速运动。

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4. Motion in a Uniform Magnetic Field | 均匀磁场中的运动

When a charged particle moves perpendicular to a uniform magnetic field, the magnetic force is always perpendicular to velocity, so it does no work and the particle moves in a circle. The centripetal force is provided by the magnetic force:

当带电粒子垂直于均匀磁场运动时，磁力始终与速度垂直，因此不做功，粒子做圆周运动。向心力由磁力提供：

BQv = mv² / r

This gives the radius of the circular path: r = mv / (BQ). The period is T = 2πm / (BQ), which is independent of speed, so all identical particles with the same charge-to-mass ratio rotate at the same frequency.

由此得到圆周半径 r = mv / (BQ)。周期 T = 2πm / (BQ) 与速度无关，因此具有相同荷质比的同类粒子旋转频率相同。

This property is exploited in particle accelerators and mass spectrometers. The cyclotron frequency is f = BQ / (2πm).

这一性质被用于粒子加速器和质谱仪。回旋加速器的频率为 f = BQ / (2πm)。

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5. Velocity Selector | 速度选择器

A velocity selector uses perpendicular electric and magnetic fields. A charged particle entering the region experiences an electric force qE in one direction and a magnetic force qvB in the opposite direction. When the forces balance, the particle travels in a straight line:

速度选择器使用相互垂直的电场和磁场。带电粒子进入该区域后，受到一个方向的电场力 qE 和相反方向的磁力 qvB。当两力平衡时，粒子沿直线运动：

qE = qvB

Therefore only particles with speed v = E / B pass through undeflected. This speed is independent of charge and mass, which makes the device useful for selecting a specific velocity before a mass spectrometer.

因此只有速度 v = E / B 的粒子才能不偏转地通过。该速度与电荷量和质量无关，这使得它常用于质谱仪前选择特定速度。

If the particle’s speed is too high, the magnetic force dominates and it bends one way; if too low, the electric force dominates and it bends the other way.

如果粒子速度过高，磁力占主导并向一侧偏转；如果速度过低，电场

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