📚 Electric Fields and Capacitance | 电场与电容
This article covers the International A-level physics topic of electric fields and capacitance for AQA. It develops the ideas of Coulomb's law, electric potential, uniform fields, capacitance, charging and discharging, and energy storage in capacitors. The focus is on definitions, equations, graphical analysis, and practical applications.
本文涵盖 AQA 国际 A-level 物理中电场与电容主题。文章逐步讲解库仑定律、电势、匀强电场、电容、充放电过程以及电容器储能，重点关注定义、公式、图像分析和实际应用。
1. Coulomb's Law and Electric Field Strength | 库仑定律与电场强度
Coulomb's law states that the force between two point charges Q₁ and Q₂ separated by distance r in a vacuum is directly proportional to the product of the charges and inversely proportional to the square of the distance.
库仑定律指出，真空中两个点电荷 Q₁ 和 Q₂ 相距 r 时，作用力与电荷乘积成正比，与距离平方成反比。
F = Q₁Q₂ / (4πε₀r²)
The electric field strength E at a point is defined as the force per unit positive charge placed at that point. It is a vector quantity with unit N C⁻¹ or V m⁻¹.
电场强度 E 定义为放在该点的单位正电荷所受的力。它是矢量，单位为 N C⁻¹ 或 V m⁻¹。
E = F / q
d the field lines point radially away from a positive charge or towards a negative charge.
对于点电荷，场强为 E = Q / (4πε₀r²)，电场线从正电荷沿径向向外，或指向负电荷。
2. Electric Potential and Work Done | 电势与做功
ll test charge from infinity to that point. The unit is the volt (V), where 1 V = 1 J C⁻¹.
电势 V 是单位正电荷从无穷远移到该点所做的功。单位是伏特（V），1 V = 1 J C⁻¹。
V = W / Q
potential is a scalar, so the potentials from different charges are added algebraically without vector addition.
对于点电荷，电势为 V = Q / (4πε₀r)。与电场不同，电势是标量，因此不同电荷产生的电势按代数量相加，无需矢量合成。
V = Q / (4πε₀r)
em, so W = QΔV. This links energy transfer to the movement of charge in an electric field.
两点之间的电势差是单位电荷在两点间移动时所做的功，因此 W = QΔV。这建立了电场中电荷移动与能量转移的联系。
3. Uniform Electric Fields and Parallel Plates | 匀强电场与平行板
A uniform electric field exists between two parallel conducting plates connected to a potential difference. Field lines are straight, parallel and evenly spaced, showing constant direction and magnitude.
两块连接电势差的平行导体板之间产生匀强电场。电场线为直线、平行且等间距，表明方向与大小处处相同。
E = V / d
Here V is the potential difference between the plates and d is their separation. The field strength is constant everywhere between the plates, ignoring edge effects.
其中 V 是两板间电势差，d 是板间距。忽略边缘效应时，板间各处场强恒定。
d particle enters the field perpendicular to the plates, its motion can be analysed using projectile equations because the acceleration is constant.
匀强电场中的电荷 q 受到恒力 F = qE。若带电粒子垂直于极板进入电场，由于加速度恒定，其运动可用抛体方程进行分析。
F = qE
4. Capacitance: Definition and Units | 电容：定义与单位
A capacitor is a device that stores charge and energy. Its capacitance C is defined as the charge stored per unit potential difference across it.
电容器是储存电荷和能量的器件。电容 C 定义为电容器两端每单位电势差所储存的电荷。
C = Q / V
The SI unit of capacitance is the farad (F). One farad is one coulomb per volt, but most practical capacitors are in microfarads (µF), nanofarads (nF) or picofarads (pF).
电容的国际单位是法拉（F）。1 法拉等于 1 库仑每伏特，但实际电容器大多为微法（µF）、纳法（nF）或皮法（pF）。
tional to separation d, given by C = ε₀ A / d, where ε₀ is the permittivity of free space.
对于真空平行板电容器，电容与板面积 A 成正比，与板间距 d 成反比，即 C = ε₀ A / d，其中 ε₀ 是真空介电常数。
C = ε₀ A / d
5. Capacitors in Series and Parallel | 电容器的串联与并联
In parallel, capacitors have the same potential difference across them. The total charge stored is the sum of the individual charges, so the total capacitance is the sum of the individual capacitances.
并联时各电容器两端电势差相同。总电荷为各电荷之和，因此总电容为各电容之和。
Ctotal = C₁ + C₂ + C₃
In series, capacitors store the same charge because charge cannot pass through the gap between the plates. The total potential difference is the sum of the individual voltages, so the reciprocal formula applies.
串联时各电容器储存的电荷相同，因为电荷不能穿过极板之间的间隙。总电势差为各电压之和，因此适用倒数公式。
1/Ctotal = 1/C₁ + 1/C₂ + 1/C₃
These rules are opposite to those for resistors: series resistors add directly, while series capacitors combine reciprocally, and parallel capacitors add directly while parallel resistors combine reciprocally.
这些规则与电阻相反：串联电阻直接相加，而串联电容按倒数规律组合；并联电容直接相加，而并联电阻按倒数规律组合。
6. Energy Stored by a Capacitor | 电容器储存的能量
When a capacitor is charged, work is done by the battery to separate positive and negative charge. This work is stored as electric potential energy in the capacitor.
电容器充电时，电池做功分离正负电荷。这些功以电势能形式储存在电容器中。
W = ½ QV = ½ CV² = ½ Q²/C
The factor ½ arises because the potential difference rises linearly from zero to V as charge is transferred, so the average potential difference during charging is V/2.
系数 ½ 来自电势差随电荷转移从零线性增加到 V，因此充电过程中的平均电势差为 V/2。
The energy is stored in the electric field between the plates. In a uniform field, the energy per unit volume, called energy density, is given by ½ ε₀ E².
能量储存在两极板之间的电场中。在匀强电场中，单位体积的能量（能量密度）为 ½ ε₀ E²。
energy density = ½ ε₀ E²
7. Charging a Capacitor Through a Resistor | 电容器经电阻充电
When a capacitor is charged through a resistor from a cell of emf E, the charge Q on the capacitor increases gradually. Initially the current is maximum, and finally it becomes zero when the capacitor voltage equals the cell emf.
电容器通过电阻从电动势为 E 的电源充电时，电容器上的电荷 Q 逐渐增加。开始时电流最大，当电容器电压等于电源电动势时电流最终为零。
Q = Q₀ (1 − e−t/RC)
e series resistance, and C is the capacitance. The product RC is called the time constant τ.
其中 Q₀ = CE 是最终电荷，R 是串联电阻，C 是电容。乘积 RC 称为时间常数 τ。
τ = RC
The charging current follows I = I₀ e−t/RC urrent decays exponentially while the charge rises asymptotically towards its final value.
充电电流遵循 I = I₀ e−t/RC，其中 I₀ = E/R。因此电流按指数衰减，而电荷渐近上升至最终值。
I = I₀ e−t/RC
8. Discharging a Capacitor: Exponential Decay | 电容器放电：指数衰减
When a charged capacitor is discharged through a resistor, the charge, voltage and current all decrease exponentially with time from their initial values.
当已充电的电容器通过电阻放电时，电荷、电压和电流都从初始值按指数规律随时间减小。
Q = Q₀ e−t/RC
V = V₀ e−t/RC
I = I₀ e−t/RC
The negative sign in the exponent indicates decay. The rate of discharge depends only on the product RC, not on the initial charge or voltage.
指数中的负号表示衰减。放电速率只取决于乘积 RC，与初始电荷或电压无关。
After one time constant t = RC, the charge falls to about 37% because e⁻¹ ≈ 0.37. After about 5RC, the capacitor is considered almost fully discharged.
经过一个时间常数 t = RC 后，电荷降至初始值的约 37%，因为 e⁻¹ ≈ 0.37。约 5RC 后
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