Coulomb's law · 库仑定律
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| Coulomb's law/ˈkuːlɒmz lɔː/ | 库仑定律 | kù lún dìng lǜ |
| permittivity of free space/ˌpɜːmɪˈtɪvəti ɒv friː speɪs/ | 真空电容率 | zhēn kōng diàn róng lǜ |
| repulsive/rɪˈpʌlsɪv/ | 排斥 | pái chì |
| attractive/əˈtræktɪv/ | 吸引 | xī yǐn |
| equipotential/ˌiːkwɪpəˈtenʃl/ | 等势面 | děng shì miàn |
The force that beats gravity by a factor of $10^{39}$
- Inside a hydrogen atom, the proton pulls on the electron electrically with about $10^{-8}\ \text{N}$, and gravitationally with about $10^{-48}\ \text{N}$.
- Gravity is weaker by a factor of ten thousand billion billion billion billion. It shapes galaxies only because charges come in two signs and cancel, while mass comes in one sign and adds.
- Both forces obey the same shape of law. Learning one gives you most of the other, which is why these two topics sit next to each other.
- This lesson is Coulomb's law 库仑定律, the inverse-square behaviour, and how to treat charged spheres.
比引力强 $10^{39}$ 倍的力
- 在氢原子内部,质子对电子的电吸引力约为 $10^{-8}\ \text{N}$,而引力约为 $10^{-48}\ \text{N}$。
- 引力弱了一万亿亿亿亿倍。它之所以塑造星系,只是因为电荷有两种符号会抵消,而质量只有一种符号会累加。
- 两种力遵守同样形状的定律。学会一个就等于学会了另一个的大部分,这正是这两个单元相邻的原因。
- 这一课讲库仑定律(Coulomb's law)、平方反比的行为,以及怎样处理带电球。
Coulomb's law
- For two point charges $Q_1$ and $Q_2$ a distance $r$ apart in free space, each feels a force of size:
- Here $\varepsilon_0 = 8.85 \times 10^{-12}\ \text{F/m}$ is the permittivity of free space 真空电容率, and $\dfrac{1}{4\pi\varepsilon_0} \approx 9.0 \times 10^9\ \text{N m}^2\text{/C}^2$, which is the number to substitute.
- The force acts along the line joining the charges: repulsive 排斥 for like charges, attractive 吸引 for opposite ones.
Same sign apart, opposite signs together
库仑定律
- 真空中相距 $r$ 的两个点电荷 $Q_1$ 和 $Q_2$,各自受到大小为下式的力:
- 其中 $\varepsilon_0 = 8.85 \times 10^{-12}\ \text{F/m}$ 是真空电容率(permittivity of free space),而 $\dfrac{1}{4\pi\varepsilon_0} \approx 9.0 \times 10^9\ \text{N m}^2\text{/C}^2$,这才是代入时要用的数。
- 力沿两电荷的连线作用:同号排斥(repulsive),异号吸引(attractive)。
同号分开,异号相聚
Coulomb's law · 库仑定律
F ∝ Qq / r²
Like charges repel, unlike attract — and the force follows 1/r². · 同号电荷相斥,异号相吸——力服从 1/r²。
Coulomb's law gives the force between two point charges as: · 库仑定律给出两个点电荷之间的力是:
Proportional to each charge, inversely proportional to the distance squared. · 与每个电荷成正比,与距离的平方成反比。
Two charges of the same sign repel each other. · 两个同号的电荷互相排斥。
Like charges repel; opposite charges attract — both along the line joining them. · 同号电荷相斥;异号电荷相吸——都沿连接它们的直线。
Stating the law in words
- "State Coulomb's law" wants a sentence, not a formula: the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of their separation.
- Both proportionalities are needed for the marks, and the charges must be point charges, or spheres treated as point charges at their centres.
- Adding "and acts along the line joining them" is worth having; it is what makes the force a vector with a definite direction.
用文字陈述定律
- "陈述库仑定律"要的是一句话,不是公式:两个点电荷之间的力与两电荷的乘积成正比,与它们间距的平方成反比。
- 两个正比反比关系都要有才能得分,而且必须是点电荷,或被当作球心处点电荷的球。
- 加上"并沿两者连线作用"是值得的;正是它让这个力成为有确定方向的矢量。
Stating Coulomb's law in words needs which elements? Select all · 所有 that apply. · 用文字陈述库仑定律需要哪些要素?选出所有适用的。
The law covers like and opposite charges alike; the sign decides only whether the force repels or attracts. · 该定律同样适用于同号和异号电荷;符号只决定力是排斥还是吸引。
An inverse-square law
- The force falls as $\dfrac{1}{r^2}$, so treble the separation and the force drops to a ninth; double it and the force quarters.
- This is the same shape as Newton's law of gravitation, $F = Gm_1m_2/r^2$. The difference is that charge has two signs, so the electric force can repel as well as attract, while gravity only ever attracts.
- Questions often exploit the ratio directly: if the separation changes from $r$ to $2r$, you need no constants at all, only the factor of four.
平方反比定律
- 力按 $\dfrac{1}{r^2}$ 衰减,所以间距变为三倍,力降为九分之一;间距加倍,力变为四分之一。
- 这与牛顿万有引力定律 $F = Gm_1m_2/r^2$ 是同一种形状。区别在于电荷有两种符号,所以电力既能吸引也能排斥,而引力永远只吸引。
- 题目常常直接利用这个比值:如果间距从 $r$ 变成 $2r$,你根本不需要任何常量,只需要那个 4 倍。
Two charges repel with $36\ \text{N}$ at separation $r$. What is the force at separation $3r$? · 两个电荷在距离 $r$ 时以 $36\ \text{N}$ 互相排斥。在距离 $3r$ 时的力是多少?
Inverse-square: $\dfrac{36}{3^{2}} = \dfrac{36}{9} = 4\ \text{N}$. · 平方反比:$\dfrac{36}{3^{2}} = \dfrac{36}{9} = 4\ \text{N}$。
Coulomb's law is an ____-square law (the force is proportional to 1/r²). · 库仑定律是一个平方 ____ 定律(力与 1/r² 成正比)。
Force · 力 $\propto \dfrac{1}{r^{2}}$ — the same distance dependence as gravity. · 力 $\propto \dfrac{1}{r^{2}}$——与引力相同的距离依赖关系。
Two charges repel with a force F. The separation is trebled. What is the new force? · 两个电荷以力 F 相斥。间距变为三倍。新的力是多少?
An inverse-square law: three times the distance means nine times weaker. No constants are needed for a ratio question like this. · 平方反比定律:距离三倍意味着弱九倍。这类比值题根本不需要任何常量。
Worked example: two point charges
- Find the force between point charges of $+2.0\ \text{nC}$ and $+3.0\ \text{nC}$ placed $4.0\ \text{cm}$ apart.
- The charges are both positive, so the force is repulsive, and saying so is part of the answer.
- Watch the unit conversions: nanocoulombs to coulombs, centimetres to metres, and the distance squared. Those three are where the marks are lost.
例题:两个点电荷
- 求相距 $4.0\ \text{cm}$ 的 $+2.0\ \text{nC}$ 与 $+3.0\ \text{nC}$ 两个点电荷之间的力。
- 两个电荷都是正的,所以力是排斥的,而说出这一点也是答案的一部分。
- 注意单位换算:纳库到库、厘米到米,以及距离要平方。这三处正是丢分的地方。
Two point charges of +2.0 nC and +3.0 nC are 4.0 cm apart. What is the force between them, in units of 10^-5 N? · 两个 +2.0 nC 和 +3.0 nC 的点电荷相距 4.0 cm。它们之间的力是多少,以 10^-5 N 为单位?
F = 9.0e9 x 2.0e-9 x 3.0e-9 / 0.040^2 = 3.4e-5 N, and it is repulsive because both charges are positive. Saying which is part of the answer. · F = 9.0e9 x 2.0e-9 x 3.0e-9 / 0.040^2 = 3.4e-5 N,而且是排斥的,因为两个电荷都为正。说出是哪一种也是答案的一部分。
Worked example: charge from an equilibrium
- A charged sphere Y of mass $2.0\ \text{g}$ hangs on an insulating thread beside a fixed charged sphere X. It settles with the thread at $12°$ to the vertical, its centre $5.0\ \text{cm}$ from X's. Find the electric force, and the charge if both are equal.
- Three forces act on Y: its weight downwards, the tension along the thread, and the electric force horizontally, away from X because like charges repel.
- Resolving: $T\cos\theta = mg$ and $T\sin\theta = F$, so $F = mg\tan\theta = (2.0 \times 10^{-3})(9.81)\tan 12° = 4.2 \times 10^{-3}\ \text{N}$.
- Then $F = Q^2/(4\pi\varepsilon_0 r^2)$ gives $Q = \sqrt{4\pi\varepsilon_0 F r^2} = 3.4 \times 10^{-8}\ \text{C}$.
- A closed triangle of the three forces is equally acceptable, but the answer must show all three.
例题:由平衡求电荷
- 一个质量 $2.0\ \text{g}$ 的带电球 Y 用绝缘线挂在固定的带电球 X 旁边。它稳定时线与竖直方向成 $12°$,球心距 X 的球心 $5.0\ \text{cm}$。求电场力,以及两球电荷相等时的电荷量。
- Y 受三个力:向下的重力、沿线的张力,以及水平方向背离 X 的电场力——因为同号电荷排斥。
- 分解:$T\cos\theta = mg$ 且 $T\sin\theta = F$,所以 $F = mg\tan\theta = (2.0 \times 10^{-3})(9.81)\tan 12° = 4.2 \times 10^{-3}\ \text{N}$。
- 再由 $F = Q^2/(4\pi\varepsilon_0 r^2)$ 得 $Q = \sqrt{4\pi\varepsilon_0 F r^2} = 3.4 \times 10^{-8}\ \text{C}$。
- 画三力封闭三角形同样可以接受,但答案必须体现全部三个力。
Put the steps of finding the charge on two equally charged hanging spheres in order. · 把求两个等量带电悬挂球电荷量的步骤按顺序排列。
A closed triangle of the three forces is equally acceptable, but the answer must show all three forces either way. · 画三力封闭三角形同样可以,但无论哪种方法,答案都必须体现全部三个力。
Spheres and conductors
- For a point outside a spherical conductor, the charge may be treated as a point charge at its centre, so $r$ is measured from the centre, not the surface.
- Inside a hollow charged conductor the field is zero, so the whole conductor is an equipotential 等势面.
- The trap is measuring $r$ from the surface of a sphere. Read the question for whether it gives a centre separation or a radius plus a gap.
球与导体
- 对球形导体外部的一点,可以把电荷当作位于球心的点电荷,所以 $r$ 从球心量起,不是从表面。
- 空心带电导体内部场为零,所以整个导体是一个等势面(equipotential)。
- 陷阱是从球面量 $r$。读题看清它给的是球心间距,还是半径加上一段间隙。
Select all · 所有 the true statements. · 选出所有正确的说法。
Like charges repel (not attract). The other three are all correct. · 同号电荷相斥(不是相吸)。其余三个都正确。
Marks that slip away
- Stating the law needs both proportionalities, and the charges must be point charges.
- Say repulsive or attractive. A magnitude with no direction is an incomplete answer.
- Measure $r$ between the centres of spheres, and convert cm to m before squaring.
- Two identical objects with the same sign always repel. Writing "attract" for two like charges is a marked error.
容易丢掉的分
- 陈述定律需要两个比例关系,而且必须是点电荷。
- 要说排斥还是吸引。只有大小没有方向的答案不完整。
- 球与球之间的 $r$ 从球心量起,而且要在平方之前把厘米换成米。
- 两个同号的相同物体总是排斥。对两个同号电荷写"吸引"是会被扣分的错误。
You've got it
- Coulomb's law: $F = \dfrac{Q_1Q_2}{4\pi\varepsilon_0 r^2}$, along the line joining the charges, repulsive for like and attractive for opposite
- in words: directly proportional to the product of the charges and inversely proportional to the square of the separation, for point charges
- it is an inverse-square law, the same shape as gravitation, but charge has two signs so it can repel
- a sphere acts as a point charge at its centre for points outside, and the field inside a hollow conductor is zero
你掌握了
- 库仑定律:$F = \dfrac{Q_1Q_2}{4\pi\varepsilon_0 r^2}$,沿两电荷连线,同号排斥、异号吸引
- 用文字说:对点电荷,力与两电荷的乘积成正比、与间距的平方成反比
- 它是平方反比定律,与万有引力同形,但电荷有两种符号所以能排斥
- 对外部各点,球表现为位于球心的点电荷;空心导体内部场为零