# 电学系统

> AP 物理 2 · AP 物理 2 CED 第3单元
> 来源: https://www.owlsprep.com/zh/study/ap-physics-2-u3-electric-systems/

本模块为准备AP物理2考试，讲解了电学系统中的系统边界定义、电荷守恒、导体电荷重新分布、多电荷电势能，以及高斯定律处理包围电荷的内容。

**先修:** 电荷守恒是一条基本物理定律; 孤立点电荷的库仑定律; 外场中单个电荷的电势能

## 学习目标

- 定义电学系统，区分封闭系统与开放系统
- 将电荷守恒应用于导体上的电荷重新分布
- 计算多电荷系统的总电势能
- 运用高斯定律求空心导体上的感应电荷

## 什么是电学系统？

电学系统是指由带电物体、导体以及相关电场组成的任意明确集合，由我们为分析选取的明确闭合曲面界定边界。与分析孤立电荷不同，研究电学系统需要追踪穿过系统边界的物理量，应用守恒定律，并计算整个集合的净性质。本内容占AP物理2考试总分的约3-5%，会在选择题和自由作答题部分同时出现。

## 电学系统中的电荷守恒

所有电学系统的分析都从电荷守恒开始，这一基本定律指出：电荷既不能被创造也不能被消灭，只能转移或重新排列。系统根据其边界分类如下：

$$\text{Closed System: } \sum Q_{\text{initial}} = \sum Q_{\text{final}}$$

$$\text{Open System: } \Delta Q_{\text{system}} = Q_{\text{in}} - Q_{\text{out}}$$

考试中常见的应用是两个导体球接触后的电荷重新分布。电荷在导体上可以自由移动，因此系统达到静电平衡时，两个导体球的电势相等。对于相同导体（半径相同、电容相同），电荷会平均分配。

**例题:** 三个放在绝缘支架上的相同导体球，初始电荷分别为 $+6 \mu\text{C}$, $-4 \mu\text{C}$, 和 $+2 \mu\text{C}$ 分别。S球A touches Sphere B, then they are separated. Then Sphere B touches Sphere C, then they are separated. What is the final charge on Sphere B?

1. 这是一个封闭系统（没有电荷进入或离开这组球体），因此每一步总电荷都守恒。
2. A与B接触后：这一对的总电荷为 $+6 \mu\text{C} + (-4 \mu\text{C}) = +2 \mu\text{C}$。由于球体相同，电荷平均分配：
3. $$Q_{A2} = Q_{B2} = +1 \mu\text{C}$$
4. B与C接触后：这一对的总电荷为 $+1 \mu\text{C} + (+2 \mu\text{C}) = +3 \mu\text{C}$。Again, identical spheres split charge equally:
5. $$Q_{B3} = Q_{C3} = +1.5 \mu\text{C}$$
6. Final charge on Sphere B is $+1.5 \mu\text{C}$.

> **tip**
>
> 如果题目没有明确说明导体相同，你不能平均分配电荷，应该使用等电势规则 $V_1 = V_2$ 来求电荷比。

## 多电荷系统的电势能

点电荷系统的总电势能等于将所有电荷从无限分离的初始静止状态组装到当前位置所需的总功。计算时，要将每一对独特电荷的势能相加，因为电势能是标量。

$$U_{\text{total}} = \frac{1}{4\pi\epsilon_0} \sum_{i<j} \frac{q_i q_j}{r_{ij}} = k \sum_{i<j} \frac{q_i q_j}{r_{ij}}$$

where $k = 8.99 \times 10^9 \text{ Nm}^2/\text{C}^2$, $q_i$ and $q_j$ are the charges of the pair, $r_{ij}$ is the distance between them, and the $i<j$ convention ensures we count each pair only once, avoiding double-counting. A negative total potential energy means the system is bound: net work is done by the electric field during assembly, so you must add external energy to pull all charges apart to infinity. A positive total means the system is unbound, with net repulsive interactions.

**例题:** Three point charges $+q$, $+q$, and $-q$ are placed at the vertices of an equilateral triangle of side length $s$. What is the total electric potential energy of the system?

1. For 3 charges, there are $\frac{3(3-1)}{2} = 3$ unique pairs, so we calculate the potential energy for each.
2. Pair 1 ($+q, +q$, separation $s$):
3. $$U_1 = k \frac{(+q)(+q)}{s} = \frac{kq^2}{s}$$
4. Pair 2 ($+q, -q$, separation $s$):
5. $$U_2 = k \frac{(+q)(-q)}{s} = -\frac{kq^2}{s}$$
6. Pair 3 ($+q, -q$, separation $s$):
7. $$U_3 = k \frac{(+q)(-q)}{s} = -\frac{kq^2}{s}$$
8. Sum the three potential energies:
9. $$U_{\text{total}} = \frac{kq^2}{s} - \frac{kq^2}{s} - \frac{kq^2}{s} = -\frac{kq^2}{s}$$
10. The negative sign confirms this is a bound system, as expected with two attractive interactions and one repulsive interaction.

> **tip**
>
> If you count interactions for each charge individually (e.g., each charge interacts with every other charge), you will get twice the correct total. Remember to divide your result by 2 if you do not use the $i<j$ counting convention.

## Gauss's Law for Enclosed Charge in Electric Systems

Gauss's law connects the net electric flux through a closed Gaussian surface (our system boundary) to the net charge enclosed by that surface. This is the primary tool for finding induced charge on conducting surfaces in electrostatic systems.

$$\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enclosed}}}{\epsilon_0}$$

A key property of this law is that only charge inside the Gaussian surface contributes to the net flux. Any charge outside the surface produces zero net flux, because every electric field line that enters the surface also exits it. For conductors in electrostatic equilibrium, the electric field inside the conducting material is always zero, which lets us solve for induced charge by placing a Gaussian surface inside the conductor material.

**例题:** A neutral hollow conducting spherical shell has a point charge of $+Q$ placed at the center of the inner cavity. What is the charge on the inner surface of the shell, and what is the charge on the outer surface?

1. Choose a Gaussian surface that lies entirely within the conducting material of the shell, between the inner cavity surface and the outer surface of the shell.
2. For a conductor in electrostatic equilibrium, the electric field everywhere inside the conductor material is zero, so the net flux through the Gaussian surface is zero.
3. By Gauss's law, $\Phi_E = 0 = \frac{Q_{\text{enclosed}}}{\epsilon_0}$, so total enclosed charge is zero. The point charge at the center is $+Q$, so the inner surface must carry $-Q$ to give a total enclosed charge of $+Q + (-Q) = 0$.
4. The shell is originally neutral, so total charge of the shell is zero. If inner surface has $-Q$, the outer surface must carry $+Q$ to give a total shell charge of zero.

> **tip**
>
> Always place your Gaussian surface inside the conductor material when solving for induced charge. Never place it inside the cavity or outside the shell, as this will not give you the zero electric field condition you need to solve for enclosed charge.

## 常见错误

- **错误做法:** 接触后将电荷平均分配给两个非相同导体
  - 原因: 学生记住了相同球体的情况，错误地将其推广到任意两个导体
  - 正确做法: 在平均分配电荷前，始终确认题目说明导体相同；对于非相同导体，使用 $V_1 = V_2$ 求电荷比。
- **错误做法:** 计算3个及以上电荷系统的总势能时重复计数电荷对
  - 原因: 学生逐个电荷计数相互作用，导致每对被记录两次
  - 正确做法: For $n$ charges, count exactly $\frac{n(n-1)}{2}$ unique pairs before summing potential energy.
- **错误做法:** Including charge outside the Gaussian surface when calculating $Q_{\text{enclosed}}$ for Gauss's law
  - 原因: Students confuse total charge in the entire problem with charge inside the defined system boundary
  - 正确做法: Only add up charges that lie strictly inside your Gaussian surface; ignore all charges outside entirely.
- **错误做法:** Assigning a non-zero net charge to a neutral conductor after induced charge separation
  - 原因: Students forget induction only separates charge, it does not create new charge
  - 正确做法: For any originally neutral conductor, the sum of charge on all its surfaces must equal zero after induction.
- **错误做法:** Assuming charge redistributes when two charged insulating spheres are brought into contact
  - 原因: Students generalize conductor behavior to insulators, where charge is fixed in place
  - 正确做法: Charge does not move on insulators, so the charge of each sphere remains unchanged after contact.

## 速查表

| 类别 | 公式 | 说明 |
| --- | --- | --- |
| Conservation of Charge (Closed System) | $\sum Q_{\text{initial}} = \sum Q_{\text{final}}$ | Applies when no charge crosses the system boundary |
| Conservation of Charge (Open System) | $\Delta Q_{\text{system}} = Q_{\text{in}} - Q_{\text{out}}$ | Applies when charge can enter/leave the system |
| Charge Redistribution (Identical Conductors) | $Q_1 = Q_2 = \frac{Q_{\text{total}}}{2}$ | Only for identical conductors after contact at equilibrium |
| Multi-Charge Potential Energy | $U_{\text{total}} = k \sum_{i<j} \frac{q_i q_j}{r_{ij}}$ | Count each unique pair only once; $k = 1/(4\pi\epsilon_0)$ |
| Gauss's Law | $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enclosed}}}{\epsilon_0}$ | Only charge inside the Gaussian surface contributes to net flux |
| Induced Charge (Hollow Conductor) | $Q_{\text{inner}} = -Q_{\text{cavity}}$ | Applies for any hollow conductor with charge inside its cavity |
| Electric Field Outside Conducting Sphere | $E = \frac{k Q_{\text{outer}}}{r^2}$ | Matches the field of a point charge equal to the outer surface charge |

## 下一步

Mastering electric systems is the critical foundation for the next topics in Unit 3, including electric potential of charged conductors, Gauss's law applications to symmetric charge distributions, and capacitance of multi-conductor systems. Without being able to correctly apply conservation of charge and account for induced charge on conductor surfaces, you will struggle to correctly calculate capacitance or potential difference between conductors, a heavily tested topic on the AP Physics 2 exam. This topic also feeds into later units, including DC circuits, where conservation of charge is the basis for Kirchhoff's junction rule, and electromagnetism, where Gauss's law for charge is extended to other electromagnetic quantities.

- [电荷与电场力](https://www.owlsprep.com/zh/study/ap-physics-2-u3-charge-and-electric-force/)
- [电场](https://www.owlsprep.com/zh/study/ap-physics-2-u3-electric-field/)
- [电势能与电势](https://www.owlsprep.com/zh/study/ap-physics-2-u3-potential-and-electric-potential-energy/)

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来自 [OwlsPrep](https://www.owlsprep.com) —— A-Level / IB / AP / IGCSE 免费学习指南，依据官方考纲编写。原页面：https://www.owlsprep.com/zh/study/ap-physics-2-u3-electric-systems/