普物二总结——电学部分#

Chap 32/33 The Steady Magnetic Field#

32-1 Basic phenomena#

history

first hard disk 1957:50 platters

Basic Phenomena of Magnetism#

• attract small bits of metal
• have two poles
• like poles repel, and unlike poles attract
• oersted experiment: $B,i\to F$
• Solenoid is similar to a bar magnet
• Interaction between electric currents

the magnetic field line distribution is like electric field of a electdipole pair.

Magnetic Monopoles#

since no monopole has ever been found $\displaystyle \oint \vec B\cdot \mathrm d \vec A=0$

Magnetic field#

(Ampere) molecular current: he bind solenoid and a magnet

electric charge in motion is the source of Magnetic Fields

Ampere’s Law(1820.12.4)#

Ampere’s Law

• Current element $i\mathrm d \vec s$

$$\mathrm d F_{12}\propto \frac{i_1i_2\mathrm d s_1\mathrm d s_2}{r_{12}^2}=\frac{\mu _0}{4\pi } \cdot \frac{i_2\mathrm d \vec s_2\times (i_1\mathrm d \vec s_1\times \hat r_{12})}{r_{12}^2}$$

$\mu_0 =4\pi\times10^{-7}(N\cdot A^{-2})$constant

the consider two examples

eg1

$$\begin{aligned} &d\vec{F}_{12}=\frac{\mu_0}{4\pi}\frac{i_2d\vec{s}_2\times(i_1d\vec{s}_1\times\hat{r}_{12})}{r_{12}^2} \\ &d\vec{s}_1\perp\hat{r}_{12} \\ &\therefore dF_{12}=\frac{\mu_0}{4\pi}\frac{i_1i_2ds_1ds_2}{r_{12}^2}\\&d\vec{F}_{21}=\frac{\mu_0}{4\pi}\frac{i_1d\vec{s}_1\times(i_2d\vec{s}_2\times\hat{r}_{21})}{r_{21}^2}\\&d\vec{s}_2\perp\hat{r}_{21}\\&\therefore dF_{21}=\frac{\mu_0}{4\pi}\frac{i_1i_2ds_1ds_2}{r_{12}^2}\\&\mathrm d F_{12}= \mathrm d F_{21}\end{aligned}$$

$$\mathrm d F_{12}=0\\\mathrm d F_{21}=\frac{\mu_0}{4\pi}\frac{i_1i_2\mathrm d s_1\mathrm d s_2}{r_{12}^2}$$

The Magnetic Induction Strength#

• Coulomb’s Law

• Ampere’s Law $$d\vec{F}_{12}=\frac{\mu_0}{4\pi}\frac{i_2d\vec{s}_2\times(i_1d\vec{s}_1\times\hat{r}_{12})}{r_{12}^2}$$

let us call $i_2\mathrm d \vec s_2$ the element of a test electric current

the Unit $T$(Tesla=$1N/(m\cdot A)=10^4 \mathrm{~Gauss}$)

32-2 The magneticn field of a Current-Carrying loop#

Biot-Savart Law

$$\vec{B}=\frac{\mu_0}{4\pi}\oint_L\frac{id\vec{s}\times\hat{r}}{r^2}$$

$$\begin{aligned}&B=\int_{A_{1}}^{A_{2}}dB=\frac{\mu_{0}}{4\pi}\int_{A_{1}}^{A_{2}}\frac{i\sin\theta dx}{r^{2}}\\&r_{0}=r\sin(\pi-\theta)=r\sin\theta, r=\frac{r_{0}}{\sin\theta}\\&x=-r_{0}ctg\theta,\quad dx=\frac{r_{0}d\theta}{\sin^{2}\theta}\\&B=\int_{\theta_{1}}^{\theta_{2}}\frac{\mu_{0}i}{4\pi}\frac{\sin\theta\cdot\frac{r_{0}d\theta}{\sin^{2}\theta}}{r_{0}^{2}}=\frac{\mu_{0}i}{4\pi r_{0}}\int_{\theta_{1}}^{\theta_{2}}\sin\theta d\theta\\&=\frac{\mu_{0}i}{4\pi r_{0}}(\cos\theta_{1}-\cos\theta_{2})\\&=\frac{\mu _0i}{2\pi r_0}\propto \frac{1}{r_0}\end{aligned}$$

$$\begin{aligned} &\left|d\vec{B}\right|=\left|d\vec{B}^{\prime}\right| \\ &d{\vec{B}}={\frac{\mu_{0}}{4\pi}}{\frac{id{\vec{s}}\times{\hat{r}}}{r^{2}}} \\ &dB_{x}=dB\cdot\cos\alpha \\ &dB=\frac{\mu_{0}}{4\pi}\frac{ids}{r^{2}}\sin\theta \\ &\theta=\frac{\pi}{2}, \sin\theta=1, r=r_{0}/\sin\alpha \\ &B_{x}=\int dB\cos\alpha \\ &B=\frac{\mu_{0}i}{4\pi}\oint\frac{\sin^{2}\alpha}{r_{0}^{2}}\cos\alpha ds\\&=\frac{\mu_{0}i}{4\pi r_{0}^{2}}\sin^{2}\alpha\cos\alpha\cdot2\pi R\\ &B=\frac{\mu_0}{2} \frac{iR^2}{(R^2+r_0^2)^{\frac{3}{2}}}\xrightarrow{r_0=0}B=\frac{\mu_0 i}{2R}\\\ &\xrightarrow {r_0>>R} B=\frac{\mu_0 i R^2}{2r_0^3} \end{aligned}$$

we can define magnetic dipole moment $\mu$ (just like $\vec p= q\vec l$)

$$B=\frac{\mu_0iR^2}{2r_0^3}=\frac{\mu_0i\pi R^2}{2\pi r_0^3}=\frac{\mu_0iA}{2\pi r_0^3}\\\begin{aligned}&\text{Define: }\mu=iA=i\pi R^2(\vec \mu =i\vec A)\\&B=\frac{\mu_{0}}{2}\frac{iR^{2}}{r_{0}^{3}}=\frac{\mu_{0}}{2\pi}\frac{i\pi R^{2}}{r_{0}^{3}}=\frac{\mu_{0}}{2\pi}\frac{\mu}{r_{0}^{3}}\end{aligned}$$

p757 sample33-5

$$\mathrm d B=\frac{\mu _0 di}{2\pi d}=\frac{\mu _0 \frac{i}{a} dx}{2\pi d}\\ d=\frac{R}{\cos \theta }\Rightarrow B_x=\int dB\cos \theta=\frac{\mu_0i}{2\pi a R}\int \cos^2\theta dx$$

then

$dx=\mathrm d (R\tan \theta)=R\frac{\mathrm d}{\cos^2\theta }$

$$B_x=\frac{\mu_0 i}{2\pi a}\int_{-\theta}^{\theta}d\theta =\frac{\mu _0 i}{\pi a}\alpha =\frac{\mu_0 i}{\pi a}\arctan \frac{a}{2R}$$

$R>>a:(\alpha \to \tan \alpha)B=\frac{\mu_0 i}{2\pi R}$

$R\to 0,B=\frac{\mu_0i}{2a}$​

Bohr model of the hydrogen atom

$$\begin{aligned}&a_0=0.529A=5.29\times10^{-11}m\\&\nu=6.63\times10^{15}Hz\\&i=e\nu=1.60\times10^{-19}\times6.63\times10^{15}=1.63\times10^{-3}A\\&B=\frac{\mu_0i}{2R}=\frac{4\pi\times10^{-7}\times1.06\times10^{-3}}{2\times5.29\times10^{-11}}=12.6T\\&\mu_\mathrm{B}=iA=1.63\times10^{-3}\times\pi\times(5.29\times10^{-11})^2\\&=0.923\times10^{-23}A\cdot m^2\end{aligned}$$

called as Bohr Magnon

B of Solenoid#

Solenoid

A constant magnetic field can (in principle) be produced by an $\infty$​ sheet of current. In practice, however, a constant magnetic field is often produced by a solenoid.

Parameters:$i,n,R,L$,we assume $R<<L$

for a circular loop

then

use

with

• $L\to \infty ,\beta _1=0,\beta_2=\pi$

  $B=\frac{1}{2}\mu _0ni(1+1)=\mu_0ni$

• $\beta _1=0,\beta_2=\frac{\pi}{2}$

  $B=\frac{1}{2}\mu_0 n i(1-0)=\frac{1}{2}\mu_0 ni$

The field outside the ideal solenoid is zero.

egsolenoid

$$B=\frac12 \mu_{0}ni(\cos\beta_{1}-\cos\beta_{2})$$

With this

$$\begin{aligned}&ni=\frac{Ni}{L}\Rightarrow\frac{jLdr}{L}=jdr=\frac{Ni}{2l(R_{2}-R_{1})} dr\\&\cos\beta_{2}=-\cos\beta_{1},\cos\beta_{1}=\frac{l}{\sqrt{l^{2}+r^{2}}}\end{aligned}$$$$\begin{aligned} &dB=\frac{1}{2} \mu_{0} \frac{Ni}{2l(R_{2}-R_{1})}\cdot\frac{2l}{\sqrt{l^{2}+r^{2}}} dr \\ &B=\mu_{0}jl\int_{R_{1}}^{R_{2}}\frac{dr}{\sqrt{l^{2}+r^{2}}} \\ &=\mu_{0}jl\ln\frac{R_{2}+\sqrt{R_{2}^{2}+l^{2}}}{R_{1}+\sqrt{R_{1}^{2}+l^{2}}} \end{aligned}$$

in practice

we define $\gamma=\frac{l}{R_1},\alpha =\frac{R_2}{R_1}$

$$B_0=\mu_0hR_1\gamma \ln \frac{\alpha +\sqrt {\alpha^2+\gamma^2}}{1+\sqrt{1+\gamma^2}}$$

32-3 Gauss Law and Ampere’s Loop Law for B#

let’s first review

for $\vec E$

• Gauss law:

  $$\oint\vec{E}\bullet d\vec{A}=\frac1{\varepsilon_0}\sum q, \nabla\bullet\vec{E}=\frac{\rho_e}{\varepsilon_0}$$

• Loop law

  $$\oint \vec E\bullet d\vec l =0,\nabla \times \vec E=0(\vec E=-\nabla V)$$

for $\vec B$

• Gauss law:

  $$\oint\vec{B}\bullet d\vec{A}=0, \nabla\bullet\vec{B}=0$$

• Loop law

  $$\oint \vec B\bullet d\vec l =\mu _0\sum i,\nabla \times \vec B=\mu_0\vec J$$

fact…

$$\oint \vec B\mathrm d \vec l =\mu_0\sum i+\mu_0\varepsilon_0\frac{\mathrm d}{\mathrm dt}\int _S\vec E\mathrm d \vec S$$

The differential form is

$$\nabla \times \vec B =\mu_0\vec J+\mu_0\varepsilon _0\frac{\partial \vec E}{\partial t}$$

egs

[e.g.1]

inf long wire $R$,i uniform distribution

$$B\cdot 2\pi r=\mu_0i\cdot \frac{r^2}{\pi R^2}\Rightarrow B=\frac{\mu_0 i r}{2\pi R^2}(r<R)\propto r$$$$B\cdot 2\pi r=\mu_0i\Rightarrow B=\frac{\mu_0 i }{2\pi r}(r>R)\propto \frac{1}{r}$$

[e.g.2]

Consider an $\infty$​ sheet of current described by n wires/length each carrying current i into the screen as shown. Calculate the B field.

for symmetrydirection on the screen

we catch a w*w square

$$\oint \vec B \bullet \mathrm d\vec l=2Bw=\mu_0 n w i\Rightarrow B=\frac{1}{2}\mu_0 ni$$

[e.g.3]

calculate B for $\infty$​ solenoid

view it as two sheets

$$\oint \vec B\bullet \mathrm d \vec l =Bw=\mu _0 n w i\Rightarrow B=\mu_0 ni$$

[e.g.4]

the B of Toroid ($N$ total turns with current i)

$$\oint \vec B\bullet \mathrm d \vec l=B2\pi r=\mu_0Ni\to B=\mu _0ni(n=\frac{N}{2\pi r})$$

::: tip[Application]

• Power door locks

• Magnetic cranes

• Electronic Switch “relay

:::

32-4 The magnetic force on a carrying-current wire#

remember

EG32-5,P738

$$F_{2}=F_{\perp}=\int_{0}^{\pi}iBR\mathrm d \theta \sin\theta=2iBR$$$$F=F_1+F_2+F_3=iB(2L+2R)$$

for two parallel conductors

we define $i=1A$ to be the current that make two $1m$ apart parallel conductors have force density $2\times 10^{-7}N/m$

for convenience sake, we define $\hat n$ the unit normal vector of current loop

this holds for arbitrary shape loop

for magnetic dipole

we define $\vec \tau=iA(\vec n \times \vec B)$

then

remind that

we have

MRI (Magnetic Resonance Imaging)

Proton Spin and Magnetic Moment:

• A single proton possesses a positive charge $+|e|$ and an intrinsic angular momentum, known as “spin.”
• Naively imagining the charge circulating in a loop gives rise to a magnetic dipole moment $\mu$.

Behavior in an External Magnetic Field ( B ):

• Classically: Torques will be present unless the magnetic moment $\vec{\mu}$ is aligned with or against the magnetic field ( B ).

• Quantum Mechanics (QM): The spin is always aligned either with or against the magnetic field ( B ).

  • Anti-aligned State: Energy $U_2 = \mu B$
  • Aligned State: Energy $U_1 = -\mu B$

• Energy Difference: $\Delta U = U_2 - U_1 = 2\mu B$

$$\begin{aligned}&\mu_{proton}=1.36\times10^{-26}\mathrm{~A~m}^2\quad B=1\text{ Tesla }(=10^4\mathrm{~Gauss})\\&\Delta U=2\mu B=2.7\times10^{-26}\mathrm{J}\end{aligned}$$

$h\nu =\Delta U\Rightarrow \nu=\frac{2.7\times 10^{-26}}{6.6\times 10^{-34}Js}=41\mathrm{MHz}$

more applications

app

1. Magnetic Force on a Current Loop:
   • When a loop of wire carrying an electric current is placed in a magnetic field, the field exerts a torque on the loop, attempting to align the loop’s magnetic dipole moment with the field.
2. Structure of a Galvanometer:
   • Inside a galvanometer, there is a rotating coil attached to a pivot.
   • To return the pointer to its equilibrium position, a spring is included in the galvanometer, which creates a torque in the opposite direction to the rotation of the coil.

motor is almost the same ,we just skip it

• fixed voltagemotors
• fixed currentmotors

32-5 The motion of a charge in a magnetic field#

we define Lorentz Force as

we can find that

$\vec F_L=q\vec v \times \vec B$is the microscopic des.

$\mathrm d \vec F_A = id \vec s \times \vec B$is the macroscopic des.

from this image we can confirm that

then let’s talk about the motion of a charged part. in a magnetic field

• $\vec v\perp \vec B$

  • $$\vec F =q\vec c\times \vec B\\qvB=n\frac{mv^2}{R},R=\frac{mv}{qB}$$

  • period : $T=\frac{2\pi R}{v}=\frac{2\pi m}{qB}$

  • frequency : $f=\frac{qB}{2\pi m}$

• $v_\perp =v\sin \theta ,v_{||} =v\cos \theta$​

  • 
  • Period : $T=\frac{2\pi R}{v\perp}$
  • $h=v_{||}T =\frac{2\pi mv_{||}}{qB}$

appmirror

$$R=\frac{mv_\perp}{qB}\\ h=\frac{2\pi mv }{qB}\\ \theta \to 0,v_\perp \to v\theta ,v_{||}\to v$$

in uniform magnetic field

the path is just circles(directions following the signal of charge)

the detailed application is

• convex lens
• magnetic mirror

appbottle

$$R=\frac{mv_perp}{qB}$$

the magnetic bottle is to confine the hot ionized gases and control the thermonuclear fusion

when taking the strong B, and we can have a small R constraint

discovery of electron by J.J.Thomson in 1897

$$\begin{align*} \vec F=q\vec E+q\vec v\times \vec B\\eE=evB\Rightarrow v=\frac{E}{B}\\y=\frac{1}{2}\frac{eE}{m}\left(\frac{L}{v}\right)^2=\frac{eEL^2}{2mv^2}\\\Rightarrow\frac{e}{m}=\frac{2yE}{B^2L^2} =1.759\times 10^{11}C/kg \end{align*}$$

mass spectrometer

$$R=\frac{mv}{qB}\\\frac{1}{2}mv^2=qV \\\Rightarrow \frac{m}{q}=\frac{R^2B^2}{2V}$$

detailedly we accelerate electrons in a known potential $U=qv$

with looking at the $m/q$ value we can distinguish different particles and approximately calculate their propotion

• paleoceanography
• space exploration
• detect chemical and bio. Weapons(anthrax)

movement measurement

$$R=\frac{mv}{qB}=\frac{\sqrt {2mK}}{qB}$$

the cyclotron

$$qvB=m\frac{v^2}{R}\\ f=\frac{qB}{2\pi m}\\ v_M=BR\frac{q}{m}\\ E_K=\frac{1}{2}mv_M^2 =\frac{B^2R^2q^2}{2m}$$

with relaticity effect

$$\begin{align} m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}\\ T=\frac{2\pi m_0}{qB}\times \frac{1}{\sqrt {1-\frac{v^2}{c^2}}}\\ B\sqrt{1-\frac{v^2}{c^2}}=\mathrm {const} \end{align}$$

for proton

$$\frac{v}{c}=0.3 \rightarrow \sqrt{1-\frac{v^2}{c^2}}=0.95$$

we can have synchrotron withe much larger magnitude B and N

The Hall Effect#

when at equilibrium

(remember $j\cdot b\cdot d\equiv i$ Please)

we define the charge drift speed as $v$,and the volume density of charge is $n$,the Hall resistance $R_H$(this is a definition with replacement ,not a real concept )

the quantized hall effect

when detect hall effect on two-dimensional hall effect , it;s displayed not linearly

$$\frac{\Delta _H}{i}=\frac{B}{nqd}$$

with const $n,q,d$,$R_H\propto B$

1986 Nobel Prize $\frac{h}{e^2}=25812.806\Omega$ we can define it as new standard resistance unit since it’s made of some basic const in late physic

App of Hall effect#

• Measure B
• Measure i P
• Transfer DC->AC

a specific field is ABS: work as B tensor,detect shaft rotation speed and beam oscillating voltage

Chap 34 The Faraday’s Law of Induction#

General Review#

• Electrostatics
  • motion of $q$ in $\vec E$
  • $\displaystyle \int \vec E\cdot d\vec A =\frac{\sum q_{cls}}{\varepsilon}$
• Magnetostatics
  • motion of $q$ in $\vec B$
• Electrodynamics
  • $\frac{d\vec B}{d t}\neq 0\Rightarrow \vec E$
  • AC circuit,inductors,transformers
  • $\frac{d\vec E}{d t}\neq 0\Rightarrow \vec B$

34-1/2 Basic Phenomena & Faraday’s Law of Induction#

just treat $\varepsilon$‘s sign with right hand rule

34-3 Lenz’s Law#

The induced current will appear in such a direction that it opposes the change in flux that produced it.

laminated materials can reduce eddy currents

app : magnetic Induction

• E-M Cannon: alternating voltage to a solenoid ,then the flux changes -> current -> force
• Tape/Hard Drive/ZIP Readout : Tiny coil responds to change in flux as the magnetic domains (encoding 0’s or 1’s) go by.
• Credit Card Readerswipe -> large signal
• Magnetic Levitation (Maglev) Trains: eddy currents produce field in opposite direction

34-4 Motional emf and Induced emf#

• Motional emf: $\frac{d S}{d t}\to \frac{d\Phi}{dt}\to \varepsilon$
• Induced emf: $\frac{d B}{d t}\to \frac{d\Phi}{dt}\to \varepsilon$

Motional emf#

Lorentz force results in a motional emf

Non electrostatic force

Motional emf

Lorentz force can’t do work to electron

P781 34-4

Sol1:

$$d\varepsilon =(\vec v\times \vec B)\cdot d\vec r =-Bvdr\\ \varepsilon =-\frac{1}{2}B\omega R^2$$

Sol2:

$$\varepsilon =\frac{-d\Phi_B}{dt}=-\frac{1}{2}\frac{dB(\frac{1}{2}R^2\theta)}{dt}$$

direction +o -a

Appand Motors

$$\Phi_B=BA\cos \omega t\\\varepsilon =BA\omega \sin \omega t$$

Induced emf#

• Vortex electric field

a increasing magnetic field pass through the loop will generate a ringing electric field.

it just implide that

Es is set up by charges, Ein is set up by changing magnetic field. Both kinds of electric field exert on charges.

The betatron

Betatron produces a pulse rather than a continuous beam.

34-5. Induction & Relative Motion#

Chapter 35(36) Inductance and Magnetic properties of materials#

35-1 Inductance#

Mutual Inductance#

i1 change s2 induced emf $\varepsilon _2$

i2 change s1 induced emf $\varepsilon _1$

$M_{12},M_{21}$ are called inductance constant (Unit: Henry)

AppInductance

• Transformers
• Airport Metal Detectors
• Pacemaker

Self-Inductance#

L —— self-inductance

how to calculate self-inductance?

$i,\vec B,N\to L$

Toroid of rectangular

$$\begin{gathered} \oint\vec{B}\bullet d\vec{l}=\mu_0N\boldsymbol{i} \\ B=\frac{\mu_0\boldsymbol{i}N}{2\pi r} \\ \boldsymbol{\Phi}_B=\int\int\vec{B}\bullet d\vec{A}=\int_a^b\frac{\mu_0\boldsymbol{i}N}{2\pi r}\boldsymbol{h}\mathrm{d}r \\ =\frac{\mu_0iNh}{2\pi}\int_a^b\frac{\mathrm{d}r}r=\frac{\mu_0iNh}{2\pi}\ln\frac ba\quad\therefore L=\frac{N\boldsymbol{\Phi}_B}i=\frac{\mu_0N^2\boldsymbol{h}}{2\pi}\ln\frac ba \end{gathered}$$