Force on a current-carrying conductor
Introduced| English | Meaning |
|---|---|
| magnetic flux density/mæɡˈnetɪk flʌks ˈdensɪti/ | The force per unit current per unit length on a wire at right angles to the field. |
| Fleming's left-hand rule/ˈflemɪŋz left hænd ruːl/ | A rule linking conventional current, magnetic field and force directions. |
| tesla/ˈteslə/ | The unit of magnetic flux density, equal to one weber per square metre. |
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.
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.
The force on a current-carrying wire in a magnetic field is:
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Largest when the wire is perpendicular to the field ($\sin 90^{\circ} = 1$, so $F = BIL$).
The force on the wire is zero when it lies along the field.
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With the wire parallel to the field, $\theta = 0$ and $\sin\theta = 0$, so there is no force.
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?
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$F = BIL = 0.50 \times 2.0 \times 0.10 = 0.10\ \text{N}$.
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.
Magnetic flux density is measured in the ____.
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$1\ \text{T} = 1\ \dfrac{\text{N}}{\text{A}\cdot\text{m}}$ — the force per unit current per unit length.
What is magnetic flux density?
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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.
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
Feel the force on the wire
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.
In Fleming's left-hand rule, the thumb shows the:
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First finger = Field, second finger = Current, thumb = force/Motion.
Match each finger of Fleming's left-hand rule to what it represents.
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First-Field, seCond-Current, thuMb-Motion. The left hand is for the motor effect; the right hand is for the generator effect.
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.
A $0.10\ \text{m}$ wire carries $2.0\ \text{A}$ at $30^{\circ}$ to a $0.25\ \text{T}$ field. What is the force on it, in newtons?
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$F = BIL\sin 30^{\circ} = 0.25 \times 2.0 \times 0.10 \times 0.5 = 0.025\ \text{N}$, half the perpendicular value. Parallel to the field the force would be zero.
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.
Two parallel wires carrying currents in the same direction attract each other.
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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.
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.
Put the steps of measuring a magnetic flux density with a current balance in order.
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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.
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.
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