Ch.9 Electromagnetic Induction

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Overview

Induced Currents

Faraday discovered that a changing magnetic field created an induced current in a circuit.

Motional emf

We first consider a stationary magnet with a moving circuit.
Suppose we have a conductor of length ll with velocity v⃗\vec{v} moving through a uniform magnetic field B⃗\vec{B}. The charges inside the conductor experience a force of F⃗B=qv⃗×B⃗\vec{F}_B=q\vec{v}\times\vec{B} with magnitude FB=qvBF_B=qvB, separating the positive and negative charges, generating an electric field.
They stop separating when FE=qEF_E=qE cancels out with FBF_B, or
E=vBE=vB
So, this creates an electric field with magnitude E=vBE=vB
Assuming the positive charges are above and the negative charges are below, we can find the potential difference as
ΔV=Vtop−Vbottom=−∫0lEydy=−∫0l(−vB)dy=vlB\Delta V=V_\text{top}-V_\text{bottom}=-\int_0^lE_ydy=-\int_0^l(-vB)dy=vlB
Thus, the motion of a the wire through a magnetic field induces a potential difference, the motional emf, of ε=vlB\varepsilon=vlB

Consider a moving conductor attached to a stationary circuit moving at a constant velocity in a constant perpendicular magnetic field.
If the total resistance is RR, we have an induced current according to Ohm's law
I=εR=vlBRI=\frac{\varepsilon}{R}=\frac{vlB}{R}
The magnetic field exerts a force F⃗mag\vec{F}_\text{mag} to the left on the conductor, so we need an external force to the right to keep it at a constant velocity. The required force is
Fpull=Fmag=IlB=(vlBR)lB=vl2B2RF_\text{pull}=F_\text{mag}=IlB=\left(\frac{vlB}{R}\right)lB=\frac{vl^2B^2}{R}

The power exerted by a force pushing or pulling an object with velocity vv is P=FvP=Fv. So, the power exerted by a the pull force on the conductor is
Pinput=v2l2B2RP_\text{input}=\frac{v^2l^2B^2}{R}
The circuit also dissipates energy through resistance and heat at a rate of Pdissipated=IR2=v2l2B2RP_\text{dissipated}=IR^2=\frac{v^2l^2B^2}{R}
These are the same, meaning it is the energy provided by the external force that separates the charges and creates electric energy. Such a device is a generator.

Eddy Currents

Consider pulling a nonmagnetic (no force at rest) metal through a magnetic field. This exerts a magnetic force on the charge carriers in the metal, but there are no wires defining the cirucit path, so two "whirlpools" of current begin to circulate, called eddy currents
The magnetic force on an eddy current is opposite that of the velocity, so an external force is needed to pull a metal out or to push one in. Though often undesirable, there are some uses.
Train car braking systems use a powerful electromagnet to generate an eddy current in the rail to brake the car. It also heats the rail instead of the break. Magnetic brakes are very efficient.

Magnetic Flux

The magnetic flux is defined analogously to electric flux.
Φm=A⃗⋅B⃗=ABcos⁡θ\Phi_\text{m}=\vec{A}\cdot\vec{B}=AB\cos\theta
with SI unit weber (1 Wb=1 Tm21\text{ Wb}=1\text{ Tm}^2)
In a nonuniform field, consider each small dAdA to get
Φm=∫areaB⃗⋅dA⃗\Phi_\text{m}=\int_\text{area}\vec{B}\cdot d\vec{A}

Lenz's Law

Lenz's Law states that a closed, conducting loop has an induced current iff the magnetic flux through the loop is changing, with direction opposite of the change of flux

Changes in magnetic flux can happen if:

Faraday's Law

A changing magnetic field creates an induced emfε\varepsilon, which with a complete circuit having resistance RR, creates an induced current
Iinduced=εRI_\text{induced}=\frac{\varepsilon}{R}
Faraday's Law lets us determine the emf.
ε=−dΦmdt\varepsilon=-\frac{d\Phi_\text{m}}{dt}
with the direction such that it follows Lenz's Law.
For an NN coil wire, the induced emfs act as batteries in series, adding the induced emf.
ε=−NdΦmdt\varepsilon=-N\frac{d\Phi_\text{m}}{dt}
Faraday's Law tells us that an induced emf can be created through any change in magnetic flux, regardless of how it was created
ε=−ddtΦm=−(B⃗⋅dA⃗dt+A⃗⋅dB⃗dt)\varepsilon=-\frac{d}{dt}\Phi_\text{m}=-\left(\vec{B}\cdot\frac{d\vec{A}}{dt}+\vec{A}\cdot\frac{d\vec{B}}{dt}\right)

Induced Fields

When a magnetic field changes, it induces an electric field, called the induced electric field, that is the mechanism that creates the moving charges in an induced current in a stationary loop. This kind of electric field caused by changes in B⃗\vec{B} is called a non-Coulomb electric field, as opposed to a Coulomb electric field created by charges.
This field is non-conservative, so there is no associated potential energy; thus, there is no association between this field and electric potential. However, we can use ε=W/q\varepsilon=W/q.
For a small movement dsds, the work done is dW=F⃗⋅ds⃗=qE⃗⋅ds⃗dW=\vec{F}\cdot d\vec{s}=q\vec{E}\cdot d\vec{s} around a closed curve, giving Wclosed curve=q∮E⃗⋅ds⃗W_\text{closed curve}=q\oint\vec{E}\cdot d\vec{s}, so
ε=Wclosed curveq=∮E⃗⋅ds⃗\varepsilon=\frac{W_\text{closed curve}}{q}=\oint\vec{E}\cdot d\vec{s}
If we let only the magnetic field change and restrict ourselves to situations where the loop is perpendicular to the magnetic field, we have Faraday's law as ε=∣dΦm/dt∣=A∣dB/dt∣\varepsilon=|d\Phi_m/dt|=A|dB/dt|, so
∮E⃗⋅ds⃗=A∣dBdt∣\oint\vec{E}\cdot d\vec{s}=A\left|\frac{dB}{dt}\right|
We can evaluate this for one loop of the field inside a solenoid (r<Rr<R) to get
∮E⃗⋅ds⃗=2πrE=(πr2)∣dBdt∣⟹Einside=r2∣dBdt∣\oint\vec{E}\cdot d\vec{s}=2\pi rE=(\pi r^2)\left|\frac{dB}{dt}\right|\implies E_\text{inside}=\frac{r}{2}\left|\frac{dB}{dt}\right|

Electromagnetic Wave

Maxwell hypothesized the existance of induced magnetic fields, in symmetry with induced electric fields. This suggested the possibility of electric fields inducing magnetic fields, which induced electric fields, creating a self-sustaining system free of charges or currents. This would be possible if they were in the form of electromagnetic waves. They must be a transverse wave, where E⃗⊥B⃗\vec{E}\perp\vec{B} and both E⃗\vec{E} and B⃗\vec{B} are perpendicular to v⃗\vec{v}, and calculated the velocity to be the speed of light. This lead to the conclusion that light is an electromagnetic wave.

Induced Currents Applications

Generators

Something (e.g. a wind turbine, water, etc.) turns a coil of wire in a constant magnetic field, changing the magnetic flux and inducing a current. Rotating slip rings press against brushes to remove the induced current. The generated emf resembles a sine wave.

The flux through the coil is
Φm=A⃗⋅B⃗=ABcos⁡θ=ABcos⁡ωt\Phi_m=\vec{A}\cdot\vec{B}=AB\cos\theta=AB\cos\omega t
By Faraday's Law, the induced emf is
εcoil=−NdΦmdt=−ABNddt(cos⁡ωt)=ωABNsin⁡ωt\varepsilon_\text{coil}=-N\frac{d\Phi_m}{dt}=-ABN\frac{d}{dt}(\cos\omega t)=\omega ABN\sin\omega t
Since the emf alternates in sign, the current alternates in direction, generating what we call a AC voltage

Transformers

Two coils are wrapped around an iron core. An oscillating voltage V1cos⁡ωtV_1\cos\omega t drives the primary coil with N1N_1 turns. This generates a magnetic field through the iron core, which passes through the secondary coil, inducing an emf and generating an oscillating voltage V2cos⁡ωtV_2\cos\omega t
B∝1/N1B\propto1/N_1 (by coil's inductance), and emfsec∝N2emf_\text{sec}\propto N_2 by Faraday's Law, giving
V2=N2N1V1V_2=\frac{N_2}{N_1}V_1
Based on the ratio N2/N1N_2/N_1, the voltage V2V_2 can be transformed to a higher or lower voltage than V1V_1. Thus, this device is a transformer
Step-up transformers convert with extremely high voltage to reduce loss due to wire resistance during transfer. A step-down transformer converts this to a household voltage.

Metal Detectors

A metal detector has a transmitter coil with a high-frequency alternating current and receiver coil. The current generates a magnetic field along its axis, creating an induced current in the receiver.
If a metal is placed between them, the current generates eddy currents in the metal, which counteracts the magnetic field, reducing the current induced in the receiver coil. This signifies metal. Eddy currents cannot flow in insulators, so this only detects metals.

Inductors

Just like capacitors store energy in the electric field, inductors store energy in the magnetic field.

A coil in a circuit is called an inductor with the potential difference across being the induced emf. An ideal inductor has no resistance.
The inductanceLL is the flux-to-current ratio:
L=ΦmIL=\frac{\Phi_m}{I}
with SI unit henry defined as
1 H≡1 Wb/A=1 T m2/A1\text{ H}\equiv1\text{ Wb/A}=1\text{ T m}^2\text{/A}

Since it relies on magnetic flux, it is only interesting if the current is changing.
We use Faraday's Law and the definition of inductance to find
∣εcoil∣=∣dΦmdt∣=L∣dIdt∣|\varepsilon_\text{coil}|=\left|\frac{d\Phi_m}{dt}\right|=L\left|\frac{dI}{dt}\right|
The induced current in the coil opposes the change in the existing current, making it difficult to actually change the current through an inductor.
In the direction of the current, the potential difference across the inductor is
ΔVL=−LdIdt\Delta V_L=-L\frac{dI}{dt}
in like with the convention for the potential difference across a resistor.
In short, increasing current means potential decrease; decreasing current means potential increase in the direction of current.

When a switch is suddenly opened in a circuit with a resistor, a massive potential difference is created, which can cause a spark across the opened switch.

Energy and Power

From P=IΔVP=I\Delta V, we have
Pelec=−ILdIdtP_\text{elec}=-IL\frac{dI}{dt}
indicating an increasing current causes the circuit to lose electrical energy, instead being stored in the (magnetic field of the) inductor at a rate
dULdt=ILdIdt\frac{dU_L}{dt}=IL\frac{dI}{dt}
Integrating, we have
∫0ULdUL=∫0IILdI⟹UL=12LI2\int_0^{U_L}dU_L=\int_0^IILdI\implies U_L=\frac{1}{2}LI^2
From L=μ0N2A/lL=\mu_0N^2A/l and B=μ0NI/lB=\mu_0NI/l, we also have
UL=12μ0AlB2U_L=\frac{1}{2\mu_0}AlB^2
Since AlAl is the volume, we have the magnetic field energy density as
uB=12μ0B2u_B=\frac{1}{2\mu_0}B^2

LC Circuits

Telecommunication is based on oscillating electromagnetic signals produced by a circuit containing an inductor and a capacitor, called a LC Circuit
Like a block attached to a spring, because an inductor resists change in current, when the capacitor's charge runs out, the inductor still produces a current, flipping the excess charge on the capacitor.

Starting with Kirchoff's Law, and clockwise II, we have
ΔVC+ΔVL=0\Delta V_C+\Delta V_L=0
The potential difference across a capacitor is given by ΔVC=Q/C\Delta V_C=Q/C and across an inductor is given by ΔVL=−LdI/dt\Delta V_L=-LdI/dt
Since the current is charge leaving the capacitor, we have
I=−dQdt⟹dIdt=−d2Qdt2I=-\frac{dQ}{dt}\implies \frac{dI}{dt}=-\frac{d^2Q}{dt^2}
yielding
d2Qdt2=−1LCQC⟹Q=Q0cos⁡(ωt)\frac{d^2Q}{dt^2}=-\frac{1}{LC}\frac{Q}{C}\implies Q=Q_0\cos\left(\omega t\right)
where ω=1/LC\omega=\sqrt{1/LC}
We also know the current, which is
I=−dQdt=Imaxsin⁡(ωt)I=-\frac{dQ}{dt}=I_\text{max}\sin\left(\omega t\right)
where Imax=ωQ0I_\text{max}=\omega Q_0

LR Circuits

If a circuit instead had an inductor and resistor (and maybe a battery) it is a LR Circuit
Suppose we had a circuit connected to a battery for a long time, making a steady current I0I_0, then quickly switch it so the battery is no longer in the circuit (without causing a spark). Then we have
ΔVres+ΔVL=0⟹−IR−LdIdt=0\Delta V_\text{res}+\Delta V_L=0\implies -IR-L\frac{dI}{dt}=0
Then we rearrange and integrate
∫0t−RLdt=∫I0IdII⟹−tL/R=ln⁡(II0)⟹I=I0e−t/(L/R)\int_0^t-\frac{R}{L}dt=\int_{I_0}^I\frac{dI}{I}\implies-\frac{t}{L/R}=\ln\left(\frac{I}{I_0}\right)\implies I=I_0e^{-t/(L/R)}
We have time constant τ=L/R\tau=L/R with dimensions of time