Electric Potential Energy · 电势能
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| work/wɜːk/ | 功 | gōng |
| electric potential energy/ɪˈlektrɪk pəˈtenʃl ˈenədʒi/ | 电势能 | diàn shì néng |
Squeeze two like charges together and you bottle up energy
- Push two like charges closer and the field fights you every step.
- The work 功 you do doesn't vanish — it is stored.
- Release the charges and that stored energy pushes them apart, gaining speed.
- Stored energy that depends only on position is potential energy.
把两个同种电荷挤到一起,你就储存了能量
- 把两个同种电荷推近,场每一步都在跟你较劲。
- 你做的功并没有消失——它被储存起来。
- 松手,储存的能量就把它们推开,获得速度。
- 只取决于位置的储存能量就是势能。
Energy to assemble charges
- Electric potential energy 电势能 $U$ is the work to bring the charges from far away to here.
- For two point charges: $U = \dfrac{k\,q_1 q_2}{r}$.
- Note it is $1/r$, not $1/r^2$ — energy, not force.
- $U$ is a scalar (just a number with a sign), so it is easier than force.
装配电荷所需的能量
- 电势能 $U$ 是把电荷从很远处带到此处所做的功。
- 对两个点电荷:$U = \dfrac{k\,q_1 q_2}{r}$。
- 注意是 $1/r$,不是 $1/r^2$——这是能量,不是力。
- $U$ 是标量(只是一个带符号的数),所以比力更好处理。
The potential energy of two point charges is proportional to: · 两个点电荷的电势能与以下哪项成正比：
$U = kq_1q_2/r$ — energy goes as $1/r$ (force goes as $1/r^2$). · $U = kq_1q_2/r$——能量随$1/r$变化（力随$1/r^2$变化）。
Find $U$ (in J) for a $+1\ \text{nC}$ and $+1\ \text{nC}$ charge · 电荷 $1\ \text{m}$ apart. Use $k = 9\times10^9$. · 求 $U$ (单位为焦耳) 对于一个 $+1\ \text{nC}$ 和 $+1\ \text{nC}$ 电荷 $1\ \text{m}$ 相距。使用 $k = 9\times10^9$.
$U = \dfrac{(9\times10^9)(10^{-9})(10^{-9})}{1} = 9\times10^{-9}\ \text{J}$.
The sign tells the story
- Like charges: $U > 0$ — you had to push them together, energy is stored.
- Opposite charges: $U < 0$ — they pulled together, energy is "owed" (bound).
- A more negative $U$ means a more tightly bound pair.
- We choose $U = 0$ when the charges are infinitely far apart.
符号讲述整个故事
- 同种电荷:$U > 0$——你必须把它们推到一起,能量被储存。
- 异种电荷:$U < 0$——它们自己拉到一起,能量是"欠着的"(束缚)。
- $U$ 越负,这对电荷束缚得越紧。
- 我们规定电荷相距无穷远时 $U = 0$。
Two opposite charges have a potential energy that is: · 两个异号电荷的电势能是：
Opposite charges attract, so $U = kq_1q_2/r < 0$ — a bound pair. · 异号电荷相互吸引，因此$U = kq_1q_2/r < 0$——形成束缚态。
We usually set the potential energy to zero when the charges are infinitely ____. · 我们通常设定当电荷无穷____时电势能为零。
The reference $U = 0$ is taken at infinite separation. · 参考$U = 0$取在无穷远处分离。
Work and force from energy
- The work done by the field equals the drop in energy: $W_{\text{field}} = -\Delta U$.
- So moving to lower $U$ releases energy as motion.
- In calculus form the force is the slope of $U$: $F = -\dfrac{dU}{dr}$.
- Energy and force are two views of the same interaction.
从能量得出功与力
- 场做的功等于能量的减少:$W_{\text{field}} = -\Delta U$。
- 所以移到更低的 $U$ 会以运动形式释放能量。
- 用微积分形式,力是 $U$ 的斜率:$F = -\dfrac{dU}{dr}$。
- 能量和力是同一相互作用的两种视角。
Electric potential energy · 电势能
The potential energy of two charges falls off as one over the distance between them. · 两个电荷的电势能随它们之间距离的倒数而衰减。
The work done by the electric field equals the decrease in potential energy, $W = -\Delta U$. · 电场做的功等于势能的减少量$W = -\Delta U$。
A drop in $U$ releases energy as work: $W = -\Delta U$. · $U$的下降释放能量做功：$W = -\Delta U$。
Many charges add up
- For several charges, add the energy of every pair.
- Because $U$ is a scalar, you just add numbers — no vectors.
- $U_{\text{total}} = \sum_{\text{pairs}} \dfrac{k\,q_i q_j}{r_{ij}}$.
- Keep the signs: a mix of $+$ and $-$ can give a net negative energy.
多个电荷相加
- 有几个电荷时,把每一对的能量加起来。
- 因为 $U$ 是标量,你只需把数字相加——没有矢量。
- $U_{\text{total}} = \sum_{\text{pairs}} \dfrac{k\,q_i q_j}{r_{ij}}$。
- 保留符号:$+$ 和 $-$ 混合可能给出净负能量。
Select all · 所有 true statements about electric potential energy. · 选择关于电势能的所有正确陈述。
$U$ is a signed scalar, summed over pairs; like charges give $U>0$. It is never a vector. · $U$是带符号的标量，按对累加；同号电荷给出$U>0$。它绝非矢量。
How much energy to bring a $+2\ \text{nC}$ charge to $3\ \text{cm}$ from a $+3\ \text{nC}$ charge?
- $U = \dfrac{k\,q_1 q_2}{r} = \dfrac{(9\times10^9)(2\times10^{-9})(3\times10^{-9})}{0.03}$.
- $U \approx 1.8\times10^{-6}\ \text{J}$ — positive, since both are positive.
把一个 $+2\ \text{nC}$ 电荷带到距一个 $+3\ \text{nC}$ 电荷 $3\ \text{cm}$ 处需要多少能量?
- $U = \dfrac{k\,q_1 q_2}{r} = \dfrac{(9\times10^9)(2\times10^{-9})(3\times10^{-9})}{0.03}$。
- $U \approx 1.8\times10^{-6}\ \text{J}$——为正,因为两者都是正的。
$U$ is a scalar with a sign, not a force and not a vector. A negative $U$ doesn't mean a negative force — it means the pair is bound (energy would be needed to pull them apart).
$U$ 是一个带符号的标量,不是力也不是矢量。负的 $U$ 不意味着负的力——它意味着这对电荷是束缚的(需要能量才能把它们拉开)。
Electric potential energy is the work to assemble charges: $U = k q_1 q_2 / r$ (a signed scalar, $1/r$). Like charges store $U>0$; opposite charges give $U<0$ (bound). The field's work is $W = -\Delta U$, and $F = -dU/dr$.
电势能是装配电荷所做的功:$U = k q_1 q_2 / r$(一个带符号的标量,$1/r$)。同种电荷储存 $U>0$;异种电荷给出 $U<0$(束缚)。场做的功为 $W = -\Delta U$,而 $F = -dU/dr$。