Electric Potential · 电势
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| volt/vəʊlt/ | 伏特 | fú tè |
| electric potential/ɪˈlektrɪk pəˈtenʃl/ | 电势 | diàn shì |
| equipotential/ˌiːkwɪpəˈtenʃl/ | 等势面 | děng shì miàn |
| gradient/ˈɡreɪdɪənt/ | 梯度 | tī dù |
The "height" of the electric landscape
- Imagine the field as a hilly landscape and charge as a ball.
- Potential tells you the "height" at each point — energy per unit charge.
- A $+$ charge rolls downhill (to low potential); a $-$ charge rolls uphill.
- Because it is energy per charge, it is a scalar — no arrows to juggle.
电学地形的"高度"
- 把场想象成起伏的地形,把电荷想象成一个球。
- 电势告诉你每一点的"高度"——单位电荷的能量。
- $+$ 电荷向下滚(到低电势);$-$ 电荷向上滚。
- 因为它是单位电荷的能量,所以是标量——没有箭头要摆弄。
Volts: energy per charge
- The electric potential 电势 $V$ is the potential energy per unit charge: $V = \dfrac{U}{q}$.
- Its unit is the volt 伏特 (1 V $= 1$ J/C).
- For a point charge: $V = \dfrac{kQ}{r}$ — positive near $+$, negative near $-$.
- The energy of a charge placed there is simply $U = qV$.
伏特:单位电荷的能量
- 电势 $V$ 是单位电荷的势能:$V = \dfrac{U}{q}$。
- 它的单位是伏特(1 V $= 1$ J/C)。
- 对点电荷:$V = \dfrac{kQ}{r}$——$+$ 附近为正,$-$ 附近为负。
- 放在那里的电荷的能量就是 $U = qV$。
Electric potential · 电势 $V$ is: · 电势$V$是：
$V = U/q$ — potential energy per unit charge, in volts. · $V = U/q$——单位电荷的电势能，单位为伏特。
Scalars add the easy way
- Potential is a scalar, so many charges just add their $V$ values.
- $V = \sum \dfrac{kQ_i}{r_i}$ — no components, no angles.
- This is why potential is often easier to compute than the field.
- Find $V$ everywhere first, then get the field from it.
标量以简单方式相加
- 电势是标量,所以多个电荷只需把它们的 $V$ 值相加。
- $V = \sum \dfrac{kQ_i}{r_i}$——没有分量,没有角度。
- 这就是为什么电势常常比场更容易计算。
- 先求处处的 $V$,再从它得出场。
Potentials from several charges add as simple scalars — no vectors needed. · 多个电荷产生的电势作为简单标量相加——无需矢量。
$V$ is a scalar: $V = \sum kQ_i/r_i$, just add the numbers. · $V$是标量：$V = \sum kQ_i/r_i$，只需将数值相加。
Two charges give $+50\ \text{V}$ and $-20\ \text{V}$ at a point. Find the total $V$ (in V). · 两个电荷在一点产生$+50\ \text{V}$和$-20\ \text{V}$。求总$V$（单位为V）。
Scalars add: $50 + (-20) = 30\ \text{V}$. · 标量相加：$50 + (-20) = 30\ \text{V}$。
Equipotentials map the field
- An equipotential 等势面 is a line (or surface) of constant $V$.
- No work is done moving a charge along an equipotential.
- Equipotentials are always perpendicular to field lines.
- For a point charge they are circles; for a uniform field, parallel lines.
等势线描绘场
- 等势线是一条 $V$ 恒定的线(或面)。
- 沿等势线移动电荷不做功。
- 等势线始终垂直于电场线。
- 对点电荷它们是圆;对均匀场,是平行线。
Field and equipotentials · 电场与等势面
See how the field lines from a charge cross its circular equipotentials at right angles. · 观察电荷的电场线如何与其圆形等势面垂直相交。
Equipotential surfaces are always ____ to the electric field lines. · 等势面始终与电场线____。
Field lines cross equipotentials at right angles. · 电场线与等势面垂直相交。
Field is the slope of potential
- The field points downhill, from high $V$ to low $V$.
- Its size is the steepness: $E = -\dfrac{dV}{dx}$ (a gradient 梯度).
- Closely packed equipotentials mean a steep slope — a strong field.
- So $E$ (a vector) and $V$ (a scalar) carry the same information.
场是电势的斜率
- 场指向下坡,从高 $V$ 到低 $V$。
- 它的大小就是陡度:$E = -\dfrac{dV}{dx}$(一个梯度)。
- 等势线挨得越密,斜率越陡——场越强。
- 所以 $E$(矢量)和 $V$(标量)携带相同的信息。
Where equipotentials are packed closely together, the field is: · 等势面密集的地方，电场是：
A steep potential slope ($E = -dV/dx$) means a strong field. · 陡峭的电势梯度（$E = -dV/dx$）意味着强电场。
Select all · 所有 true statements about $V$ and $E$. · 选择关于$V$和$E$的所有正确陈述。
$V$ scalar, $E$ vector, $E = -dV/dx$. But $V=0$ does NOT force $E=0$. · $V$ 标量，$E$ 矢量，$E = -dV/dx$。但 $V=0$ 并不强制推出 $E=0$。
Two charges give $V = +30\ \text{V}$ and $V = -12\ \text{V}$ at a point. What is the total potential?
- Potential is a scalar, so just add: $V = 30 + (-12)$.
- $V = +18\ \text{V}$ — no vectors needed.
两个电荷在某点给出 $V = +30\ \text{V}$ 和 $V = -12\ \text{V}$。总电势是多少?
- 电势是标量,所以直接相加:$V = 30 + (-12)$。
- $V = +18\ \text{V}$——不需要矢量。
$V$ is a scalar; $E$ is a vector. They are not the same: at the midpoint between two equal opposite charges, $V = 0$ but $E$ is not zero. Never assume "$V=0$ means $E=0$".
$V$ 是标量;$E$ 是矢量。它们不是一回事:在两个等量异号电荷的中点,$V = 0$ 但 $E$ 不为零。绝不要以为"$V=0$ 就意味着 $E=0$"。
Electric potential is energy per charge, $V = U/q$ (volts), a scalar you can add: $V = \sum kQ_i/r_i$. Equipotentials are constant-$V$ surfaces, perpendicular to the field, and the field is the slope $E = -dV/dx$.
电势是单位电荷的能量,$V = U/q$(伏特),一个可相加的标量:$V = \sum kQ_i/r_i$。等势线是 $V$ 恒定的面,垂直于场,而场是斜率 $E = -dV/dx$。