Electromagnetic Induction

Put an iron pan on an induction hob and it is hot within seconds, yet the glass top under it stays cool, and a glass bowl placed there would not warm up at all. Under the glass is a coil carrying a rapidly alternating current. Its changing magnetic field induces currents in the metal base of the pan, and those currents heat the pan directly. Electric guitars, wireless phone chargers, transformers and every power-station generator use the same idea: a changing magnetic flux makes an emf. This unit is about how to calculate that emf, its direction, and the forces and energy that come with it.

Magnetic flux 磁通量 measures how much magnetic field passes through a surface. For a uniform field through a flat loop it is the dot product $\Phi_B=\vec{B}\cdot\vec{A}=BA\cos\theta$; in general it is the surface integral 面积分

The area vector 面积矢量 is perpendicular to the surface (outward from a closed one), and the sign of the flux comes from the dot product. Flux changes if $B$ changes, the area changes, or the loop turns – keep all three routes in mind, because each one is an exam question.

Worked example (flux from a long wire). A long straight wire carries $I=20\ \text{A}$. A rectangular loop lies in the same plane, with its sides of length $b=0.30\ \text{m}$ parallel to the wire, its near side $a=0.010\ \text{m}$ away and its far side $c=0.060\ \text{m}$ away. The field $B=\dfrac{\mu_0 I}{2\pi r}$ changes across the loop, so cut it into strips of width $dr$ and area $b\,dr$:

If the current falls steadily to zero in $0.10\ \text{s}$, the average emf in the loop is $2.2\times10^{-5}\ \text{V}$.

A changing flux induces an emf 电动势 – Faraday's law 法拉第定律:

With constant area, $\varepsilon=-A\,\dfrac{dB_\perp}{dt}$; with constant field, $\varepsilon=-B\,\dfrac{dA_\perp}{dt}$. A coil of $N$ turns multiplies the single-loop emf: $|\varepsilon_{\text{sol}}|=N\left|\dfrac{d\Phi_B}{dt}\right|$.

Lenz's law 楞次定律 is the minus sign: the induced current 感应电流 flows so that its own magnetic field opposes the change in flux that created it. Push a magnet toward a loop and the loop pushes back; pull it away and the loop pulls it in. Use the right-hand rule 右手定则 to turn "oppose the change" into a current direction.

A rod of length $L$ sliding at speed $v$, with the rod, velocity and magnetic field mutually perpendicular, is the special case worth memorising – the motional emf 动生电动势 $\varepsilon=BLv$.

Worked example. A $0.20\ \text{m}$ rod slides at $3.0\ \text{m/s}$ across a $0.50\ \text{T}$ field: $\varepsilon=BLv=0.50(0.20)(3.0)=0.30\ \text{V}$. Equivalently, if a single loop's flux drops from $0.020\ \text{Wb}$ to $0.008\ \text{Wb}$ in $0.030\ \text{s}$, the average emf is $\varepsilon=\dfrac{0.012}{0.030}=0.40\ \text{V}$.

Faraday's law is also the third of Maxwell's equations 麦克斯韦方程组, in a deeper form: a changing magnetic flux creates a circulating electric field, $\oint\vec{E}\cdot d\vec{l}=-\dfrac{d\Phi_B}{dt}$ – that field is what pushes the charges around the loop. Together, Maxwell's equations predict electromagnetic waves 电磁波 travelling at $c=1/\sqrt{\varepsilon_0\mu_0}$ (you should know this connection, but AP will not ask you to derive it).

Once an induced current flows, the external field exerts forces on it ($\vec{F}=\int I\,d\vec{l}\times\vec{B}$) – and by Lenz's law those forces always resist the motion that causes the induction. Only the segments of the loop actually inside the field feel a force, which can make a loop accelerate, rotate, or brake. This is the origin of eddy currents 涡电流 braking, and you can apply Newton's second law to a moving loop or rod like any other mechanics problem.

The classic setup is a rod sliding on conducting rails connected by a resistor:

Worked example (the full chain). Rails $L=0.20\ \text{m}$ apart with resistance $R=0.60\ \Omega$ sit in a $0.50\ \text{T}$ field into the page. The rod is pushed at a constant $3.0\ \text{m/s}$. Then: $\varepsilon=BLv=0.30\ \text{V}$; $I=\varepsilon/R=0.50\ \text{A}$; the field pushes back on the rod with $F=BIL=0.50(0.50)(0.20)=0.050\ \text{N}$. The pushing force does work at $P=Fv=0.15\ \text{W}$ – exactly the $P=I^2R=0.15\ \text{W}$ dissipated in the resistor. Mechanical work becomes electrical energy: that is a generator 发电机, and energy is conserved. Released with no push, the rod slows exponentially: $ma=-\dfrac{B^2L^2}{R}v$.

Worked example (falling bar, terminal velocity). A bar of mass $m=0.020\ \text{kg}$ and length $L=0.25\ \text{m}$ falls from rest down vertical rails joined at the top by $R=0.50\ \Omega$, in a horizontal field $B=0.40\ \text{T}$. At speed $v$ the current is $\dfrac{BLv}{R}$, so the upward magnetic force is $\dfrac{B^2L^2v}{R}$. Newton's second law:

At terminal velocity the two forces balance: $v_T=\dfrac{mgR}{B^2L^2}=\dfrac{(0.020)(9.8)(0.50)}{(0.40)^2(0.25)^2}=9.8\ \text{m/s}$. Separating variables gives $v(t)=v_T\big(1-e^{-t/\tau}\big)$ with $\tau=\dfrac{mR}{B^2L^2}=1.0\ \text{s}$. At terminal velocity gravity does work at $mgv_T=1.9\ \text{W}$, which equals $I^2R$ with $I=1.96\ \text{A}$: all the lost gravitational energy heats the resistor.

Loops moving through a field region. The emf depends on the change in flux, not on the flux itself. When a loop enters a field region, the flux grows and a current flows; while the loop is wholly inside a uniform field, the flux is constant, so there is no current and no magnetic force, even though the loop is full of field. Only the edge that is in the field feels a force, and it always opposes the motion.

Worked example (rotating ring). A ring of area $A=0.50\ \text{m}^2$ and resistance $R=2.0\ \Omega$ spins at $\omega=10\ \text{rad/s}$ about a diameter perpendicular to a uniform field $B=0.50\ \text{T}$. Then $\Phi_B=BA\cos\omega t$ and $\varepsilon=-\dfrac{d\Phi_B}{dt}=BA\omega\sin\omega t$, so $\varepsilon_{\max}=BA\omega=2.5\ \text{V}$ and $I_{\max}=1.25\ \text{A}$. The emf is largest when the ring's plane is parallel to the field, where the flux is zero but changing fastest. Spin it twice as fast and $\varepsilon_{\max}$ doubles, so the power $\varepsilon^2/R$ is four times as large.

Exam skill. FRQs walk this exact chain: flux $\to$ emf $\to$ current $\to$ force $\to$ Newton's second law. Write each link separately and check the direction with Lenz's law at the end.

Inductance 电感 is a conductor's tendency to oppose a change in its own current: changing current changes its own flux, which self-induces an emf. From Faraday's law,

An inductor 电感器 is a circuit element built to have large inductance – usually a solenoid 螺线管, where geometry gives

with $N$ total turns, area $A$, length $\ell$, and the magnetic permeability 磁导率 of the core. (Straight wires are modelled as having zero inductance.) An inductor carrying current $I$ stores energy in its magnetic field:

which can later be dissipated in a resistor or moved into a capacitor – conservation of energy applies as usual.

In an RL circuit RL电路 starting with zero current, Kirchhoff's loop rule for a battery $\varepsilon$, resistor $R$, and inductor $L$ in series gives a differential equation 微分方程:

The time constant 时间常数 $\tau$ sets the pace: after one $\tau$ a rising current reaches about $63\%$ of its final value (a decaying one falls to $37\%$); it is also how long the change would take at the initial rate. Learn the two limits – they answer most conceptual questions:

• Just after the switch closes ($t=0$): current cannot jump, so the inductor momentarily blocks it, self-inducing an emf equal and opposite to the applied potential difference.
• Long after ($t\gg\tau$): the current is steady, $dI/dt=0$, and the inductor behaves as a plain wire – the exact opposite of a capacitor.

Worked example. $\varepsilon=12\ \text{V}$, $R=6.0\ \Omega$, $L=3.0\ \text{H}$: final current $\varepsilon/R=2.0\ \text{A}$, $\tau=L/R=0.50\ \text{s}$. At $t=0.50\ \text{s}$ the current is $2.0(1-e^{-1})\approx1.3\ \text{A}$. At $t=0$ the inductor's potential difference is the full $12\ \text{V}$; as $t\to\infty$ it falls to zero. The current approaches its final value exponentially, and the inductor voltage decays exponentially. Stored magnetic energy follows $U_B=\tfrac12 L[I(t)]^2$, so it is not a single exponential during this current growth.

Worked example (opening the switch). In the circuit above, a $30\ \Omega$ resistor is also connected in parallel with the inductor. Long after closing the switch, the inductor carries $2.0\ \text{A}$ and acts as a wire, so no current flows in the $30\ \Omega$ resistor. Now open the switch. The inductor's current cannot change instantly, so the full $2.0\ \text{A}$ now flows round the inductor–$30\ \Omega$ loop. The potential difference across that resistor is $(2.0)(30)=60\ \text{V}$, five times the battery's emf. The current then decays with $\tau=L/R=3.0/30=0.10\ \text{s}$, and the resistor dissipates all of the stored $\tfrac12LI^2=\tfrac12(3.0)(2.0)^2=6.0\ \text{J}$. This is why opening a switch on a large inductor can make a spark.

Exam skill (measuring $L$). Past FRQs ask for a straight-line graph that gives an inductance. Measure the time constant for several series resistances $R$: since $\tau=L/R$, a graph of $\tau$ against $1/R$ is a straight line through the origin with slope $L$. From one current–time curve, plot $\ln\!\big(1-I/I_{\max}\big)$ against $t$ instead: the slope is $-R/L$.

An LC circuit LC电路 has no resistance, so nothing dissipates energy: it oscillates 振荡, sloshing energy between the capacitor's electric field and the inductor's magnetic field. The loop rule gives

the same equation as a mass on a spring – simple harmonic motion 简谐运动 with charge playing the role of displacement and

The total energy $\dfrac{q^2}{2C}+\tfrac12LI^2$ stays constant: all in the capacitor at maximum charge, all in the inductor at maximum current.

Worked example. $L=2.0\ \text{H}$, $C=8.0\ \mu\text{F}$: $\omega=\dfrac{1}{\sqrt{2.0(8.0\times10^{-6})}}=250\ \text{rad/s}$, period $T=2\pi/\omega\approx0.025\ \text{s}$. If the capacitor starts charged to $12\ \text{V}$, then $U=\tfrac12CV^2=5.8\times10^{-4}\ \text{J}$, and the maximum current follows from $\tfrac12LI_{\max}^2=U$: $I_{\max}=\sqrt{2U/L}=0.024\ \text{A}$ – conservation of energy 能量守恒, no calculus needed.

• Compute flux $\Phi_B=\int \vec B\cdot d\vec A$ and get the induced EMF from Faraday's law $\varepsilon=-\tfrac{d\Phi_B}{dt}$.
• Use Lenz's law (the minus sign) to fix the direction: the induced current opposes the change in flux.
• Flux changes three ways — changing $B$, changing area, or changing angle — identify which and differentiate.
• For a rod of length $L$ moving at speed $v$ with the rod, motion and field mutually perpendicular, the motional EMF is $BLv$.
• An inductor stores energy $\tfrac12 LI^2$ and resists changes in current (RL time constant $\tau=L/R$).

Common mistakes

• Using the full $B$ in the flux when the field is at an angle. $\Phi_B = BA\cos\theta$, with $\theta$ between $\vec B$ and the normal to the loop.
• Saying the induced current opposes the field. It opposes the change in flux: if the flux falls, the induced field points the same way as the original.
• Treating an inductor like a capacitor at $t = 0$. Just after switching on from zero current, an inductor carries no current (like a break); after a long time it acts as a plain wire.
• Forgetting energy in an $LC$ circuit swaps between $\tfrac12 CV^2$ and $\tfrac12 LI^2$ with the total constant.
• Saying a loop wholly inside a uniform field has a current because it is "in a field". Only a changing flux induces an emf.
• Using $B$ at one distance for the flux from a long wire. The field changes across the loop, so integrate strips: the answer has a $\ln$.
• Thinking an inductor's current stops the moment a switch opens. It keeps flowing, through whatever path is left, and decays with $\tau=L/R$ of that path.