磁学与电磁学

Drop a strong magnet down a copper pipe and it falls slowly, as if through honey, though copper is not magnetic. Pull a magnet away from a coil and a current flows that tries to pull it back. This unit explains both, and the rule behind them: magnetism never gives energy away for free.

Different materials respond very differently to a magnetic field, and the course names three. Ferromagnetic 铁磁性 materials (iron, nickel, cobalt) can develop strongly aligned magnetic domains 磁畴 and retain some alignment, so suitable materials can be permanent magnets 永磁体. Paramagnetic 顺磁性 materials (aluminium, titanium) align only weakly and do not stay aligned. Diamagnetic 抗磁性 materials - in fact all materials have some diamagnetism - align weakly opposite the field. This behaviour comes from a material property, the permeability 磁导率: free space has a fixed permeability of free space 真空磁导率 $\mu_0$, while the permeability of matter differs from it and is not even constant for a given material.

A magnetic field 磁场 $\vec{B}$ surrounds magnets and moving charges. Magnetic field lines 磁感线 run from a magnet's north pole 北极 to its south pole 南极 outside the magnet, and denser lines mean a stronger field. Magnetic poles always come in pairs – cutting a magnet in half makes two smaller magnets, never an isolated pole 磁极.

A charge moving through a magnetic field feels a magnetic force 磁力:

where $\theta$ is the angle between the velocity and the field. The force is perpendicular 垂直 to both $\vec{v}$ and $\vec{B}$ (use the right-hand rule 右手定则 for a positive charge and reverse it for a negative charge), so it changes direction but not speed – a charge moving perpendicular to a uniform magnetic field 匀强磁场 travels in a circle. A stationary 静止的 charge, or one moving parallel 平行 to the field, feels no magnetic force.

Worked example. A proton ($q=1.6\times10^{-19}\ \text{C}$, $m=1.67\times10^{-27}\ \text{kg}$) enters a $0.50\ \text{T}$ field at $2.0\times10^{5}\ \text{m/s}$, at right angles to the field. The magnetic force is

This force is the centripetal force 向心力, so it bends the proton into a circle of radius Setting $qvB=\dfrac{mv^2}{r}$ and cancelling gives that neat $r=mv/(qB)$ – the principle behind mass spectrometers 质谱仪.

Because a current is moving charge, a magnetic field pushes on a current-carrying wire 载流导线:

A current also creates its own magnetic field: circular field lines wrap around a straight wire (right-hand rule), and a coil (solenoid 螺线管) makes a field like a bar magnet 条形磁铁. This is how electromagnets 电磁铁 and motors 电动机 work.

Worked example. A $0.30\ \text{m}$ length of wire carries $4.0\ \text{A}$ at right angles to a $0.20\ \text{T}$ field. The force on it is $F=BIL=0.20\times4.0\times0.30=0.24\ \text{N}$ – the push that turns a motor's coil.

Worked example (the field around a wire). A long straight wire carries $10\ \text{A}$. The field $2.0\ \text{cm}$ away is

about twice the Earth's field, which is why a compass 指南针 near a current-carrying wire swings. The field falls off as $1/r$: twice as far away, half as strong.

Changing magnetic flux linkage 磁链 produces an emf 电动势 – electromagnetic induction 电磁感应. A current flows if the conducting circuit is closed. In a uniform field, magnetic flux 磁通量 is $\Phi=BA\cos\theta$, where $\theta$ is measured between the field and the loop's normal 法线. For $N$ turns sharing the same flux, Faraday's law 法拉第定律 gives the average induced emf 感应电动势:

The flux changes if the field, the area, or the loop's orientation changes. Lenz's law 楞次定律 (the minus sign) says the induced current 感应电流 produces a magnetic field that opposes the flux change that caused it – the basis of generators 发电机.

Worked example. The magnetic flux through a single loop drops from $0.020\ \text{Wb}$ to $0.008\ \text{Wb}$ in $0.030\ \text{s}$. The average induced emf is

A coil of $N$ turns would give $N$ times this – which is why generators and transformers 变压器 use many-turn coils.

Finding the direction of an induced current is a chain of four questions. Push the north pole of a magnet toward a coil:

1. Which way is the flux through the coil? Field lines leave a north pole, so the flux through the coil points away from the magnet.
2. Is it increasing or decreasing? Increasing, because the magnet is getting closer.
3. What does the induced current do? It makes its own field opposing the change, pointing back toward the magnet.
4. So the coil face nearest the magnet becomes a north pole, and it repels the approaching magnet.

You must push to move the magnet in; that work becomes the electrical energy in the coil. If the induced current helped the motion instead, you would get energy for nothing. That is why the falling magnet in the copper pipe is slowed.

A conducting rod 导体棒 of length $L$ sliding at speed $v$ across a field $B$ sweeps out area, so the flux through the circuit changes. This motional emf 动生电动势 is $\varepsilon=BLv$.

Worked example (a rod on rails). A $0.50\ \text{m}$ rod slides at $4.0\ \text{m/s}$ along rails in a $0.20\ \text{T}$ field. The circuit has a resistance of $2.0\ \Omega$.

• $\varepsilon=BLv=(0.20)(0.50)(4.0)=0.40\ \text{V}$, so $I=\varepsilon/R=0.20\ \text{A}$.
• The current in the rod feels a force $F=BIL=(0.20)(0.20)(0.50)=0.020\ \text{N}$, opposing the motion (Lenz again).
• To keep the rod moving steadily you must push with $0.020\ \text{N}$, a power of $Fv=0.080\ \text{W}$. The resistor 电阻器 dissipates 耗散 $I^2R=(0.20)^2(2.0)=0.080\ \text{W}$. The two match: every joule of electrical energy came from your push.

Answer each before reading the answers below. Electron: $q=1.6\times10^{-19}\ \text{C}$, $m=9.11\times10^{-31}\ \text{kg}$.

1. An electron moves at $3.0\times10^6\ \text{m/s}$ at right angles to a $0.010\ \text{T}$ field. Find the force on it and the radius of its circle.
2. A $0.40\ \text{m}$ wire carries $3.0\ \text{A}$ at $30°$ to a $0.50\ \text{T}$ field. Find the force on it.
3. The flux through each turn of a 50-turn coil falls by $0.0040\ \text{Wb}$ in $0.020\ \text{s}$. Find the average induced emf.
4. The north pole of a magnet is pulled away from a coil. What pole forms on the coil face nearest the magnet, and why?
5. A student says: "A strong magnet held still inside a coil induces a large current." Correct the student.

1. $F=qvB=(1.6\times10^{-19})(3.0\times10^6)(0.010)=4.8\times10^{-15}\ \text{N}$. $r=\dfrac{mv}{qB}=\dfrac{(9.11\times10^{-31})(3.0\times10^6)}{(1.6\times10^{-19})(0.010)}=1.7\times10^{-3}\ \text{m}$.
2. $F=BIL\sin\theta=(0.50)(3.0)(0.40)\sin30°=0.30\ \text{N}$.
3. $\varepsilon=N\dfrac{\Delta\Phi}{\Delta t}=50\times\dfrac{0.0040}{0.020}=10\ \text{V}$.
4. A south pole. The flux from the magnet is decreasing, so the induced current tries to keep it: the coil attracts the departing north pole, opposing the change.
5. Induction needs a changing flux. A magnet held still gives a constant flux, so the induced emf and current are zero, however strong the magnet.

• The magnetic force $F=qvB\sin\theta$ is perpendicular to the velocity, so it changes direction (a circle) but not speed; a stationary charge or one moving along the field feels no force.
• Use $F=BIL$ for the force on a current-carrying wire (the motor effect).
• A current creates a magnetic field (circles around a wire; a solenoid acts like a bar magnet).
• Induction needs a changing flux $\Phi=BA$ — a stationary magnet in a coil induces nothing.
• Faraday: $\varepsilon=\Delta\Phi/\Delta t$ (times $N$ turns); Lenz: the induced current opposes the change (energy conservation).

Common mistakes

• Using the right-hand rule for a negative charge without reversing the answer. For an electron, the force is opposite to the direction the rule gives.
• Saying a magnetic force speeds up or slows down a charge. It is always perpendicular to the velocity, so it changes only the direction; it does no work.
• Expecting an emf from a large but steady flux. The induced emf depends on the rate of change of flux, $\varepsilon = -\Delta\Phi_B/\Delta t$.
• Using $B$ alone instead of the flux. $\Phi_B = BA\cos\theta$: the emf also changes if the loop's area or angle changes.
• Reading Lenz's law as "the induced field opposes the field". It opposes the change in flux: if the flux is falling, the induced field points the same way as the original field.
• Forgetting $\sin\theta$ in $F = qvB\sin\theta$ or $F = BIL\sin\theta$. Motion or current parallel to the field feels no force.