Force on a current-carrying conductor · 载流导体受到的力
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| magnetic flux density/mæɡˈnetɪk flʌks ˈdensɪti/ | 磁通密度 | cí tōng mì dù |
| Fleming's left-hand rule/ˈflemɪŋz left hænd ruːl/ | 弗莱明左手定则 | fú lái míng zuǒ shǒu dìng zé |
| tesla/ˈteslə/ | 特斯拉 | tè sī lā |
The definition hiding inside a motor
- A wire carrying a current across a magnetic field jumps sideways. Every electric motor ever built is that jump, arranged to keep repeating.
- Look at the force closely and something else falls out of it. Nobody can measure a magnetic field directly, but this force can be measured, and it is proportional to $B$.
- So the motor effect is not only an application; it is how the strength of a magnetic field is defined.
- This lesson is $F = BIL\sin\theta$, the definition of magnetic flux density 磁通密度, and Fleming's left-hand rule 弗莱明左手定则.
藏在电动机里的那个定义
- 一根载着电流横越磁场的导线会侧向跳开。有史以来的每一台电动机都是这一跳,只是被安排成不断重复。
- 仔细看这个力,还能从中得到别的东西。没有人能直接测量磁场,但这个力可以测量,而且它正比于 $B$。
- 所以电动机效应不只是一个应用;它就是磁场强度被定义的方式。
- 这一课讲 $F = BIL\sin\theta$、磁通密度(magnetic flux density)的定义,以及弗莱明左手定则。
The force on a current-carrying wire
- A wire of length $L$ carrying current $I$ at angle $\theta$ to a field of flux density $B$ feels a force:
- The force is largest at right angles, where $\sin\theta = 1$ and $F = BIL$, and zero when the wire lies along the field, where $\sin\theta = 0$.
- The force is at right angles to both the current and the field, which is why a motor turns rather than pulling itself apart.
载流导线所受的力
- 长为 $L$、载有电流 $I$、与磁通密度为 $B$ 的场成 $\theta$ 角的导线受到的力:
- 力在垂直时最大,那时 $\sin\theta = 1$、$F = BIL$;当导线沿着磁场时力为零,那时 $\sin\theta = 0$。
- 力同时垂直于电流和磁场,这就是电动机会转动而不是把自己扯开的原因。
The force on a current-carrying wire in a magnetic field is: · 磁场中载流导线受到的力是:
Largest when the wire is perpendicular to the field ($\sin 90^{\circ} = 1$, so $F = BIL$). · 当导线垂直于场时最大($\sin 90^{\circ} = 1$,所以 $F = BIL$)。
The force on the wire is zero when it lies along the field. · 当导线沿场方向放置时,它受到的力为零。
With the wire parallel to the field, $\theta = 0$ and $\sin\theta = 0$, so there is no force. · 当导线平行于场时,$\theta = 0$,$\sin\theta = 0$,所以没有力。
A $0.10\ \text{m}$ wire carries $2.0\ \text{A}$ at right angles to a $0.50\ \text{T}$ field. What is the force? · 一根 $0.10\ \text{m}$ 的导线载有 $2.0\ \text{A}$ 的电流,与 $0.50\ \text{T}$ 的场成直角。力是多少?
$F = BIL = 0.50 \times 2.0 \times 0.10 = 0.10\ \text{N}$. · $F = BIL = 0.50 \times 2.0 \times 0.10 = 0.10\ \text{N}$。
Magnetic flux density
- Rearranged for a wire at right angles, the same equation defines $B$:
- Magnetic flux density is the force per unit current per unit length on a wire at right angles to the field.
- Its unit is the tesla 特斯拉: $1\ \text{T} = 1\ \text{N/(A m)}$. One tesla is a strong field; the Earth's is about 50 microtesla.
磁通密度
- 对垂直放置的导线移项后,同一个式子定义了 $B$:
- 磁通密度是垂直于磁场的导线上单位电流、单位长度所受的力。
- 它的单位是特斯拉(tesla):$1\ \text{T} = 1\ \text{N/(A m)}$。一特斯拉是很强的场;地球磁场约为 50 微特斯拉。
Magnetic flux density is measured in the ____. · 磁通密度的单位是 ____。
$1\ \text{T} = 1\ \dfrac{\text{N}}{\text{A}\cdot\text{m}}$ — the force per unit current per unit length. · $1\ \text{T} = 1\ \dfrac{\text{N}}{\text{A}\cdot\text{m}}$——每单位电流、每单位长度上的力。
What is magnetic flux density? · 什么是磁通密度?
B = F/(IL) for a wire at right angles, in tesla. The definition comes straight from the motor-effect force, which is how B is measured. · 垂直导线的 B = F/(IL),单位特斯拉。这个定义直接来自电动机效应的力,而这正是测量 B 的方式。
Fleming's left-hand rule
- Hold the thumb and first two fingers of your left hand mutually at right angles.
- First finger is the Field, seCond finger is the Current, thuMb is the force, or Motion.
- Left hand for the motor effect, where a current produces motion. The right hand is for the generator effect, where motion produces a current, and mixing them up reverses every answer.
Three quantities, three directions, all mutually perpendicular
弗莱明左手定则
- 把左手的拇指和前两指互相垂直伸开。
- First finger(食指)是 Field(磁场),seCond finger(中指)是 Current(电流),thuMb(拇指)是力,即 Motion(运动)。
- 电动机效应用左手——电流产生运动。右手用于发电机效应——运动产生电流,而把两者混淆会让每个答案都反过来。
三个量、三个方向,两两垂直
Feel the force on the wire · 感受导线上的力
A current in a magnetic field feels a force F = BIL at right angles to both — reverse the current or flip the magnet and the force jumps the other way. · 磁场中的电流受到一个力 F = BIL,方向与电流和磁场都成直角——反转电流或翻转磁铁,力就朝相反方向跳动。
In Fleming's left-hand rule, the thumb shows the: · 在弗莱明左手定则中,拇指表示:
First finger = Field, second finger = Current, thumb = force/Motion. · 食指 = 场(Field),中指 = 电流(Current),拇指 = 力/运动(Motion)。
Match each finger of Fleming's left-hand rule to what it represents. · 把弗莱明左手定则中的每根手指与它代表的量配对。
First-Field, seCond-Current, thuMb-Motion. The left hand is for the motor effect; the right hand is for the generator effect. · 食指-磁场、中指-电流、拇指-运动。左手用于电动机效应;右手用于发电机效应。
Worked example: force on a wire
- A wire of length $0.10\ \text{m}$ carries $2.0\ \text{A}$ at right angles to a field of flux density $0.25\ \text{T}$. Find the force, and then the force if the wire is turned to $30°$ to the field.
- At right angles: $F = BIL = (0.25)(2.0)(0.10) = 0.050\ \text{N}$.
- At $30°$: $F = BIL\sin 30° = 0.050 \times 0.5 = 0.025\ \text{N}$, half as much.
- If the wire were turned parallel to the field the force would be zero. The angle is measured between the wire and the field, not between the wire and anything else.
例题:导线所受的力
- 一根长 $0.10\ \text{m}$ 的导线载有 $2.0\ \text{A}$ 电流,垂直于磁通密度 $0.25\ \text{T}$ 的场。求这个力,再求导线转到与场成 $30°$ 时的力。
- 垂直时:$F = BIL = (0.25)(2.0)(0.10) = 0.050\ \text{N}$。
- $30°$ 时:$F = BIL\sin 30° = 0.050 \times 0.5 = 0.025\ \text{N}$,是原来的一半。
- 如果把导线转到与场平行,力就是零。这个角是在导线和磁场之间量的,不是导线与别的什么之间。
A 0.10 m wire carries 2.0 A at 30 degrees to a 0.25 T field. What is the force on it, in newtons? · 一根 0.10 m 的导线载 2.0 A 电流,与 0.25 T 的场成 30 度。它受到的力是多少牛顿?
F = BIL sin30 = 0.25 x 2.0 x 0.10 x 0.5 = 0.025 N, half the perpendicular value. Parallel to the field the force would be zero. · F = BIL sin30 = 0.25 x 2.0 x 0.10 x 0.5 = 0.025 N,是垂直时的一半。与场平行时力为零。
Two parallel currents
- Each wire sits in the other's field, so each feels a force. Apply the left-hand rule twice and the result is symmetric.
- Currents in the same direction attract. Currents in opposite directions repel.
- This is worth remembering because it is the reverse of what charges do: like charges repel, but like currents attract.
两条平行电流
- 每根导线都处在另一根的磁场中,所以各自受力。用两次左手定则,结果是对称的。
- 同方向的电流相吸。反方向的电流相斥。
- 这值得记住,因为它与电荷的行为相反:同号电荷相斥,而同向电流相吸。
Two parallel wires carrying currents in the same direction attract each other. · 两根载有同方向电流的平行导线互相吸引。
Each wire sits in the other's field; applying the left-hand rule to each gives attraction. It is the opposite of like charges, which repel. · 每根导线都处在另一根的场中;对各自用左手定则可得吸引。这与同号电荷相斥恰好相反。
Worked example: measuring a field
- A stiff wire on a top-pan balance carries a current through a magnetic field. Explain how this measures $B$.
- The wire is placed at right angles to the field, so the force is $F = BIL$.
- The magnet sits on the balance, and by Newton's third law the force on the magnet is equal and opposite to the force on the wire, so the balance reading changes by $F/g$ in kilograms.
- Then $B = F/(IL)$, using the measured force, the current from an ammeter and the length of wire actually in the field.
- The last detail is the one that catches people: $L$ is the length inside the field, not the whole wire.
例题:测量磁场
- 一根置于电子天平上方的硬导线载着电流穿过磁场。解释这怎样测出 $B$。
- 导线垂直于磁场放置,所以力是 $F = BIL$。
- 磁铁放在天平上,由牛顿第三定律,磁铁受到的力与导线受到的力大小相等方向相反,所以天平读数变化 $F/g$ 千克。
- 于是 $B = F/(IL)$,用测得的力、由安培计读出的电流,以及实际处在磁场中的那段导线长度。
- 最后这个细节最容易绊人:$L$ 是场内那一段的长度,不是整根导线。
Put the steps of measuring a magnetic flux density with a current balance in order. · 把用电流天平测量磁通密度的步骤按顺序排列。
Newton's third law is what lets the balance feel the reaction to the force on the wire. L is the length inside the field, not the whole wire. · 牛顿第三定律让天平感受到导线受力的反作用力。L 是场内那一段的长度,不是整根导线。
Marks that slip away
- $\theta$ is the angle between the wire and the field. The force is zero when they are parallel, not maximum.
- $L$ is the length of wire in the field.
- Left hand for the motor effect, right hand for the generator effect. Say which you are using.
- Parallel currents in the same direction attract, which is the opposite of like charges.
容易丢掉的分
- $\theta$ 是导线与磁场之间的夹角。两者平行时力为零,不是最大。
- $L$ 是处在磁场中那段导线的长度。
- 电动机效应用左手,发电机效应用右手。要说清你在用哪一个。
- 同方向的平行电流相吸,这与同号电荷恰好相反。
You've got it
- $F = BIL\sin\theta$: maximum at right angles, zero along the field, and the force is perpendicular to both current and field
- rearranged, it defines magnetic flux density: the force per unit current per unit length on a wire at right angles, in tesla
- Fleming's left-hand rule: First finger Field, seCond finger Current, thuMb Motion, for the motor effect
- parallel currents attract when in the same direction and repel when opposite
你掌握了
- $F = BIL\sin\theta$:垂直时最大,沿场方向时为零,而力同时垂直于电流和磁场
- 移项后它定义了磁通密度:垂直导线上单位电流、单位长度所受的力,单位特斯拉
- 弗莱明左手定则:食指磁场、中指电流、拇指运动,用于电动机效应
- 平行电流同向时相吸,反向时相斥