导体与电容器

AP 物理 C:电磁学 · 第 10 主题

训练

讲义词汇表

10.1

导体的静电学

大纲

Learning Objective

Essential Knowledge

10.1.A
Describe the charge distribution within a conductor.

10.1.A.1 An ideal conductor is a material in which electrons are able to move freely.
10.1.A.2 When a conductor is in electrostatic equilibrium, mutual repulsion of excess charge carriers results in those charge carriers residing entirely on the surface of the conductor.
- 10.1.A.2.i In a conductor with a negative net charge, excess electrons reside on the surface of the conductor.
- 10.1.A.2.ii In a conductor with a positive net charge, the surface becomes deficient in electrons, and can be modeled as if positive charge carriers reside on the surface of the conductor.
10.1.A.3 Excess charges will move to the surface of a conductor to create a state of electrostatic equilibrium within the conductor.
- 10.1.A.3.i The time interval over which charges reach electrostatic equilibrium within a conductor is so short as to be negligible.
- 10.1.A.3.ii When a conductor reaches electrostatic equilibrium, all points on the surface of the conductor have the same electric potential, and the conductor becomes an equipotential surface.
- 10.1.A.3.iii The charge density on the surface of a conductor will be greater where there are points or edges compared to planar areas.
10.1.A.4 All excess charges reside on the surface of a conductor, which means there is no net charge in the interior of the conductor, and the electric field is zero within the conductor.
10.1.A.5 The electric field is perpendicular to the outer surface of a conductor.
10.1.A.6 A conductor can be polarized in the presence of an external electric field. This is a consequence of the conductor remaining an equipotential surface.
10.1.A.7 Electrostatic shielding is the process of surrounding an area with a closed, conducting shell to create a region inside the conductor that is free from external electric fields.

来源:美国大学理事会 AP 课程与考试说明

在一个理想的导体(conductor)里,电子自由移动。在一个上放额外的电荷,电荷互相排斥,直到几乎瞬间,它们稳定进入静电平衡(electrostatic equilibrium)。在那个状态导体有四个你必须能够陈述和使用的性质:

导体内部的电场是零。若它不是,自由电子仍会移动。
所有多余的电荷坐在表面上。(负净电荷 = 表面上的额外电子;正 = 那里的电子短缺。)
恰好在外面的场垂直于(perpendicular)表面,大小为 $E=\sigma/\varepsilon_0$。任何平行分量会把电荷沿表面横向推。
整个导体是一个等势面(equipotential surface):每个点,内部和在表面上,都处于相同的电势。

电荷密度在表面急剧弯曲的地方——在点和边缘——最大,所以外面的场在那里最强。在一个外部场里一个导体极化:电荷在它的表面上移动,以便内部保持无场而物体保持一个等势面。用一个闭合的导电壳包围一个区域完全把外部场挡在外——静电屏蔽(electrostatic shielding),法拉第笼(Faraday cage)背后的思想。

词汇表训练

英文	中文	拼音
conductor	导体	dǎo tǐ
electrostatic equilibrium	静电平衡	jìng diàn píng héng
perpendicular	垂直	chuí zhí
equipotential surface	等势面	děng shì miàn
electrostatic shielding	静电屏蔽	jìng diàn píng bì
Faraday cage	法拉第笼	fǎ lā dì lóng

练习卷

10.2

导体间电荷的重新分布

大纲

Learning Objective

Essential Knowledge

10.2.A
Describe the movement of charge and the resulting interactions when conductors physically contact each other.

10.2.A.1 When conductors are in electrical contact, charges will be redistributed such that the surfaces of each conductor are at the same electric potential.
10.2.A.2 Ground is an idealized reference point that has zero electric potential and can absorb or provide an infinite amount of charge without changing its electric potential.
10.2.A.3 Charge can be induced on a conductor by grounding the conductor in the presence of an external electric field.

来源:美国大学理事会 AP 课程与考试说明

当两个导体触碰(或被连线在一起)时,电荷在它们之间流动,直到两个表面达到相同电势——那是停止条件,不是"相等的电荷"。一个更大的球在相同的电势容纳更多电荷($V=kQ/R$),所以它取更大的份额。

Worked example. 一个携带 $+6.0\ \mu\text{C}$ 的半径 $R$ 的小球触碰一个遥远的半径 $2R$ 的球,然后它们分离。相等的电势要求 $\dfrac{kq_1}{R}=\dfrac{kq_2}{2R}$,所以 $q_2=2q_1$。以 $q_1+q_2=6.0\ \mu\text{C}$:$q_1=2.0\ \mu\text{C}$ 而 $q_2=4.0\ \mu\text{C}$。

接地(ground)是一个理想化的零电势参考,它能吸收或供应任何量的电荷。在一个外部电荷附近时给一个导体接地使导体带一个净的感应电荷(induced charge):外部场把一个符号的电荷推向地,而在移除外部电荷之前切断接地线困住其余的。

词汇表训练

英文	中文	拼音
ground	接地	jiē dì
induced charge	感应电荷	gǎn yìng diàn hè
work	功	gōng

练习卷

10.3

电容器

大纲

Learning Objective

Essential Knowledge

10.3.A
Describe the physical properties of a parallel-plate capacitor.

10.3.A.1 A parallel-plate capacitor consists of two separated parallel conducting surfaces that can hold equal amounts of charge with opposite signs.
10.3.A.2 Capacitance relates the magnitude of the charge stored on each plate to the electric potential difference created by the separation of those charges.
- Equation: $C = \dfrac{Q}{\Delta V}$
- 10.3.A.2.i The capacitance of a capacitor depends only on the physical properties of the capacitor, such as the capacitor's shape and the material used to separate the plates.
- 10.3.A.2.ii The capacitance of a parallel-plate capacitor is proportional to the area of one of its plates and inversely proportional to the distance between its plates. The constant of proportionality is the product of the dielectric constant, $\kappa$, of the material between the plates and the electric permittivity of free space, $\varepsilon_0$.
  - Equation: $C = \dfrac{\kappa \varepsilon_0 A}{d}$
10.3.A.3 The electric field between two charged parallel plates with uniformly distributed electric charge, such as in a parallel-plate capacitor, is constant in both magnitude and direction, except near the edges of the plates.
- 10.3.A.3.i The magnitude of the electric field between two charged parallel plates, where the plate separation is much smaller than the dimensions of the plates, can be determined by applying Gauss's law and the principle of superposition.
  - Equation: $E = \dfrac{Q}{\varepsilon_0 A}$
- 10.3.A.3.ii The electric field is proportional to the surface charge density on either plate of the capacitor.
- 10.3.A.3.iii A charged particle between two oppositely charged parallel plates undergoes constant acceleration, and therefore its motion shares characteristics with the projectile motion of an object with mass in the gravitational field near Earth's surface.
10.3.A.4 The electric potential energy stored in a capacitor is equal to the work done by an external force to separate that amount of charge on the capacitor.
10.3.A.5 The electric potential energy stored in a capacitor is described by the equation $U_C = \dfrac{1}{2} Q \Delta V$.

Boundary statement: While other shapes are also able to separate charges, AP Physics C: Electricity & Magnetism only expects the quantitative analysis and description of parallel-plate capacitors, concentric spherical capacitors, and coaxial cylindrical capacitors.

来源:美国大学理事会 AP 课程与考试说明

电容器充电（RC）

一个电容器(capacitor)在被一个间隙分开的两个导体上储存电荷:一个板上 $+Q$、另一个上 $-Q$。它的电容(capacitance)把储存的电荷与板之间的电势差关联:

$$C=\frac{Q}{\Delta V}.$$

电容只取决于几何和间隙里的材料——从不取决于 $Q$ 或 $\Delta V$ 本身。

Worked derivation (parallel plates). 对于一个板面积 $A$ 和小间隙 $d$ 的平行板电容器(parallel-plate capacitor):高斯定律加叠加(superposition)给出板之间的一个均匀(uniform)场,$E=\dfrac{\sigma}{\varepsilon_0}=\dfrac{Q}{\varepsilon_0 A}$(每个板单独贡献 $\sigma/2\varepsilon_0$;在板之间两者相加,外面它们抵消)。一个均匀场意味着 $\Delta V=Ed=\dfrac{Qd}{\varepsilon_0 A}$,所以

$$C=\frac{Q}{\Delta V}=\frac{\varepsilon_0 A}{d}.$$

这个 $E\to\Delta V\to C$ 链是一个标准的 FRQ 推导——把它作为三个步骤学,并引用每一个。同样的方法处理 AP 期望的另外两个形状:同心球($C=4\pi\varepsilon_0\dfrac{ab}{b-a}$)和一个长度 $L$ 的同轴圆柱,那里 $E=\dfrac{\lambda}{2\pi\varepsilon_0 r}$ 给出 $\Delta V=\dfrac{\lambda}{2\pi\varepsilon_0}\ln\dfrac{b}{a}$ 因而 $C=\dfrac{2\pi\varepsilon_0 L}{\ln(b/a)}$。

Worked example. 面积 $A=0.020\ \text{m}^2$ 和间隙 $d=1.0\ \text{mm}$ 的板:$C=\dfrac{8.85\times10^{-12}(0.020)}{1.0\times10^{-3}}=1.8\times10^{-10}\ \text{F}$。充电到 $100\ \text{V}$ 它容纳 $Q=CV=1.8\times10^{-8}\ \text{C}$。

因为板之间的场是均匀的,那里的一个带电粒子感受一个恒定的力,所以它以恒定加速度移动——恰好像重力里的抛体运动(projectile motion):横向恒定速率、朝一个板均匀加速。

板之间的一个带电粒子遵循一条抛物线,像一个抛体

Worked example. 一个电子在板中间以 $v_0=2.0\times10^{7}\ \text{m/s}$、平行于它们进入。场是 $E=1.0\times10^{3}\ \text{N/C}$ 而板长 $4.0\ \text{cm}$。加速度:$a=\dfrac{eE}{m}=\dfrac{(1.6\times10^{-19})(1.0\times10^{3})}{9.11\times10^{-31}}=1.8\times10^{14}\ \text{m/s}^2$。板之间的时间:$t=\dfrac{0.040}{2.0\times10^{7}}=2.0\times10^{-9}\ \text{s}$。偏转:$y=\tfrac12at^{2}=\tfrac12(1.8\times10^{14})(2.0\times10^{-9})^{2}\approx3.5\times10^{-4}\ \text{m}$ ——约 $0.35\ \text{mm}$ 朝正板。

储存电荷需要功(work):一个外力必须逆着已经在那里的电荷的场移动每一小块电荷。总功最终成为储存的势能,

$$U_C=\tfrac12\,Q\,\Delta V=\tfrac12 C(\Delta V)^2=\frac{Q^2}{2C}.$$

储存在一个电容器里的能量是它的电荷-电压线下的面积

因子 $\tfrac12$ 是 $Q$–$\Delta V$ 线下三角形的面积:第一个移过的电荷几乎不花费,最后一个花费完整的 $\Delta V$。

电容器与电阻相反地组合:在并联(parallel)里电容相加($C_{\text{eq}}=C_1+C_2$,相同的 $\Delta V$,电荷相加),而在串联(series)里倒数相加($\tfrac{1}{C_{\text{eq}}}=\tfrac{1}{C_1}+\tfrac{1}{C_2}$,相同的 $Q$,电压相加)。

串联的电容器携带相同的电荷,而它们的电势差相加

Worked example. 一个 $2\ \mu\text{F}$ 和一个 $4\ \mu\text{F}$ 电容器串联:$\tfrac{1}{C}=\tfrac12+\tfrac14$,所以 $C=\tfrac43\ \mu\text{F}$。同一对并联:$6\ \mu\text{F}$。

Assorted capacitors: devices that store charge and energy in an electric field between conductors

Electrolytic capacitors on a circuit board: capacitance stores energy as U = ½CV²

探索

Charge and discharge a capacitor

A capacitor stores charge on two plates, filling and emptying exponentially with time constant $\tau=RC$. Bigger $R$ or $C$ slows it down.

词汇表训练

英文	中文	拼音
capacitor	电容器	diàn róng qì
capacitance	电容	diàn róng
parallel-plate capacitor	平行板电容器	píng xíng bǎn diàn róng qì
superposition	叠加	dié jiā
uniform	均匀	jūn yún
projectile motion	抛体运动	pāo tǐ yùn dòng
parallel	并联	bìng lián
series	串联	chuàn lián

练习卷

10.4

电介质

大纲

Learning Objective

Essential Knowledge

10.4.A
Describe how a dielectric inserted between the plates of a capacitor changes the properties of the capacitor.

10.4.A.1 In a dielectric material, electric charges are not as free to move as they are in a conductor. Instead, the material becomes polarized in the presence of an external electric field.
10.4.A.2 The dielectric constant of a material relates the electric permittivity of that material to the permittivity of free space.
- Equation: $\kappa = \dfrac{\varepsilon}{\varepsilon_0}$
10.4.A.3 The electric field created by a polarized dielectric is opposite in direction to the external field.
10.4.A.4 The electric field between the plates of an isolated parallel-plate capacitor decreases when a dielectric is placed between the plates.
- Equation: $\kappa = \dfrac{E_0}{E}$
10.4.A.5 The insertion of a dielectric into a capacitor may change the capacitance of the capacitor.
- Equation: $C = \kappa C_0$

来源:美国大学理事会 AP 课程与考试说明

一个电介质(dielectric)是一个绝缘材料。它的电荷不能行进,但在一个外部场里每个分子略微伸展或转动——材料变得极化(polarized)。排列的分子创造它们自己的与施加的相反的小场,所以材料内部的净场下降:

$$E=\frac{E_0}{\kappa},\qquad \kappa=\frac{\varepsilon}{\varepsilon_0}\ \ (\kappa>1),$$

其中 $\kappa$ 是介电常数(dielectric constant),材料的介电常数与真空介电常数(permittivity of free space)之比。

一个极化的电介质创造一个反对施加场的内部场

用电介质填充一个电容器使它的电容成倍增加:

$$C=\kappa\,C_0\qquad\left(\text{parallel plates: } C=\frac{\kappa\varepsilon_0 A}{d}\right).$$

接下来发生什么取决于什么保持固定——一个最爱的考试陷阱:

	电池保持连接($\Delta V$ 固定)	电池先移除($Q$ 固定)
电荷 $Q$	升到 $\kappa Q_0$	不变
电压 $\Delta V$	不变	降到 $\Delta V_0/\kappa$
场 $E$	不变	降到 $E_0/\kappa$
能量 $U$	升到 $\kappa U_0$	降到 $U_0/\kappa$

Worked example. 一个 $100\ \text{pF}$ 的电容器被充电到 $12\ \text{V}$、断开,然后填充一个 $\kappa=3$ 的电介质。$Q$ 被困住,所以 $\Delta V$ 降到 $4\ \text{V}$ 而储存的能量降到三分之一——缺失的能量进入把电介质拉进去。相反重新连接到 $12\ \text{V}$ 电池,电容器会容纳三倍的电荷和三倍的能量。

Exam skill. 总是通过写下哪个量被保持固定($Q$ 或 $\Delta V$)来开始电介质问题,然后让 $C=\kappa C_0$ 通过 $Q=C\Delta V$ 和 $U=\tfrac12 C(\Delta V)^2$ 驱动其他一切。

探索

Add a dielectric

A dielectric between the plates raises the capacitance, so the capacitor holds more charge at the same voltage. Watch the charge build faster.

词汇表训练

英文	中文	拼音
dielectric	电介质	diàn jiè zhì
polarized	极化	jí huà
dielectric constant	介电常数	jiè diàn cháng shù
permittivity of free space	真空介电常数	zhēn kōng jiè diàn cháng shù

练习卷

10.4

考试技巧

在静电平衡中一个导体有内部 $\vec E=0$ 而所有多余的电荷在表面上。
用一个匹配对称的高斯面(球、圆柱、药盒)应用高斯定律 $\oint \vec E\cdot d\vec A=\tfrac{q_{enc}}{\varepsilon_0}$。
整个导体是一个等势面,而表面场垂直于它。
对于一个电容器用 $C=\tfrac{Q}{V}$、能量 $U=\tfrac12 CV^2$,以及一个电介质如何提高 $C$。
挑选高斯面使 $\vec E$ 在每个部分恒定且平行(或零)。

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