Magnetic fields

Iron filings trace the field lines around bar magnets.

A magnetic field 磁场 is a region where a moving charge (or a current 电流) feels a force 力. It is made by:

• moving charges (usually a current in a wire), or
• permanent magnets 永磁体 (where it comes from tiny atomic currents).

Field lines

• field lines 场线 point from N to S outside a magnet, and S to N inside (so they form closed loops).
• lines never cross; closer lines mean a stronger field.

The field of a bar magnet: lines run from N to S, strongest near the poles

Iron filings around a bar magnet line up along the field, showing its real shape

Patterns to know:

• bar magnet 条形磁铁 — curved lines from N to S outside, strongest near the poles.
• long straight wire — circles around the wire; the direction comes from the right-hand grip rule 右手定则 (thumb along the current, fingers curl the way the field points).
• flat circular coil — the field through the centre is at right angles to the coil; the coil acts like a small bar magnet.
• long solenoid 螺线管 — the field inside is nearly uniform along the axis, like a stretched bar magnet; outside it falls off fast.

Field around a long straight wire

Field of a solenoid

An iron core 铁芯 inside a solenoid greatly increases the field, because the iron's atomic magnets line up and add to it. This is why electromagnets 电磁铁 and transformers 变压器 have iron cores.

A current $I$ in a wire of length $L$ in a magnetic field of flux density $B$ feels a force

where $\theta$ is the angle between the wire and the field. The force is largest when the wire is at right angles to the field ($F = BIL$) and zero when the wire is along the field.

Worked example. A wire of length $0.20\ \text{m}$ carries a current of $3.0\ \text{A}$ at right angles to a magnetic field of flux density $0.50\ \text{T}$. Find the force on it.

Magnetic flux density

This equation also defines the magnetic flux density 磁通密度 $B$:

So $B$ is the force per unit current per unit length on a wire at right angles to the field. Unit: tesla 特斯拉, $\text{T} = \text{N A}^{-1}\ \text{m}^{-1}$.

Direction — Fleming's left-hand rule

Use the left hand (Fleming's left-hand rule 弗莱明左手定则): first finger = Field, second finger = Current, thumb = force (thrust). Hold the three at right angles.

Fleming's left-hand rule: thumb = force, first finger = field, second finger = current

A charge $Q$ moving at velocity 速度 $v$ through a field feels

with $\theta$ the angle between $v$ and $B$. Same left-hand rule (the second finger is the motion of a positive charge — reverse it for a negative charge). The force is largest when $v$ is at right angles to $B$, and zero when $v$ is along $B$.

Circular motion in a uniform field

A charge moving at right angles to a uniform field 匀强场 feels a force at right angles to both $v$ and $B$. This force does no work (always at right angles to the motion), so the kinetic energy 动能 and speed stay constant — the particle moves in a circle. Set the magnetic force equal to the centripetal force 向心力:

Worked example. A proton (mass $1.7 \times 10^{-27}\ \text{kg}$, charge $1.6 \times 10^{-19}\ \text{C}$) moves at $2.0 \times 10^{6}\ \text{m s}^{-1}$ at right angles to a $0.50\ \text{T}$ field. Find the radius of its circular path.

So the radius depends on the momentum 动量 $mv$. The period is

which does not depend on the speed — a faster particle goes in a bigger circle but takes the same time per turn. If $v$ also has a part along $B$, that part is unchanged, and the path is a helix 螺旋.

Circular path of a charged particle in a magnetic field

Hall effect

A slab of conductor carrying current $I$, in a field $B$ at right angles to the current, develops a voltage across its faces — the Hall voltage 霍尔电压 $V_{\text{H}}$ (the Hall effect 霍尔效应).

The moving charges feel a magnetic force $BQv_{\text{d}}$ ($v_{\text{d}}$ is the drift velocity 漂移速度), so they build up on one face, making an electric field 电场 $E$ that opposes more build-up. At steady state $eE = Bev_{\text{d}}$, so $E = B v_{\text{d}}$. With $V_{\text{H}} = E w$ and $I = n e v_{\text{d}} w t$:

where $q$ is the carrier charge. A Hall probe 霍尔探头 uses this to measure $B$: pass a known current through a thin semiconductor 半导体 slab and read $V_{\text{H}}$ (largest when the slab is at right angles to $B$).

The Hall effect

Velocity selector

A velocity selector 速度选择器 uses crossed electric and magnetic fields to let through only one speed. With the electric force $qE$ and magnetic force $qvB$ set to oppose each other, the net force is zero only when

Particles at speed $E/B$ go straight through; faster or slower ones are deflected.

Velocity selector

Two long parallel wires each sit in the other's magnetic field. Using Fleming's left-hand rule: parallel currents (same direction) attract; antiparallel currents (opposite directions) repel. This is the basis of the SI definition of the ampere.

Magnetic flux

The magnetic flux 磁通量 $\Phi$ through a flat area $A$ at right angles to $B$ is

If the area's normal is at angle $\theta$ to $B$, use $\Phi = B A \cos\theta$. Unit: weber 韦伯, $\text{Wb} = \text{T m}^{2}$. For a coil 线圈 of $N$ turns, the flux linkage 磁链 is $N\Phi = N B A$.

Faraday's and Lenz's laws

When the flux linkage through a circuit changes, an electromotive force 电动势 (e.m.f.) is induced — this is electromagnetic induction 电磁感应.

Faraday's law 法拉第定律: the induced e.m.f. equals the rate of change of flux linkage:

Worked example. A coil of $200$ turns and area $0.010\ \text{m}^{2}$ sits with its plane at right angles to a $0.50\ \text{T}$ field. The field falls steadily to zero in $0.20\ \text{s}$. Find the average induced e.m.f.

The flux linkage changes from $N\Phi = NBA = 200 \times 0.50 \times 0.010 = 1.0\ \text{Wb}$ to zero, so

Lenz's law 楞次定律: the induced e.m.f. acts to oppose the change that makes it. This is conservation of energy 能量守恒 — if it reinforced the change, energy would come from nothing. Combined:

What changes the flux?

• changing $B$ (moving a magnet near a coil),
• changing area $A$ (a rod sliding along rails),
• changing orientation (a coil turning in a field — the a.c. generator, next topic).

Demonstrations

• moving a bar magnet into a coil deflects a galvanometer 检流计; the deflection reverses when the magnet is pulled out (Lenz's law), and is larger for faster motion (Faraday's law).

Demonstrating electromagnetic induction

• a copper disc swinging into a field is quickly slowed — eddy currents 涡流 are induced that oppose the motion.

Eddy-current damping

What makes the induced e.m.f. larger

From $\varepsilon = N\,d\Phi/dt$ with $\Phi = BA$: more turns $N$, a stronger $B$, a larger area $A$, or a faster change — each gives a larger induced e.m.f.

• Force on a current $F = BIL\sin\theta$; on a moving charge $F = BQv$; find the direction with Fleming's left-hand rule.
• A charge moving perpendicular to $B$ moves in a circle ($BQv = mv^2/r$).
• Electromagnetic induction: e.m.f. $=$ rate of change of flux linkage (Faraday); its direction opposes the change (Lenz's law).