Coulomb's law
| English | Chinese | Pinyin |
|---|---|---|
| Coulomb's law | 库仑定律 | kù lún dìng lǜ |
| permittivity of free space | 真空电容率 | zhēn kōng diàn róng lǜ |
| repulsive | 排斥 | pái chì |
| attractive | 吸引 | xī yǐn |
| equipotential | 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.
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
Coulomb's law
F ∝ Qq / r²
Like charges repel, unlike attract — and the force follows 1/r².
Coulomb's law gives the force between two point charges as:
Proportional to each charge, inversely proportional to the distance squared.
Two charges of the same sign repel each other.
Like charges repel; opposite charges attract — both along the line joining them.
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.
Stating Coulomb's law in words needs which elements? Select all that apply.
The law covers like and opposite charges alike; the sign decides only whether the force repels or attracts.
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.
Two charges repel with $36\ \text{N}$ at separation $r$. What is the force at separation $3r$?
Inverse-square: $\dfrac{36}{3^{2}} = \dfrac{36}{9} = 4\ \text{N}$.
Coulomb's law is an ____-square law (the force is proportional to 1/r²).
Force $\propto \dfrac{1}{r^{2}}$ — the same distance dependence as gravity.
Two charges repel with a force F. The separation is trebled. What is the new force?
An inverse-square law: three times the distance means nine times weaker. No constants are needed for a ratio question like this.
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.
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?
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.
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.
Put the steps of finding the charge on two equally charged hanging spheres in order.
A closed triangle of the three forces is equally acceptable, but the answer must show all three forces either way.
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.
Select all the true statements.
Like charges repel (not attract). The other three are all correct.
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.
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