学习目标 9.1.A： 描述系统的电势能。
- 9.1.A.1 两个点电荷组成的系统的电势能等于外力将这两个点电荷从无穷远处移至当前位置所做的功。
- 9.1.A.2 两个带电物体之间的电势能的一般形式由以下方程给出：
- 公式： $U_E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1 q_2}{r} = k\dfrac{q_1 q_2}{r}$。
- 9.1.A.3 系统的总电势能可以通过求系统中每对带电物体之间相互作用的电势能之和来确定。
电势
AP 物理 C:电磁学 · 知识点 9
8:02
电势
看看这些数字。一节小电池标着一点五伏。汽车电池标着十二伏。高压输电线承载四十万伏。一…
英语旁白 · 英中内嵌字幕
9.1
电势能
教学大纲
来源：College Board AP 课程与考试说明
English
When you push two like charges together, you do work 功 against the electric force. That work is stored by the pair as electric potential energy 电势能:
$$U_E=\frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r}=k\frac{q_1 q_2}{r}.$$
$U_E$ is defined as the work an external force must do to bring the charges from infinitely far away to a distance $r$ apart. The signs carry the physics:
- Like charges: $U_E>0$. An outside push was needed to bring them together. If released, they fly apart, and the stored energy becomes kinetic energy 动能.
- Opposite charges: $U_E<0$. The pair is bound. You must supply $|U_E|$ of work to pull them apart to infinity.
The electric force is a conservative force 保守力: the work it does depends only on the start and end points, not the path, and equals $-\Delta U_E$.
For a system 系统 of several charges, add the potential energy of every distinct pair:
$$U_{\text{total}}=k\!\left(\frac{q_1q_2}{r_{12}}+\frac{q_1q_3}{r_{13}}+\frac{q_2q_3}{r_{23}}\right).$$
Worked example. Three $+2.0\ \mu\text{C}$ charges sit at the corners of an equilateral triangle with sides $0.30\ \text{m}$. Each pair stores $U=\dfrac{kq^2}{r}=\dfrac{(9.0\times10^{9})(2.0\times10^{-6})^2}{0.30}=0.12\ \text{J}$. There are three pairs, so $U_{\text{total}}=3\times0.12\ \text{J}=0.36\ \text{J}$. This is the work needed to assemble the triangle from infinity – and the kinetic energy the charges would share if all three were released.
探索
电势能
两个电荷的电势能随它们之间距离的倒数而衰减。
词汇
训练
| English | 中文 | 拼音 |
|---|---|---|
| work/wɜːk/ | 功 | gōng |
| electric potential energy/ɪˈlektrɪk pəˈtenʃl ˈenədʒi/ | 电势能 | diàn shì néng |
| kinetic energy/kɪˈnetɪk ˈenədʒi/ | 动能 | dòng néng |
| conservative force/kənˈsɜːvətɪv fɔːs/ | 保守力 | bǎo shǒu lì |
| system/ˈsɪstəm/ | 系统 | xì tǒng |
9.2
电势
教学大纲
学习目标 9.2.A： 描述带电物体系统产生的电势。
- 9.2.A.1 电势描述了空间中某点处单位电荷所具有的电势能。
- 9.2.A.2 电荷分布的电势表达式可以通过积分和叠加原理求得。
- 公式： $V = \dfrac{1}{4\pi\varepsilon_0}\displaystyle\int \dfrac{dq}{r}$
- 9.2.A.2.i 单个点电荷的电势为：
- 公式： $V = \dfrac{q}{4\pi\varepsilon_0 r}$。
- 9.2.A.2.ii 多个点电荷产生的电势可以通过各点电荷产生电势的标量叠加原理来确定。
- 公式： $V = \dfrac{1}{4\pi\varepsilon_0}\displaystyle\sum_i \dfrac{q_i}{r_i}$
- 9.2.A.3 两点之间的电势差是指当测试电荷在这两点之间移动时，单位电荷的电势能变化量。
- 公式： $\Delta V = \dfrac{\Delta U_E}{q}$
- 9.2.A.4 化学过程（如电池中）导致正负电荷分离也会产生电势差。
边界说明： AP物理C：电磁学仅要求学生使用微积分计算以下电荷分布及位置所产生的电势：距无限长均匀带电导线或圆柱体中心轴一定距离处的点、位于圆环轴线上某点的薄带电圆环、半圆弧或其一部分圆心处的点，以及与线电荷共线的点或位于其垂直平分线上的有限长导线或线电荷。
学习目标 9.2.B： 描述电势与电场之间的关系。
- 9.2.B.1 给定位置处任意方向的电场分量值等于该位置处电势空间变化率的负值。
- 公式： $E_x = -\dfrac{dV}{dx}$
- 9.2.B.2 两点间的电势变化可以通过对沿连接两点路径上电场与位移的点积进行积分来确定。
- 公式： $\Delta V = V_b - V_a = -\displaystyle\int_a^b \vec{E}\cdot d\vec{r}$
- 9.2.B.3 电场矢量图和等势线是描述电荷或电荷系统产生场的工具，可用于预测带电物体在电场中的运动。
- 9.2.B.3.i 等势线代表电势相等的线。这些线也被称为电势等值线。
- 9.2.B.3.ii 等值线与电场矢量垂直。可以根据电场矢量图构建电势等值线图，也可以根据等值线图构建电场图。
- 9.2.B.3.iii 电场矢量指向电势降低的方向。
- 9.2.B.3.iv 电场沿等值线方向没有分量。
来源：College Board AP 课程与考试说明
English
Electric potential 电势 is electric potential energy per unit charge at a point in space. It is measured in volts ($1\ \text{V}=1\ \text{J/C}$):
$$V=\frac{U_E}{q},\qquad V=\frac{1}{4\pi\varepsilon_0}\frac{q}{r}\ \text{(point charge)}.$$
Potential is a scalar 标量: it has a sign but no direction. This makes $V$ far easier to work with than the field $\vec{E}$. For several point charges 点电荷, use scalar superposition 叠加 – add the potentials with their signs:
$$V=\frac{1}{4\pi\varepsilon_0}\sum_i \frac{q_i}{r_i}.$$
Worked example. The potential $0.10\ \text{m}$ from a $+3.0\ \text{nC}$ point charge is $V=\dfrac{kq}{r}=\dfrac{9.0\times10^{9}(3.0\times10^{-9})}{0.10}=270\ \text{V}$. Now add a $-3.0\ \text{nC}$ charge $0.30\ \text{m}$ from the same point: it contributes $\dfrac{9.0\times10^{9}(-3.0\times10^{-9})}{0.30}=-90\ \text{V}$, so the total is $270-90=180\ \text{V}$. Just a signed sum – no vector components to resolve.
Continuous charge distributions
For a continuous charge distribution 连续电荷分布, cut it into small pieces $dq$, treat each piece as a point charge, and integrate 积分:
$$V=\frac{1}{4\pi\varepsilon_0}\int\frac{dq}{r}.$$
This is a scalar integral, so it is usually much easier than the field integral. AP expects you to handle four shapes with calculus: a thin ring (at a point on its axis), an arc (at its centre), a finite line or wire (collinear, or on its perpendicular bisector), and an infinitely long wire or cylinder (at a distance from its axis).
Worked example (ring of charge). A thin ring of radius $R$ carries total charge $Q$. For a point $P$ on the axis a distance $z$ from the centre, every element $dq$ sits at the same distance $r=\sqrt{R^2+z^2}$ from $P$. The distance is constant, so it comes out of the integral:
$$V=\frac{1}{4\pi\varepsilon_0}\int\frac{dq}{\sqrt{R^2+z^2}}=\frac{1}{4\pi\varepsilon_0}\frac{Q}{\sqrt{R^2+z^2}}.$$
On an FRQ, say that $r$ is the same for every element – that statement is the marked step. The same idea gives $V=kQ/R$ at the centre of any arc, whatever its angle.
From potential to field
Potential and field contain the same information, linked by a derivative in each direction and by a path integral:
$$E_x=-\frac{dV}{dx},\qquad \Delta V=V_b-V_a=-\int_a^b \vec{E}\cdot d\vec{r}.$$
The field is the negative gradient 梯度 of the potential: $\vec{E}$ points from high potential to low potential, "downhill". Where $V$ changes quickly, the field is strong.
Worked example. Along the $x$-axis, $V(x)=3x^{2}-2x$ (volts, with $x$ in metres). Then $E_x=-\dfrac{dV}{dx}=2-6x\ \text{V/m}$. At $x=0.50\ \text{m}$, $E_x=-1.0\ \text{V/m}$: the field there points in the $-x$ direction.
The potential difference 电势差 between two points is the change in potential energy per unit charge moved between them: $\Delta V=\Delta U_E/q$. A battery makes a potential difference chemically: reactions inside separate positive from negative charge and hold the terminals a fixed $\Delta V$ apart.
Equipotential maps
An equipotential 等势面 surface contains points that share the same potential; its intersection with a map appears as an isoline 等值线. Four rules let you read any map:
- Isolines are perpendicular 垂直 to field lines 电场线 everywhere they cross.
- $\vec{E}$ points from high $V$ to low $V$ – never along an isoline. Moving a charge along an isoline takes no work.
- Closely spaced isolines mean a strong field: $E\approx-\Delta V/\Delta x$.
- You can sketch the field map from an isoline map, and the isoline map from a field map.
Exam skill. Given a map with isolines every $10\ \text{V}$ spaced about $2\ \text{cm}$ apart, estimate $E\approx\dfrac{10}{0.02}=500\ \text{V/m}$, pointing from the higher-value line to the lower one. The work an external agent does moving a charge $q$ slowly from $A$ to $B$ is $W=q(V_B-V_A)$ – the path taken does not matter.
探索
电荷周围的电场线与电势
电势是单位电荷的能量。它随距正电荷的距离增加而降低；电场线从高电势指向低电势。
词汇
训练
| English | 中文 | 拼音 |
|---|---|---|
| Electric potential/ɪˈlektrɪk pəˈtenʃl/ | 电势 | diàn shì |
| scalar/ˈskeɪlə/ | 标量 | biāo liàng |
| point charges/pɔɪnt ˈtʃɑːdʒɪz/ | 点电荷 | diǎn diàn hè |
| superposition/ˌsuːpəpəˈzɪʃn/ | 叠加 | dié jiā |
| continuous charge distribution/kənˈtɪnjuːəs tʃɑːdʒ ˌdɪstrɪˈbjuːʃn/ | 连续电荷分布 | lián xù diàn hè fēn bù |
| integrate/ˈɪntɪɡreɪt/ | 积分 | jī fēn |
| gradient/ˈɡreɪdɪənt/ | 梯度 | tī dù |
| potential difference/pəˈtenʃl ˈdɪfrəns/ | 电势差 | diàn shì chā |
| equipotential/ˌiːkwɪpəˈtenʃl/ | 等势面 | děng shì miàn |
| isoline/ˈaɪsəliːn/ | 等值线 | děng zhí xiàn |
| perpendicular/ˌpɜːpənˈdɪkjʊlə/ | 垂直 | chuí zhí |
| field lines/fiːld laɪnz/ | 电场线 | diàn chǎng xiàn |
9.3
电能守恒
教学大纲
学习目标 9.3.A： 描述由于两个位置之间电势差而引起的系统变化。
- 9.3.A.1 当带电物体在两个具有不同电势的位置之间移动时，该物体-场系统的电势能变化由以下方程给出。
- 方程： $\Delta U_E = q\Delta V$
- 9.3.A.2 带电物体在两个具有不同电势的点之间移动会导致其动能发生变化，这一变化符合能量守恒定律。
来源：College Board AP 课程与考试说明
English
When a charge $q$ moves between two points that differ in potential by $\Delta V$, the potential energy of the charge–field system changes by
$$\Delta U_E=q\,\Delta V.$$
If only the electric force acts, total energy is conserved, so the kinetic energy changes by the opposite amount:
$$\Delta K=-\Delta U_E=-q\,\Delta V.$$
Watch the two signs together: a positive charge speeds up when it moves to lower potential, while a negative charge speeds up when it moves to higher potential. Both are just the system trading potential energy for kinetic energy. This is how particle accelerators give charged particles their energy, and it underlies energy analysis in circuits.
Worked example. A proton ($q=1.6\times10^{-19}\ \text{C}$, $m=1.67\times10^{-27}\ \text{kg}$) starts at rest and is accelerated through a potential drop of $500\ \text{V}$ (it moves to lower potential, so $\Delta V=-500\ \text{V}$ and $\Delta K=-q\,\Delta V=+8.0\times10^{-17}\ \text{J}$). Setting $\Delta K=\tfrac12 mv^2$:
$$v=\sqrt{\frac{2(8.0\times10^{-17})}{1.67\times10^{-27}}}=3.1\times10^{5}\ \text{m/s}.$$
Exam skill. Choose energy methods, not kinematics, whenever the field is non-uniform or the path curves: only the end-point potentials matter. A typical FRQ chain is $q\,\Delta V \to \Delta K \to v$, with one line of justification: "the electric force is conservative, so energy is conserved."
探索
加速还是减速？
电荷在电场中运动时，能量在动能和势能之间转换。对每种情况进行分类。
9.3
考试技巧
English
- Relate field and potential by $V=-\int \vec E\cdot d\vec l$ and $\vec E=-\nabla V$ (in 1-D, $E_x=-\tfrac{dV}{dx}$).
- Potential is a scalar — add contributions with signs, no vector components needed.
- Potential energy of a pair is $U=\tfrac{1}{4\pi\varepsilon_0}\tfrac{q_1 q_2}{r}$; use energy conservation for a charge's speed.
- Know that no work is done moving along an equipotential, which is perpendicular to $\vec E$.
- Choose the zero of potential (usually infinity) and state it.
Common mistakes
- Treating potential as a vector. Potentials from several charges add as signed numbers.
- Losing the sign in $E_x = -dV/dx$: the field points towards decreasing potential.
- Getting the sign wrong for electrons. $\Delta U = q\,\Delta V$ with $q$ negative: an electron gains kinetic energy moving to higher potential.
- Saying that $V = 0$ means $E = 0$ (or the reverse). Midway between equal and opposite charges $V = 0$ but $E \ne 0$.
该知识点的互动课程
逐步完成，配合即时检查练习。