Intro Physics II
Administrivia
Intro Physics II Physics 11b Lecture 18
Summary
!
Calculating Fields r r µ 0 I1 dlr × rˆlx 2 dB( x ) = ! Superposition l 4 Field of Loops µ = IA ! Magnetic Dipoles ! !
!
Intrinsic dipoles Torque
Ampere’s Law !
Hour Exam #2 Average 140 / 200 ! Equivalent to B/B+
Test %
Course
50 60 70 80 90
772.5 824 868.5 906 936.5
C+ B-/B B/B+ AA-/A
What We Will Learn Today
Magnetic Fields / Biot-Savart Law !
!
!
Magnetic Induction Faraday’s Law
!
!
New HW #8 due Friday at 4 PM Lab #4 begins this week
!
r
Line integral
∫
r
r B
r r B dl
!
Magnetic Force Example
!
Magnetic Induction
!
Faraday’s Law
r r B = dB
∫
µ 2(IA 3 ) 0
µ 0 I encl
1
Cardboard Speaker !
!
Ferromagnetism
Wonder of magnetism only !
!
Induction not necessary
We saw week ago
Can induce dipole moment in a material with external field
!
Speaker played even when I held magnet still
!
S
! !
N !
!In iron and other ferromagnetic materials
Speaker is due to !
Current
Stationary magnet Movable membrane attached to current-carrying coil
!
Small B-field torques all unpaired, free electrons Dipole aligns to local field Dipoles have themselves a magnetic field This serves to greatly increase the total magnetic field outside of the material
!
Force of B-field on current
r r F ( x) r r = I×B l
! !
Current oscillates, so force oscillates
!
As coil moves, there is some “backinduction”
!
Source
!
Generally small
Formalism of Ferromagnetism !Replace the µ0 in magnetism laws by µ ! ! !
“Magnetic permeability” For iron, µ can be 1000 or more Ampere and iron-core solenoid ! If surface you use in Ampere’s law contains magentically permeable material
r r B d l = µ0 I
the effect of currents !
!
Torus of radius r and thickness t !
!
Iron-core toroids and solenoids
Looks like a solenoid bent around in on itself ! B-field follows line of solenoid “axis”
Three relevant areas of B-field !
!
r r B d l = µI
Almost all real induction ∫ use ∫ devices ferromagnetic cores to multiply !
Formalism of Ferromagnetism
Inside torus ! Path A encloses 0 current Outside torus ! Path B encloses current ! !
r t
+2prNI -2prNI
Total enclosed current is 0 Interior of the windings ! B is constant along path C (symmetry) ! 2prB=m2prNI ! B=mNI !
!
∫
r r B d l = µI
2
What We Know About E and B ∫∫
Closed Surface
r r Q E ⋅ da = encl ε0
Gauss’ Law
!
r r ∫∫ B ⋅ da = 0
Closed Surface
r
No monopoles
r
∫ E ⋅ dl = 0
Closed Path
r
!
!
Energy conservation (Kirchoff)
r
∫ B ⋅ d l = µI
We know there is some connection between E and B
!
Magnetic fields caused by moving electric charges (currents) Magnetic force on electric charge proportional to q Loops of electric current cause magnetic dipole fields very similar in shape to electric dipole fields
But our “fundamental Ampere’s equations” don’t contain much enlightenment on this count Law ! We’ve so far been dealing Lorenz with static cases Force ! Static, unvarying Law charges and currents
Magnetic Flux, Redux !
! ! !
!
!
encl
Closed
rPath r r r r r r F ( x ) = q (E ( x ) + v × B( x ))
Open-Surface Flux of a Loop
We didn’t spend too much time on magnetic flux No monopoles No magnetic “charge” No magnetic flux over closed surface ! Enclosed charge is always 0
Calculate a magnetic flux the same way as electric flux !
Take each patch of area, dot B into normal, multiply by area, sum over the surface
r r r Φ B = ∫∫ B ⋅ nˆ da = ∫∫ B ⋅ da
!
Now we want to consider non-closed surfaces
Changing Current
Small circular surface in plane of large current loop
!Now change the current in the loop
!
!
!
r
Rcoil
Calculate B-field at center !
Ismall=0
!
Flux
∫
µI =− 2R 0
!
µ 0 I ( zˆ ) nˆ da − ⋅
∫ 2R
coil
2 0I da = − µ πr R 2 ∫
!
!
!
Biot-Savart
r µ 0 Idl r 2 dB = ! DirectionB = ! Into page !
r r B ⋅ da =
Coil loop is boundary of surface (circle)
µ dl ∫ 4 R 0
I
2 il
=
µ0 I 2 R il
B-field nearly constant over area Choose positive (CCW) sense for flux calc Φ=-µ0Iπr2/2R
I=I(t)
Flux also a function of time !
r
2
µ 0πr I ( t ) 2R !Suppose we examine what happens on the boundary of the surface Φ(t ) = −
Rcoil
!
Coil
No current in small coil
3
Time-Dependent Magnetic Flux If Φ changes as function of t, find that there is a current induced in the small coil !
!
Small coil is on boundary of the surface
Once Φ is settled, no more current !
!
dΦ dt
There is an “electromotive force” around the boundary of the surface
Actually, what is induced is an electric field, not a force Line integral of E-field (a voltage) on the boundary is the “EMF”
I(t) Ismall0 r R Rcoil
In fact, the voltage here is equal to the change in flux (with a minus sign)
!
! !
Ismall0 r
r r Q E ⋅ da = encl ε0 ClosedSurface
∫∫
R
r r B ⋅ da = 0
Rcoil
Sign: direction of voltage with respect to “sense” of flux calculation
µ 0πr I ( t ) 2R 2
E =
∫∫r r dΦ B E ⋅ dl = 0 − ∫ dt ClosedPath r r ∫ B ⋅ d l = µ0 I encl ClosedSurf ace
2
Φ(t ) = −
This is voltage across resistor What about Kirchoff’s rule?
What We Know About E and B
I(t)
r r dΦ B E ⋅ dl = − closed dt ∫ path
EMF (line-integral of E-field) induced on boundary of a surface is proportional to the time derivative of magnetic flux over the surface !Faraday’s Law needs to supplement Kirchoff’s Law !
1 dΦ R dt dΦ I small Rα dt dΦ Vα dt I smallα
1 dΦ R dt
!
!
Current induced is inversely proportional to resistor
Rcoil
Faraday’s Law
!
!
Current induced is inversely proportional to resistor
I smallα
!
If we put resistor on coil
Ismall0 r
If we put resistor on coil !
!
I(t)
Current is due to change of flux
I smallα !
Time-Dependent Magnetic Flux
µ 0πr I '(t ) 2 R il
ClosedPath
r r F( )
r r r r r E( ) B( )
Gauss’ Law No monopoles Energy conservation (Kirchoff) + Faraday’s Law Ampere’s Law Lorenz Force Law
How fields E and B behave (Maxwell eq’ns)
How particles respond to fields
4
EMF and Voltage !
Electric field ! !
!
Line integral around loop is not 0! Violates Kirchoff’s law!? ! Energy conservation!?
Stokes, Curl, and Divergence r r ∫ E ⋅ dl ≠ 0
Induced E-field due to changing B-field flux is a nonconservative field ! !
It is also time-dependent! E-field strength depends on dΦ/dt
!
Stokes
!
Induction
r
!
Stokes Implies
!
Integrand can’t be 0
r ∂Φ ∂Φ ∂Φ , , ∇Φ = ∂x ∂y ∂z r r ∇ × ∇Φ =
( )
r
r
r r ∇ r ×rE ≠ 0 ∇ × (∇ϕ ) ≠ 0
Lenz’ Law Lenz’s Law is a consequence of Faraday’s Law !But it helps you intuitively keep track of the signs !Lenz’ Law !
iˆ r r ∇ × ∇Φ = ∂ x ( ) ∂ ∂Φ ∂
ˆj ∂
∂y ∂Φ ∂
kˆ ∂ ∂z ∂Φ ∂
∂ ∂Φ ∂ ∂Φ ˆ ∂ ∂Φ ∂ ∂Φ ˆ ∂ ∂Φ ∂ ∂Φ iˆ − − − − j + k ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
=0
r
r r E ∫ ⋅ dl ≠ 0 r r r ∫∫ (∇ × E )⋅ da ≠ 0
!Electrostatic E field is gradient of φ
Curl of a Gradient
r
∫∫ (∇ × A)⋅ da = ∫ A ⋅ d l
Curl of Gradient is Always 0
!
!
“The induced current acts to oppose the change in flux” ! I.e. to restore the original situation
Signs in Faraday’s Law actually says this !
As we saw earlier
Lenz’s Law lets you calculate the magnitude first, then use physics to determine the sign
!
Electrodynamic (time-varying) fields cannot be written as the gradient of a scalar potential
5
Quiz
Applications of Induction !How Did the Ring-Flinger Work? !
!
!
The universe would be dark and lonely without induction
!
A metal ring placed atop the coil Coil connected to battery with switch Threw switch and the metal ring is flung
!
E&M starts to be really interesting now
!
!
Applications of induction ! ! !
Microphones
Microphones Generators Transformers
DC Magnetic Motor ! !
Stationary coil, movable membrane As airwaves impinge on membrane ! !
S N
Life would be an endless series of DC circuit problems
B-field magnitude gets stronger and weaker
!
!
Flux gets stronger and weaker during oscillation Induced voltage, measured by voltmeter, oscillates
Record voltage oscillations !
!
Air pressure oscillates Magnet oscillates forward and backwards under the pressure
!
!
Magnetic motor consists of three major parts
!
Same as the pressure oscillations
! !
Source of voltage Region of large B-field Rotating current-carrying coil ! Commutator brushes
Loops of area A starts || to Bfield
!
!
! !
Current I creates dipole ⊥ to B-field ! Into page, in this picture Torque on dipole loop: BIA Begins to rotate under torque
6
DC Magnetic Motor !
Induction Generator
Rotates under torque !
!
Construction is inverse of motor !You force the coil to rotate inside the field by providing mechanical source of power !
At top, brushes disconnect Angular momentum carries loop through small angle
!
Now opposite commutator brushes make contact !
!
!
!
As loop turns !
Current reverses direction Dipole moment switches direction
! !
Induction Generator !
!
!
!
!
!
Vin (from generator)
F=NBacoswt E=-(-NBAωsinωt)= NBAωsinωt
AC electrical voltage from ! ! !
A coil of wire Permanent magnets A steam turbine
Take output of rotating coil Vin Pass through an iron-core solenoid (windings / length = N1/L≡n1) !Wrap another set of N2 windings around the solenoid (second solenoid) !
θ=ωt B.da=BAcosθ=BAcosωt
As loop turns !
Dot product of B.da changes Flux changes Voltage induced around coil ! Current flow in coil
AC Transformer
For constant rate of rotation !
Usually, it is steam turning a turbine
!
!
N1
N2
Field in first solenoid ! B=µn I=µn I sinwt 1 1 0 Flux through second solenoid ! Φ=N BA=µn N AI sinwt 2 1 2 0 ! dΦ/dt=µn N A dI/dt 1 2 ! !
!
E=µn1N2 dI/dt Output voltage of second set of windings Not necessarily the same as input voltage!
7
Summary ! !
Ampere’s Law, modified r Magnetic Flux Redux Φ B = ∫∫ B ⋅ dar = 0 ! !
!
=0 on closed surface ≠0 on open surface
Magnetic Induction !
!
r
r
∫ B ⋅ d l = µI
encl
Closed Surface
r r r Φ B = ∫∫ B ⋅ nˆ da = ∫∫ B ⋅ da
Changing magnetic field → electric field ! Field is nonconservative ! No longer the gradient of a scalar potential Faraday’s Law
E =
r
r
∫ E ⋅ dl = −
Closed Path
dΦ B dt
8
Intro Physics II Physics 11b Lecture 18
Summary
!
Calculating Fields r r µ 0 I1 dlr × rˆlx 2 dB( x ) = ! Superposition l 4 Field of Loops µ = IA ! Magnetic Dipoles ! !
!
Intrinsic dipoles Torque
Ampere’s Law !
Hour Exam #2 Average 140 / 200 ! Equivalent to B/B+
Test %
Course
50 60 70 80 90
772.5 824 868.5 906 936.5
C+ B-/B B/B+ AA-/A
What We Will Learn Today
Magnetic Fields / Biot-Savart Law !
!
!
Magnetic Induction Faraday’s Law
!
!
New HW #8 due Friday at 4 PM Lab #4 begins this week
!
r
Line integral
∫
r
r B
r r B dl
!
Magnetic Force Example
!
Magnetic Induction
!
Faraday’s Law
r r B = dB
∫
µ 2(IA 3 ) 0
µ 0 I encl
1
Cardboard Speaker !
!
Ferromagnetism
Wonder of magnetism only !
!
Induction not necessary
We saw week ago
Can induce dipole moment in a material with external field
!
Speaker played even when I held magnet still
!
S
! !
N !
!In iron and other ferromagnetic materials
Speaker is due to !
Current
Stationary magnet Movable membrane attached to current-carrying coil
!
Small B-field torques all unpaired, free electrons Dipole aligns to local field Dipoles have themselves a magnetic field This serves to greatly increase the total magnetic field outside of the material
!
Force of B-field on current
r r F ( x) r r = I×B l
! !
Current oscillates, so force oscillates
!
As coil moves, there is some “backinduction”
!
Source
!
Generally small
Formalism of Ferromagnetism !Replace the µ0 in magnetism laws by µ ! ! !
“Magnetic permeability” For iron, µ can be 1000 or more Ampere and iron-core solenoid ! If surface you use in Ampere’s law contains magentically permeable material
r r B d l = µ0 I
the effect of currents !
!
Torus of radius r and thickness t !
!
Iron-core toroids and solenoids
Looks like a solenoid bent around in on itself ! B-field follows line of solenoid “axis”
Three relevant areas of B-field !
!
r r B d l = µI
Almost all real induction ∫ use ∫ devices ferromagnetic cores to multiply !
Formalism of Ferromagnetism
Inside torus ! Path A encloses 0 current Outside torus ! Path B encloses current ! !
r t
+2prNI -2prNI
Total enclosed current is 0 Interior of the windings ! B is constant along path C (symmetry) ! 2prB=m2prNI ! B=mNI !
!
∫
r r B d l = µI
2
What We Know About E and B ∫∫
Closed Surface
r r Q E ⋅ da = encl ε0
Gauss’ Law
!
r r ∫∫ B ⋅ da = 0
Closed Surface
r
No monopoles
r
∫ E ⋅ dl = 0
Closed Path
r
!
!
Energy conservation (Kirchoff)
r
∫ B ⋅ d l = µI
We know there is some connection between E and B
!
Magnetic fields caused by moving electric charges (currents) Magnetic force on electric charge proportional to q Loops of electric current cause magnetic dipole fields very similar in shape to electric dipole fields
But our “fundamental Ampere’s equations” don’t contain much enlightenment on this count Law ! We’ve so far been dealing Lorenz with static cases Force ! Static, unvarying Law charges and currents
Magnetic Flux, Redux !
! ! !
!
!
encl
Closed
rPath r r r r r r F ( x ) = q (E ( x ) + v × B( x ))
Open-Surface Flux of a Loop
We didn’t spend too much time on magnetic flux No monopoles No magnetic “charge” No magnetic flux over closed surface ! Enclosed charge is always 0
Calculate a magnetic flux the same way as electric flux !
Take each patch of area, dot B into normal, multiply by area, sum over the surface
r r r Φ B = ∫∫ B ⋅ nˆ da = ∫∫ B ⋅ da
!
Now we want to consider non-closed surfaces
Changing Current
Small circular surface in plane of large current loop
!Now change the current in the loop
!
!
!
r
Rcoil
Calculate B-field at center !
Ismall=0
!
Flux
∫
µI =− 2R 0
!
µ 0 I ( zˆ ) nˆ da − ⋅
∫ 2R
coil
2 0I da = − µ πr R 2 ∫
!
!
!
Biot-Savart
r µ 0 Idl r 2 dB = ! DirectionB = ! Into page !
r r B ⋅ da =
Coil loop is boundary of surface (circle)
µ dl ∫ 4 R 0
I
2 il
=
µ0 I 2 R il
B-field nearly constant over area Choose positive (CCW) sense for flux calc Φ=-µ0Iπr2/2R
I=I(t)
Flux also a function of time !
r
2
µ 0πr I ( t ) 2R !Suppose we examine what happens on the boundary of the surface Φ(t ) = −
Rcoil
!
Coil
No current in small coil
3
Time-Dependent Magnetic Flux If Φ changes as function of t, find that there is a current induced in the small coil !
!
Small coil is on boundary of the surface
Once Φ is settled, no more current !
!
dΦ dt
There is an “electromotive force” around the boundary of the surface
Actually, what is induced is an electric field, not a force Line integral of E-field (a voltage) on the boundary is the “EMF”
I(t) Ismall0 r R Rcoil
In fact, the voltage here is equal to the change in flux (with a minus sign)
!
! !
Ismall0 r
r r Q E ⋅ da = encl ε0 ClosedSurface
∫∫
R
r r B ⋅ da = 0
Rcoil
Sign: direction of voltage with respect to “sense” of flux calculation
µ 0πr I ( t ) 2R 2
E =
∫∫r r dΦ B E ⋅ dl = 0 − ∫ dt ClosedPath r r ∫ B ⋅ d l = µ0 I encl ClosedSurf ace
2
Φ(t ) = −
This is voltage across resistor What about Kirchoff’s rule?
What We Know About E and B
I(t)
r r dΦ B E ⋅ dl = − closed dt ∫ path
EMF (line-integral of E-field) induced on boundary of a surface is proportional to the time derivative of magnetic flux over the surface !Faraday’s Law needs to supplement Kirchoff’s Law !
1 dΦ R dt dΦ I small Rα dt dΦ Vα dt I smallα
1 dΦ R dt
!
!
Current induced is inversely proportional to resistor
Rcoil
Faraday’s Law
!
!
Current induced is inversely proportional to resistor
I smallα
!
If we put resistor on coil
Ismall0 r
If we put resistor on coil !
!
I(t)
Current is due to change of flux
I smallα !
Time-Dependent Magnetic Flux
µ 0πr I '(t ) 2 R il
ClosedPath
r r F( )
r r r r r E( ) B( )
Gauss’ Law No monopoles Energy conservation (Kirchoff) + Faraday’s Law Ampere’s Law Lorenz Force Law
How fields E and B behave (Maxwell eq’ns)
How particles respond to fields
4
EMF and Voltage !
Electric field ! !
!
Line integral around loop is not 0! Violates Kirchoff’s law!? ! Energy conservation!?
Stokes, Curl, and Divergence r r ∫ E ⋅ dl ≠ 0
Induced E-field due to changing B-field flux is a nonconservative field ! !
It is also time-dependent! E-field strength depends on dΦ/dt
!
Stokes
!
Induction
r
!
Stokes Implies
!
Integrand can’t be 0
r ∂Φ ∂Φ ∂Φ , , ∇Φ = ∂x ∂y ∂z r r ∇ × ∇Φ =
( )
r
r
r r ∇ r ×rE ≠ 0 ∇ × (∇ϕ ) ≠ 0
Lenz’ Law Lenz’s Law is a consequence of Faraday’s Law !But it helps you intuitively keep track of the signs !Lenz’ Law !
iˆ r r ∇ × ∇Φ = ∂ x ( ) ∂ ∂Φ ∂
ˆj ∂
∂y ∂Φ ∂
kˆ ∂ ∂z ∂Φ ∂
∂ ∂Φ ∂ ∂Φ ˆ ∂ ∂Φ ∂ ∂Φ ˆ ∂ ∂Φ ∂ ∂Φ iˆ − − − − j + k ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
=0
r
r r E ∫ ⋅ dl ≠ 0 r r r ∫∫ (∇ × E )⋅ da ≠ 0
!Electrostatic E field is gradient of φ
Curl of a Gradient
r
∫∫ (∇ × A)⋅ da = ∫ A ⋅ d l
Curl of Gradient is Always 0
!
!
“The induced current acts to oppose the change in flux” ! I.e. to restore the original situation
Signs in Faraday’s Law actually says this !
As we saw earlier
Lenz’s Law lets you calculate the magnitude first, then use physics to determine the sign
!
Electrodynamic (time-varying) fields cannot be written as the gradient of a scalar potential
5
Quiz
Applications of Induction !How Did the Ring-Flinger Work? !
!
!
The universe would be dark and lonely without induction
!
A metal ring placed atop the coil Coil connected to battery with switch Threw switch and the metal ring is flung
!
E&M starts to be really interesting now
!
!
Applications of induction ! ! !
Microphones
Microphones Generators Transformers
DC Magnetic Motor ! !
Stationary coil, movable membrane As airwaves impinge on membrane ! !
S N
Life would be an endless series of DC circuit problems
B-field magnitude gets stronger and weaker
!
!
Flux gets stronger and weaker during oscillation Induced voltage, measured by voltmeter, oscillates
Record voltage oscillations !
!
Air pressure oscillates Magnet oscillates forward and backwards under the pressure
!
!
Magnetic motor consists of three major parts
!
Same as the pressure oscillations
! !
Source of voltage Region of large B-field Rotating current-carrying coil ! Commutator brushes
Loops of area A starts || to Bfield
!
!
! !
Current I creates dipole ⊥ to B-field ! Into page, in this picture Torque on dipole loop: BIA Begins to rotate under torque
6
DC Magnetic Motor !
Induction Generator
Rotates under torque !
!
Construction is inverse of motor !You force the coil to rotate inside the field by providing mechanical source of power !
At top, brushes disconnect Angular momentum carries loop through small angle
!
Now opposite commutator brushes make contact !
!
!
!
As loop turns !
Current reverses direction Dipole moment switches direction
! !
Induction Generator !
!
!
!
!
!
Vin (from generator)
F=NBacoswt E=-(-NBAωsinωt)= NBAωsinωt
AC electrical voltage from ! ! !
A coil of wire Permanent magnets A steam turbine
Take output of rotating coil Vin Pass through an iron-core solenoid (windings / length = N1/L≡n1) !Wrap another set of N2 windings around the solenoid (second solenoid) !
θ=ωt B.da=BAcosθ=BAcosωt
As loop turns !
Dot product of B.da changes Flux changes Voltage induced around coil ! Current flow in coil
AC Transformer
For constant rate of rotation !
Usually, it is steam turning a turbine
!
!
N1
N2
Field in first solenoid ! B=µn I=µn I sinwt 1 1 0 Flux through second solenoid ! Φ=N BA=µn N AI sinwt 2 1 2 0 ! dΦ/dt=µn N A dI/dt 1 2 ! !
!
E=µn1N2 dI/dt Output voltage of second set of windings Not necessarily the same as input voltage!
7
Summary ! !
Ampere’s Law, modified r Magnetic Flux Redux Φ B = ∫∫ B ⋅ dar = 0 ! !
!
=0 on closed surface ≠0 on open surface
Magnetic Induction !
!
r
r
∫ B ⋅ d l = µI
encl
Closed Surface
r r r Φ B = ∫∫ B ⋅ nˆ da = ∫∫ B ⋅ da
Changing magnetic field → electric field ! Field is nonconservative ! No longer the gradient of a scalar potential Faraday’s Law
E =
r
r
∫ E ⋅ dl = −
Closed Path
dΦ B dt
8
