📚 AQA Physics PH04: Fields and Further Mechanics Revision Guide | AQA物理PH04：场与进阶力学复习指南
This revision guide covers the key topics of the AQA International Physics PH04 unit, focusing on fields and further mechanics. It is designed to consolidate your understanding of circular motion, simple harmonic motion, gravitational, electric and magnetic fields, electromagnetic induction, and alternating currents, helping you prepare effectively for the June 2023 examination.
本复习指南涵盖AQA国际物理PH04单元的核心主题，重点在于场与进阶力学。它旨在帮助你巩固对圆周运动、简谐运动、引力场、电场、磁场、电磁感应以及交流电的理解，从而为2023年6月考试做好充分准备。
1. Circular Motion | 圆周运动
Circular motion describes the motion of an object travelling along a circular path. The angular displacement θ is measured in radians, and the angular velocity ω is the rate of change of angular displacement, given by ω = θ/t. The linear speed v is related to the angular velocity by v = ωr, where r is the radius of the circle.
圆周运动描述物体沿圆形路径的运动。角位移θ以弧度为单位，角速度ω是角位移的变化率，公式为ω = θ/t。线速度v与角速度的关系为v = ωr，其中r是圆的半径。
For uniform circular motion, the object moves with constant speed, but its velocity changes direction continuously. This requires a centripetal acceleration directed towards the centre of the circle, given by:
对于匀速圆周运动，物体以恒定速率运动，但速度方向不断改变。这需要指向圆心的向心加速度，公式为：
a = v²/r = ω²r
The centripetal force is the resultant force causing this acceleration, calculated using F = mv²/r or F = mω²r. For a satellite orbiting a planet, the gravitational force provides the centripetal force, allowing us to equate gravitational and centripetal equations.
向心力是产生这种加速度的合力，可用F = mv²/r或F = mω²r计算。对于绕行星运行的卫星，引力提供向心力，使我们可以将引力方程与向心力方程相等。
Vertical circular motion, such as a ball on a string, involves varying speed and tension. At the top of the circle, the minimum speed required to maintain the circular path is determined by setting the centripetal acceleration equal to g, giving v = √(rg).
竖直圆周运动，如绳子上的小球，涉及速度和张力的变化。在圆周顶部，维持圆周路径所需的最小速度通过令向心加速度等于g来确定，得到v = √(rg)。
2. Simple Harmonic Motion | 简谐运动
Simple harmonic motion (SHM) is a type of periodic oscillation where the restoring force is directly proportional to the displacement and acts in the opposite direction. The defining equation is a = -ω²x, where a is acceleration, x is displacement from equilibrium, and ω is the angular frequency.
简谐运动（SHM）是一种周期性的振动，其恢复力与位移成正比且方向相反。定义方程为a = -ω²x，其中a是加速度，x是相对于平衡位置的位移，ω是角频率。
The displacement of an object in SHM can be expressed as x = A cos(ωt) or x = A sin(ωt), where A is the amplitude. The velocity and acceleration are given by v = -Aω sin(ωt) and a = -Aω² cos(ωt). The maximum speed is Aω, and the maximum acceleration is Aω².
简谐运动中物体的位移可表示为x = A cos(ωt)或x = A sin(ωt)，其中A是振幅。速度和加速度分别为v = -Aω sin(ωt)和a = -Aω² cos(ωt)。最大速度为Aω，最大加速度为Aω²。
Energy in SHM is conserved, oscillating between kinetic energy and potential energy. The total energy is given by E = ½mA²ω², which remains constant. At equilibrium, energy is entirely kinetic; at maximum displacement, it is entirely potential.
简谐运动中的能量守恒，在动能和势能之间振荡。总能量为E = ½mA²ω²，保持不变。在平衡位置，能量全部为动能；在最大位移处，能量全部为势能。
Damping reduces the amplitude over time due to resistive forces. Critical damping returns the system to equilibrium in the shortest time without oscillation. Resonance occurs when the driving frequency equals the natural frequency, causing maximum energy transfer and large amplitudes.
阻尼因阻力作用随时间减小振幅。临界阻尼使系统在最短时间内回到平衡而不会振荡。当驱动频率等于固有频率时发生共振，导致能量传递最大且振幅很大。
3. Gravitational Fields | 引力场
A gravitational field is a region where a mass experiences a force due to the presence of another mass. Newton’s law of gravitation states that the attractive force between two point masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between them:
引力场是质量因另一质量的存在而受到力的区域。牛顿万有引力定律指出，两个点质量之间的吸引力与质量的乘积成正比，与它们之间距离的平方成反比：
F = Gm₁m₂/r²
where G is the gravitational constant (6.67 × 10⁻¹¹ N m² kg⁻²). Gravitational field strength g at a point is defined as the force per unit mass, g = F/m. For a point mass M, the field strength at a distance r is given by g = GM/r².
其中G是引力常量（6.67 × 10⁻¹¹ N·m²·kg⁻²）。引力场强度g定义为单位质量所受的力，g = F/m。对于点质量M，在距离r处的场强为g = GM/r²。
Gravitational potential V at a point is the work done per unit mass in bringing a mass from infinity to that point. It is given by V = -GM/r, measured in J kg⁻¹. The potential is negative because work is done by the gravitational field as an object approaches from infinity.
某点的引力势V是将单位质量从无穷远处移至该点所做的功，公式为V = -GM/r，单位为J·kg⁻¹。由于物体从无穷远接近时引力场做功，所以势能为负。
Satellites in orbit obey Kepler’s laws. The orbital speed of a satellite in a circular orbit is found by equating gravitational force to centripetal force: GMm/r² = mv²/r, giving v = √(GM/r). The time period T satisfies T² = (4π²/GM)r³, which is Kepler’s third law.
轨道上的卫星遵循开普勒定律。圆形轨道上卫星的轨道速度通过令引力等于向心力得出：GMm/r² = mv²/r，得到v = √(GM/r)。周期T满足T² = (4π²/GM)r³，即开普勒第三定律。
Escape velocity is the minimum speed needed for an object to escape a gravitational field from a given point, given by vₑ = √(2GM/r) or vₑ = √(2gR). This is independent of the mass of the escaping object.
逃逸速度是物体从某点逃离引力场所需的最小速度，公式为vₑ = √(2GM/r)或vₑ = √(2gR)。这与逃离物体的质量无关。
4. Electric Fields | 电场
An electric field exists in a region where a charge experiences an electric force. Coulomb’s law describes the force between two point charges:
电场存在于电荷受到电力的区域。库仑定律描述两个点电荷之间的力：
F = kQ₁Q₂/r² = Q₁Q₂/(4πε₀r²)
where k = 1/(4πε₀) and ε₀ is the permittivity of free space (8.85 × 10⁻¹² F m⁻¹). The electric field strength E is defined as force per unit positive charge, E = F/Q. For a point charge Q, E = kQ/r².
其中k = 1/(4πε₀)，ε₀是真空介电常数（8.85 × 10⁻¹² F·m⁻¹）。电场强度E定义为单位正电荷所受的力，E = F/Q。对于点电荷Q，E = kQ/r²。
Electric potential V is the work done per unit charge in bringing a positive test charge from infinity to a point. For a point charge, V = kQ/r, measured in volts (J C⁻¹). Unlike gravitational potential, electric potential can be positive or negative depending on the sign of the charge.
电势V是将正试探电荷从无穷远移至某点所做的功，对于点电荷，V = kQ/r，单位为伏特（J·C⁻¹）。与引力势不同，电势可正可负，取决于电荷的正负。
In a uniform electric field, such as between two parallel plates, the field strength is related to the potential difference by E = V/d, where d is the plate separation. The force on a charge in a uniform field is F = qE.
在均匀电场中，如两块平行板之间，场强与电势差的关系为E = V/d，其中d是板间距。静止电荷在均匀电场中的力为F = qE。
Equipotential surfaces are surfaces of constant potential. Field lines are always perpendicular to equipotential surfaces, pointing from higher to lower potential for positive charges.
等势面是电势恒定的面。电场线始终垂直于等势面，对于正电荷，电场线指向电势降低的方向。
5. Magnetic Fields | 磁场
A magnetic field is a region where a moving charge or a current-carrying conductor experiences a magnetic force. Magnetic flux density B is a measure of the strength of the magnetic field, with the unit tesla (T). The force on a straight conductor of length L carrying current I in a magnetic field is given by:
磁场是移动电荷或载流导线受到磁力的区域。磁通密度B是磁场强度的量度，单位为特斯拉（T）。长度为L、电流为I的直导线在磁场中受到的力为：
F = BIL sinθ
where θ is the angle between the wire and the magnetic field direction. When θ = 90°, the force is maximum, F = BIL.
其中θ是导线与磁场方向之间的夹角。当θ = 90°时，力最大，F = BIL。
The force on a single charge q moving with velocity v in a magnetic field is F = qvB sinθ. This force is always perpendicular to both the velocity and the magnetic field, causing charged particles to follow circular paths. The radius of the path is given by r = mv/(qB).
单个电荷q以速度v在磁场中运动所受的力为F = qvB sinθ。该力始终垂直于速度和磁场，导致带电粒子做圆周运动。轨道半径为r = mv/(qB)。
Magnetic flux Φ through an area A is defined as Φ = BA cosθ, where θ is the angle between the field and the normal to the area. The unit of flux is the weber (Wb), where 1 Wb = 1 T m².
穿过面积A的磁通量Φ定义为Φ = BA cosθ，其中θ是磁场与面积法线之间的夹角。磁通量的单位是韦伯（Wb），1 Wb = 1 T·m²。
The direction of the force on a current-carrying conductor can be found using Fleming’s left-hand rule. For a charge moving in a magnetic field, the right-hand rule determines the direction of the force.
载流导线受力方向可用弗莱明左手定则判断。对于在磁场中运动的电荷，可用右手定则确定力的方向。
6. Electromagnetic Induction | 电磁感应
Electromagnetic induction is the process of generating an electromotive force (emf) in a conductor when the magnetic flux through it changes. Faraday’s law states that the induced emf is equal to the rate of change of magnetic flux linkage:
电磁感应是当穿过导体的磁通量变化时在导体中产生电动势的过程。法拉第定律指出，感应电动势等于磁通链的变化率：
ε = -N dΦ/dt
where N is the number of turns in the coil and Φ is the magnetic flux through each turn. The negative sign comes from Lenz’s law, which states that the induced current flows in a direction that opposes the change in flux that produced it.
其中N是线圈匝数，Φ是穿过每一匝的磁通量。负号来自楞次定律，即感应电流的方向总是反抗产生它的磁通量变化。
For a coil of area A rotating in a uniform magnetic field with angular velocity ω, the induced emf is given by ε = BANω sin(ωt). This produces an alternating current.
对于在均匀磁场中以角速度ω旋转的面积为A的线圈，感应电动势为ε = BANω sin(ωt)。这产生交流电。
Transformers use electromagnetic induction to change the voltage of an alternating current. The ratio of voltages is equal to the ratio of the number of turns: Vₛ/Vₚ = Nₛ/Nₚ. For an ideal transformer, the power input equals the power output, so VₚIₚ = VₛIₛ.
变压器利用电磁感应改变交流电压。电压之比等于匝数之比：Vₛ/Vₚ = Nₛ/Nₚ。对于理想变压器，输入功率等于输出功率，即VₚIₚ = VₛIₛ。
Lenz’s law is a consequence of the conservation of energy. If the induced current aided the change in flux, energy would be created from nothing, violating energy conservation.
楞次定律是能量守恒的结果。如果感应电流促进磁通量的变化，能量就会凭空产生，违反能量守恒定律。
7. Alternating Currents | 交流电
An alternating current (AC) is an electric current that periodically reverses direction. A sinusoidal AC voltage can be described by V = V₀ sin(2πft), where V₀ is the peak voltage, f is the frequency, and t is time.
交流电（AC）是方向周期性反转的电流。正弦交流电压可表示为V = V₀ sin(2πft)，其中V₀是峰值电压，f是频率，t是时间。
The root-mean-square (rms) value of an alternating voltage is the direct voltage that dissipates the same power in a resistor. For a sinusoidal waveform, V_rms = V₀/√2 and I_rms = I₀/√2.
交流电压的均方根（rms）值是在电阻中耗散相同功率的直流电压。对于正弦波形，V_rms = V₀/√2，I_rms = I₀/√2。
Average power in an AC circuit is given by P = V_rms I_rms. Mains electricity in the UK is typically 230 V rms at 50 Hz, meaning the peak voltage is approximately 325 V.
交流电路中的平均功率为P = V_rms I_rms。英国市电通常为230 V（有效值），频率50 Hz，峰值电压约为325 V。
Capacitors and inductors in AC circuits cause phase differences between voltage and current. In a purely capacitive circuit, current leads voltage by 90°; in a purely inductive circuit, current lags voltage by 90°. Resistive components maintain voltage and current in phase.
交流电路中的电容和电感会导致电压与电流之间存在相位差。在纯电容电路中，电流超前电压90°；在纯电感电路中，电流滞后电压90°。电阻元件使电压与电流同相。
Rectification is the process of converting AC to DC. A half-wave rectifier allows only one half of the AC cycle to pass, while a full-wave rectifier (using four diodes in a bridge) uses both halves, producing a pulsating DC output.
整流是将交流转换为直流的过程。半波整流只允许交流周期的一半通过，而全波整流（使用四个二极管组成的桥式电路）利用两个半波，产生脉动的直流输出。
8. Comparing Gravitational and Electric Fields | 引力场与电场的比较
Gravitational and electric fields share many similarities. Both obey inverse square laws, and the field strength in both cases is defined as force per unit property (mass for gravity, charge for electricity). The potential in both cases decreases with distance, though gravitational potential is always negative while electric potential depends on charge sign.
引力场与电场有许多相似之处。两者都遵循平方反比定律，场强都定义为单位属性的力（引力为质量，电力为电荷）。两者的势都随距离减小，但引力势始终为负，而电势取决于电荷的正负。
| Property | Gravitational | Electric |
| Field strength | g = F/m = GM/r² | E = F/q = kQ/r² |
| Force law | F = Gm₁m₂/r² | F = kQ₁Q₂/r² |
| Potential | V = -GM/r | V = kQ/r |
| Force type | Always attractive | Attractive or repulsive |
| Property producing field | Mass | Charge |
The key difference is that gravitational forces are always attractive, because there is no negative mass. Electric forces can be attractive or repulsive because charges can be positive or negative. This means electric field lines can begin on positive charges and end on negative charges, while gravitational field lines always point toward mass.
关键区别是引力总是吸引力，因为没有负质量。电力可以吸引或排斥，因为电荷有正负。这意味着电场线从正电荷开始，在负电荷结束，而引力场线总是指向质量。
Another difference is the strength of the forces. Gravitational forces are extremely weak compared to electric forces. The gravitational constant G is very small, while the Coulomb constant k is very large, so electric forces dominate at the atomic scale.
另一个区别是力的强度。引力与电力相比极其微弱。引力常量G非常小，而库仑常量k非常大，因此在原子尺度上电力占据主导。
Both fields are conservative, meaning the work done moving a particle around a closed path is zero. This allows the definition of potential and potential energy in both fields.
两者都是保守场，这意味着粒子沿闭合路径移动所做的功为零。这允许在两个场中定义势和势能。
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