Electric field of a point charge
| English | Chinese | Pinyin |
|---|---|---|
| point charge | 点电荷 | diǎn diàn hè |
| inverse-square | 平方反比 | píng fāng fǎn bǐ |
| vector sum | 矢量和 | shǐ liàng hé |
| dipole | 偶极子 | ǒu jí zi |
How strong, how far?
- Close to a single charge the field is intense; step back and it weakens fast.
- A clean formula tells us exactly how fast.
- And it mirrors gravity almost perfectly.
Field of a point charge 点电荷
- At distance $r$ from a point charge $Q$: $E = \dfrac{Q}{4\pi\varepsilon_0 r^{2}}$.
- It points out from a positive charge, in toward a negative one, and falls as $\dfrac{1}{r^{2}}$.
Electric field of a positive point charge: field lines radiate outward, and the field strength falls as $1/r^2$ — just like gravity.
The potential near a point charge varies as 1/r — positive for a positive charge, negative for a negative one
Field of a point charge
E ∝ Q/r²
The electric field of a point charge spreads out radially and obeys the inverse-square law.
The field strength at distance $r$ from a point charge $Q$ is:
From $E = F/q$ with Coulomb's force, the test charge cancels, leaving $E = \dfrac{Q}{4\pi\varepsilon_0 r^{2}}$.
The field is $100\ \dfrac{\text{N}}{\text{C}}$ at distance $r$ from a charge. What is it at $2r$?
Inverse-square: $\dfrac{100}{2^{2}} = 25\ \dfrac{\text{N}}{\text{C}}$.
The field points ____ from a positive point charge.
A positive charge pushes a positive test charge away, so its field points outward.
Just like gravity
- Compare $E = \dfrac{Q}{4\pi\varepsilon_0 r^{2}}$ with $g = \dfrac{GM}{r^{2}}$ — the same inverse-square 平方反比 shape.
- The difference: charge can be ±, so the field can attract or repel; gravity only attracts.
A Van de Graaff generator builds up a large static charge on its metal dome, making a strong electric field around it
The electric field of a point charge falls off with distance just like gravity (as 1/r²).
Both are inverse-square: $E = \dfrac{Q}{4\pi\varepsilon_0 r^{2}}$ and $g = \dfrac{GM}{r^{2}}$.
Many charges
- For several charges, the total field is the vector sum 矢量和 of each one's field.
- Add them as arrows (size and direction), not just numbers.
Field-line patterns for parallel plates, a dipole 偶极子, a point charge, and a charged sphere above an earthed plate
For several charges, the total field at a point is:
Fields are vectors, so add them with both size and direction.
You've got it
- field of a point charge: $E = \dfrac{Q}{4\pi\varepsilon_0 r^{2}}$ (out from +, in to −)
- it falls as $\dfrac{1}{r^{2}}$ — the same shape as gravity
- combine several fields by vector sum