磁场与电磁学

A magnetic field 磁场 $\vec{B}$ is a vector field 矢量场 that surrounds magnets, moving charges, and currents; it determines the magnetic force on any moving charge placed in it. Magnetic field lines 磁感线 must form closed loops: they leave the north pole, return to the south pole, and continue through the magnet. Denser lines mean a stronger field.

Closed loops are the content of Gauss's law for magnetism, the second of Maxwell's equations 麦克斯韦方程组:

The net magnetic flux through any closed surface is zero – as many field lines leave as enter. That is exactly the statement that isolated magnetic monopoles 磁单极子 do not exist. Every source of magnetism is a magnetic dipole 磁偶极子 (a north–south pair), made by circulating charge – in materials, the motion of electrons. Break a bar magnet in half and you get two smaller dipoles, never a lone pole. Like poles repel, opposite poles attract, and a free dipole – a compass 指南针 – rotates to line up with the local field. Earth's own field is approximately a dipole, which is why a compass works. Dipole fields weaken with distance.

How a material responds to an external field depends on how its internal dipoles behave:

• Ferromagnetic 铁磁性 (iron, nickel, cobalt): an external field aligns whole magnetic domains 磁畴, and the alignment survives after the field is removed – a permanent magnet.
• Paramagnetic 顺磁性 (aluminum, titanium): dipoles align weakly with the field but relax when it is removed.
• Diamagnetic 抗磁性 (all materials): the electronic structure produces a usually weak alignment opposite to the field.

The strength of a material's response is its magnetic permeability 磁导率. Free space has the constant value $\mu_0$ (the vacuum permeability); the permeability of matter differs from $\mu_0$ and is not even constant – it varies with temperature, orientation, and field strength.

A moving charge does two things: it creates a magnetic field, and it feels a force in an external one. The field it creates at a point is perpendicular 垂直 to both its velocity and the position vector from charge to point (right-hand rule again), and is largest where those two are perpendicular – zero directly ahead of or behind the motion.

The force on a charge moving in a field $\vec{B}$ is the cross product

perpendicular to both $\vec{v}$ and $\vec{B}$ – point the fingers of your right hand along $\vec{v}$, curl toward $\vec{B}$, and the thumb gives the force on a positive charge (reverse it for negative). Because $\vec{F}_B\perp\vec{v}$, the magnetic force does no work on the charge: it changes direction, never speed.

A charge moving perpendicular to a uniform field therefore travels in a circle, the magnetic force supplying the centripetal force 向心力:

Notice the period $T$ does not depend on the speed – faster particles ride larger circles in the same time.

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 $2.0\times10^{5}\ \text{m/s}$, perpendicular to it. The magnetic force $F=qvB=1.6\times10^{-14}\ \text{N}$ bends it into a circle of radius $r=\dfrac{mv}{qB}=\dfrac{1.67\times10^{-27}(2.0\times10^{5})}{1.6\times10^{-19}(0.50)}=4.2\times10^{-3}\ \text{m}$.

In a region with both an electric and a magnetic field, the two forces act independently and add as vectors. Balancing them makes a velocity selector 速度选择器: with $\vec{E}$ and $\vec{B}$ crossed, only charges with $qE=qvB$, i.e. $v=E/B$, pass straight through. The Hall effect 霍尔效应 is the same physics inside a conductor: a field with a component perpendicular to the current pushes the moving carriers sideways, charge builds up on one face, and a measurable potential difference appears across the conductor – its sign reveals whether the carriers are positive or negative.

A current is a stream of moving charges, so it creates a magnetic field. The Biot–Savart law 毕奥-萨伐尔定律 adds up the field of each current element:

Around any straight segment the field vectors are tangent 相切 to concentric circles 同心圆 centred on the wire – no component toward, away from, or along the wire. Curl your right hand around the wire with the thumb along the current: your fingers give the field direction.

The Biot–Savart integrals AP expects: a long straight wire ($B=\dfrac{\mu_0 I}{2\pi r}$, or a point on the perpendicular bisector of a finite wire), and the centre of a circular loop,

where every element $d\vec{l}$ is perpendicular to $\hat{r}$ and equally distant $R$ – say that in an FRQ derivation. An arc that is a fraction of a full circle contributes that same fraction of $\mu_0 I/2R$ at its centre.

A field also pushes on a current-carrying wire, element by element:

Worked example. Two long parallel wires a distance $d=0.10\ \text{m}$ apart each carry $5.0\ \text{A}$ in the same direction. Wire 1's field at wire 2 is $B=\dfrac{\mu_0 I}{2\pi d}=1.0\times10^{-5}\ \text{T}$, so wire 2 feels $\dfrac{F}{L}=I B=5.0\times10^{-5}\ \text{N/m}$, pulled toward wire 1. Same-direction currents attract; opposite currents repel.

For symmetric current distributions, Ampère's law 安培定律 finds the field far faster than Biot–Savart:

Choose an Amperian loop 安培环路 that matches the symmetry, so that $B$ is constant along the loop (and parallel to it) and comes out of the integral. AP applies it to long straight wires, long solenoids, and conductors carrying a current density 电流密度 (slabs and cylinders); combinations are handled by superposition 叠加.

Worked example. Find the field $0.10\ \text{m}$ from a long wire carrying $5.0\ \text{A}$. A circular loop of radius $r$ shares the field's symmetry, so $B(2\pi r)=\mu_0 I$ and

A long solenoid 螺线管 (assumed: uniform field inside, negligible field outside) is the other classic. Take a rectangular loop with one side of length $L$ inside, parallel to the axis: only that side contributes to the integral, so $BL=\mu_0 (nL) I$ and

with $n$ the turns per metre.

Worked example (cylinder). A solid cylindrical conductor of radius $R$ carries current $I$ spread uniformly over its cross-section. Inside ($r<R$), a circular loop encloses $I_{\text{enc}}=I\dfrac{r^2}{R^2}$, so $B=\dfrac{\mu_0 I r}{2\pi R^2}$ – the field grows linearly to the surface, then falls off as $1/r$ outside.

Exam skill. Every Ampère's-law answer earns its marks in the setup: name the loop, state why $B$ is constant and parallel to it (symmetry), and count $I_{\text{enc}}$ carefully before you solve. Maxwell's fourth equation adds one more idea you should know qualitatively: a changing electric field also creates a magnetic field – but AP will not ask you to compute with that term.

• The magnetic force $\vec F=q\vec v\times\vec B$ is perpendicular to velocity — use the right-hand rule and note it does no work.
• A charge in a uniform field moves in a circle with $r=\tfrac{mv}{qB}$.
• Find the field of currents with Biot–Savart $d\vec B=\tfrac{\mu_0}{4\pi}\tfrac{I\,d\vec l\times\hat r}{r^2}$ or Ampère's law $\oint \vec B\cdot d\vec l=\mu_0 I_{enc}$ when there is symmetry.
• Force on a wire is $\vec F=I\vec L\times\vec B$; keep the cross-product direction straight.
• Distinguish the force on a charge from the field the current creates.

Common mistakes

• Using the right-hand rule unchanged for a negative charge. $\vec F = q\vec v\times\vec B$: for an electron the force is opposite to the right-hand result.
• Saying a magnetic field changes a charge's speed. The force is perpendicular to $\vec v$, so it does no work.
• Using Ampère's law where the field is not symmetric. Like Gauss's law, it gives $B$ only for a loop on which $B$ is constant and tangent.
• Using the total number of turns for $n$ in $B = \mu_0 nI$. $n$ is turns per metre.