Intro Physics II
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Intro Physics II Physics 11b Lecture 17
!
New HW due Friday (4/12) at 4 PM
!
Hour Exam #2 Thursday !
Magnetic Dipoles Amperes Magnetic Induction
! !
!
Summary !
!
Calculating Fields r r µ 0 I1 dlr × rˆlx 2 dB( x ) = ! Superposition l 4 Field of Loops ! Magnetic Dipoles ! !
Intrinsic dipoles Torque
EM Review already on website! Problems list emailed
Tomorrow (TF’s do problems)
What We Will Learn Today
Magnetic Fields / Biot-Savart Law !
Reviews ! Tonight (formal ADF review)
µ = IA
r r B = dB
∫
!
Ampere’s Law
!
Magnetic Flux, Redux
!
Magnetic Induction
µ 2(IA 3 ) 0
r r r τ = µ×B
1
Field Due to a Current Loop
Complete Field of Current Loop
r µ I µ IR dB|| = 0 2 cosθdl = 0 3 dl 4πr 4πr
B θ
!
dB
z r I
θ
!
R
Integrate r r µ I B = ∫ dB|| = ∫ dl 0 2 cosθ 4πr µ IR µ IR = 0 3 ∫ dl = 0 3 (2πR ) 4πr 4πr 2µ0 I 2 (πR ) = µ 0 2(IA3 ) = 4πr 3 4π r A=Area of loop r µ 2(IA ) B= 0 4π r 3
Magnetic Dipole
Familiar?
Magnetic Dipoles !
Encountered informally already
A=Area of loop
B
!
r µ 0 2 IA (3 ) B=
HW #4:on-axis field of Edipole
!
dB
!
N
θ
1 2p 3
z
θ
0
You are really allowed to think of it the same way as two monopoles, one North, one South, stuck together !
RHR gives sense of positive direction
!
r
R
4
IA plays role of magnetic dipole (usually called µ)
I
!
!
Dipole is oriented on axis of loop This particular dipole is oriented up ! North up, South down
Bar magnets Or current loops
!
S
!
!
!
As long as you are looking at it from “far away” (compared to N-S distance) Even if it’s really a current loop
Dipole moment goes from south (negative) to north (positive)
!
2
Torque on a Loop !
Magnetic Dipoles in B Fields Result generalizes when torque of dipole at angle to field
Consider square loop to right !
!
Current runs clockwise as seen from side
B-field from top to bottom !Forces
!
!
! !
!
!
!
Force on a current (lecture 13) Sides: none ! v || B Top ! vxB is out of page ! F=IBl Bottom ! vxB into page ! F=IBl
Net effect: torque on loop !
!
F = IB l IBll S
τ = µB
τ=IBl*l/2+IBl*l/2=IBl=2=IBA=IAB=µ µB
!
l/2 N
!
N
IBllsinθ
IBll
Magnetic dipoles are effectively like electric dipoles !
IBll
!
Units are A-m2 instead of C-m Behavior very similar ! Formulae very similar
Classical Model of Intrinsic Electron Dipole Moment !
This classical model is wrong but suggestive
!
Pretend electron is a sphere of charge
!
Electron always spinning on axis
Produces a dipole field
!
Current strength depends on frequency of revolution I=qef f in turn depends on velocity of revolution and radius ! f = 1 / (2πr/v)=v/2π πr Which in turn depend on angular momentum of electron
θ
!
!
!
S
Of course, must treat with quantum mechanics
Our cheesy classical model of the electron again !An electron circulating around the nucleus is a current! !
!
τ = µB sinθ r r r τ = µ×B
Electric dipoles ! Feel net force (not only torque) when field has gradient Magnetic dipoles ! The same
!
Intrinsic Dipoles in Atoms
!
Attraction !
From side
θ:angle of dipole to field
!
r !
r0=classical electron radius (HW #6)
Fixed rotation frequency ! Rotation speed at equator ~1% of the speed of light Mystery of QM
dq
r0
Each little ring is carrying charge (current) contributing to magnetic dipole !As with electric, simply add dipoles of each ring to find total dipole moment !Because QM says e always spinning, the electron always has a magnetic dipole moment !
3
Ampere’s Law r
r
Directions of Closed Paths
r
!
∫ B ⋅ d l = ∫ B ⋅ ˆldl = ∫ B cosθdl = ∫ Bdl = B ∫ dl
I
!
!
µI = 2πrB = 2πr 0 = µ 0 I 2πr
r r ∫ B ⋅ ˆldl = ∫ B ⋅ ˆldl +
r2
+
r
top
bottom
top
=
r1
!
r r µ I B( r ) = 0 ϕˆ 2π r B-field is tangential Path: dot product ! cosθ=1 !
!
side
!
B-field: RHR
Tangential B-field Path: dot product !
side
!
bottom
µ0 I (r2dθ ) − µ 0 I (r1dθ ) 2πr2 2πr1
!
=0
Ampere’s Law
Line integral outside wire ! ! !
!
!
nˆ
cosθ=1 along top cosθ=-1 along bottom cosθ=0 perp. To Bfield ! “Sides” of box
Ampere’s Law
I
nˆ
!Line integral outside wire r ∫ B ⋅ ˆldl ! Annular box
∫ Bdl
!
!For every closed path, we must define a direction in which to circulate about it !There is an associated “normal direction” or “circulation vector” which is related to the path direction by RHR
Circle
∫ B ⋅ ˆldl + ∫ B ⋅ ˆldl
= ∫ Bdl −
I
r
Line integral around wire
Arbitrary shape Can be constructed of annular boxes Inner lines cancel (dot products of +1 and –1) Total contribution is contribution of each box ! Integral=0
!
!
I
Arbitrary shape ! Imagine an shape infinitesimally different, connected to a circle !
! !
!
Does not enclosed current
Sum has to be 0 Circle calculated already: -µ0I ! Sign: rotation sense (RHR) Arbitrary shape equal/opposite: µ0I
Over arbitrary line integral ! Outside current !
I
!
!
I
=0
Around current !
Line integral containing wire !
Put together
µ0I
Ampere’s Law
!
!
∫
r r B d l = µ0 I
(This is half a Maxwell’s equation!) Signs: Line integral taken in right-hand sense about current I ! If you took it in left-hand sense you’d get a minus sign
4
Solenoid
Solenoid !
Solenoid !
!
!
Outside solenoid, field must be weak For each current loop ! Both sides of loop nearly same distance ! Each contribution to B from one bit of loop cancelled by a contribution from other side of loop
!
!
!
!
B=µ0NI
!
Calculate line-integral
!
τ
!
Ampere’s Law Torque
Wanted force law in terms of field B !Found we needed !
!
r
Cross-product in field (Biot-Savart) Cross-product and velocity in force (Lorenz Force)
Found experimentally that there are no confirmed “magnetic charges” (e.g. a single North pole) to be source of magnetic field B !Used Biot-Savart to calculate fields in a few cases
r
r B
Force law on loop
!
!
!
Calculating field of small current loop found current loops are sources of magnetic dipole fields Calculating field of straight wire found Ampere’s Law ! Ampere’s Law and Biot-Savart equivalent ! Could have started with Ampere and derived Biot-Savart
Current-current force
Current force in B-field
Experimental fact: uncharged current-carrying wires feel forces || force: µ0I1I2/2πr2 ! Attractive if currents parallel, repulsive if anti-parallel ⊥ force: 0
Should not depend on longitudinal position, but might on radial position Our box stays at a fixed radius Line integral ! B.dl=0 on sides ! B=0 outside r r ∫ B ⋅ d l = Bl
Magnetism Summary
How We Got Here !
N=number of turns / length
Line integral must equal ! µ I 0 encl=µ0NlI
Let B be the field inside solenoid
Equal
Draw boxlike path through coil at any point
!
Total current enclosed ! NlI !
!
!
!
For box of length l
ΦB =
∫∫
r r B ⋅ da = 0
Closed Surface
Cl
∫
r r B ⋅ d l = µ 0 I encl
dP h
r r F ( x) r r = I×B l
r r
Fpath ( ) Integrate along
Dipole
l
Field lines cannot end
No “magnetic charge” (no monopoles)
=
µ 0 I1 I 2
Force (Newtons)
Magnetic-field
µ
(Tesla)
r µ 0 I dˆl × rˆlx 2 dB = l 4 Small current loop Electron in atomic “orbit” Spinning electron charge
F
r r r r r E( ) B( )
(A-m2 or J/T)
IA
r r F ( x ) ||
B
Sum contributions from all currents (Biot-Savart)
Electric currents (Amps) I
5
Electrostatics U
Potential Energy
What We Know About E and B
Integrate F.x over path
(Joules or N.m)
∆U = q∆V Φ
(Volts; J/Coulomb)
V AB = Φ A − Φ B
Voltage
q ∑ i ch arg es 4πε 0 ri
(Volts; J / C)
V
(Newtons)
Q=CV
Closed Surface
Integrate E.x over path
r r ∫ E ⋅ dl = 0
(N/Coulomb or Volts/m)
Gradient
p Dipole
r E=
(Coulomb. m)
p = qd Charges (Coulomb)
Closed Path
Coulomb’s Law
r r Q Φ ( flux ) = ∫ E ⋅ da = encl ε0
!
! !
!
!
!
!
r
r r B da
ΦB Now we want to consider non-closed surfaces
∫∫
But our “fundamental
Small circular surface in plane of large current loop
∫
µI =− 2R 0
!
Flux !
µ 0 I ( zˆ ) nˆ da − ⋅
∫ 2R
coil
2 0I da = − µ πr 2 R ∫
Biot-Savart
r µ 0 Idl r 2 dB = ! DirectionB = ! Into page !
r r B ⋅ da =
Coil loop is boundary of surface (circle)
Calculate B-field at center !
Ismall=0
Rcoil
Take each patch of area, dot B into normal, multiply by area, sum over the surface
∫∫
!
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
!
No monopoles No magnetic “charge” No magnetic flux over closed surface ! Enclosed charge is always 0
r B nˆ da
!
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
Open-Surface Flux of a Loop
Calculate a magnetic flux the same way as electric flux !
!
Energy conservation (Kirchoff)
r r r r r r F ( x ) = q (E ( x ) + v × B( x ))
We didn’t spend too much time on magnetic flux !
!
Closed rPath
Magnetic Flux, Redux
We know there is some connection between E and B
!
No monopoles
r r ∫ B ⋅ d l = µ0 I encl
q ∑ i 2 rˆi ch arg es 4πε 0 ri Gauss’ Law
q
Gauss’ Law
Closed Surface
E
E-Field
r r Q E ⋅ da = encl ε0
r r ∫∫ B ⋅ da = 0
r r F = qE
Gradient
Electric Potential
∫∫
F
Force
!
!
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
No current in small coil
6
Changing Current
Time-Dependent Magnetic Flux
!Now change the current in the loop !
If Φ changes as function of t, find that there is a current induced in the small coil !
I=I(t)
!
Flux also a function of time µ 0πr 2 !
r
Φ(t ) = −
Rcoil
2 Rcoil
Once Φ is settled, no more current !
I (t )
!
Suppose we examine what happens on the boundary of the surface
!
!
If we put resistor on coil !
Current induced is inversely proportional to resistor
I smallα I
small
1 dΦ
Rα
dΦ
dΦ Vα
In fact, the voltage here is equal to the change in flux (with a minus sign)
!
! !
This is voltage across resistor What about Kirchoff’s rule?
dΦ dt
Coil
Current induced is inversely proportional to resistor
I smallα
1 dΦ R dt
Faraday’s Law There is an “electromotive force” around the boundary of the surface
!
I(t)
Rcoil
If we put resistor on coil !
Time-Dependent Magnetic Flux
Ismall0 r
Current is due to change of flux
I smallα
!
!
Small coil is on boundary of the surface
I(t)
!
Ismall0 r
!
R Rcoil
!
Actually, what is induced is an electric field, not a force Line integral of E-field (a voltage) on the boundary is the “EMF”
Ismall0 r R Rcoil
r r dΦ B E ⋅ dl = − closed dt ∫ path
Sign: direction of voltage with respect to “sense” of flux calculation
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 !
I(t)
2
Φ(t ) = −
µ 0πr I ( t ) 2R 2
E =
µ 0πr I '(t ) 2 R il
7
What We Know About E and B r r Q E ⋅ da = encl ε0 ClosedSurface r r ∫∫ B ⋅ da = 0
∫∫
ClosedSurf ace
r
r
∫ E ⋅ dl = 0 −
ClosedPath
r
dΦ B dt
r
∫ B ⋅ dl = µ I
0 encl
ClosedPath
r r r r r r r F ( x ) = q (E ( x ) + v × B( x ))
Gauss’ Law
Summary !
!
No monopoles Energy conservation (Kirchoff) + Faraday’s Law Ampere’s Law Lorenz Force Law
How fields E and B behave (Maxwell eq’ns)
!
!
=0 on closed surface ≠0 on open surface
Magnetic Induction !
!
encl
r r Φ B = ∫∫ B ⋅ da = 0 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 =
How particles respond to fields
r
0
Restatement of Biot-Savart
Magnetic Flux Redux !
!
r
∫ B ⋅ dl = µ I
Ampere’s Law
r
r
∫ E ⋅ dl = −
Closed Path
dΦ B dt
8
Intro Physics II Physics 11b Lecture 17
!
New HW due Friday (4/12) at 4 PM
!
Hour Exam #2 Thursday !
Magnetic Dipoles Amperes Magnetic Induction
! !
!
Summary !
!
Calculating Fields r r µ 0 I1 dlr × rˆlx 2 dB( x ) = ! Superposition l 4 Field of Loops ! Magnetic Dipoles ! !
Intrinsic dipoles Torque
EM Review already on website! Problems list emailed
Tomorrow (TF’s do problems)
What We Will Learn Today
Magnetic Fields / Biot-Savart Law !
Reviews ! Tonight (formal ADF review)
µ = IA
r r B = dB
∫
!
Ampere’s Law
!
Magnetic Flux, Redux
!
Magnetic Induction
µ 2(IA 3 ) 0
r r r τ = µ×B
1
Field Due to a Current Loop
Complete Field of Current Loop
r µ I µ IR dB|| = 0 2 cosθdl = 0 3 dl 4πr 4πr
B θ
!
dB
z r I
θ
!
R
Integrate r r µ I B = ∫ dB|| = ∫ dl 0 2 cosθ 4πr µ IR µ IR = 0 3 ∫ dl = 0 3 (2πR ) 4πr 4πr 2µ0 I 2 (πR ) = µ 0 2(IA3 ) = 4πr 3 4π r A=Area of loop r µ 2(IA ) B= 0 4π r 3
Magnetic Dipole
Familiar?
Magnetic Dipoles !
Encountered informally already
A=Area of loop
B
!
r µ 0 2 IA (3 ) B=
HW #4:on-axis field of Edipole
!
dB
!
N
θ
1 2p 3
z
θ
0
You are really allowed to think of it the same way as two monopoles, one North, one South, stuck together !
RHR gives sense of positive direction
!
r
R
4
IA plays role of magnetic dipole (usually called µ)
I
!
!
Dipole is oriented on axis of loop This particular dipole is oriented up ! North up, South down
Bar magnets Or current loops
!
S
!
!
!
As long as you are looking at it from “far away” (compared to N-S distance) Even if it’s really a current loop
Dipole moment goes from south (negative) to north (positive)
!
2
Torque on a Loop !
Magnetic Dipoles in B Fields Result generalizes when torque of dipole at angle to field
Consider square loop to right !
!
Current runs clockwise as seen from side
B-field from top to bottom !Forces
!
!
! !
!
!
!
Force on a current (lecture 13) Sides: none ! v || B Top ! vxB is out of page ! F=IBl Bottom ! vxB into page ! F=IBl
Net effect: torque on loop !
!
F = IB l IBll S
τ = µB
τ=IBl*l/2+IBl*l/2=IBl=2=IBA=IAB=µ µB
!
l/2 N
!
N
IBllsinθ
IBll
Magnetic dipoles are effectively like electric dipoles !
IBll
!
Units are A-m2 instead of C-m Behavior very similar ! Formulae very similar
Classical Model of Intrinsic Electron Dipole Moment !
This classical model is wrong but suggestive
!
Pretend electron is a sphere of charge
!
Electron always spinning on axis
Produces a dipole field
!
Current strength depends on frequency of revolution I=qef f in turn depends on velocity of revolution and radius ! f = 1 / (2πr/v)=v/2π πr Which in turn depend on angular momentum of electron
θ
!
!
!
S
Of course, must treat with quantum mechanics
Our cheesy classical model of the electron again !An electron circulating around the nucleus is a current! !
!
τ = µB sinθ r r r τ = µ×B
Electric dipoles ! Feel net force (not only torque) when field has gradient Magnetic dipoles ! The same
!
Intrinsic Dipoles in Atoms
!
Attraction !
From side
θ:angle of dipole to field
!
r !
r0=classical electron radius (HW #6)
Fixed rotation frequency ! Rotation speed at equator ~1% of the speed of light Mystery of QM
dq
r0
Each little ring is carrying charge (current) contributing to magnetic dipole !As with electric, simply add dipoles of each ring to find total dipole moment !Because QM says e always spinning, the electron always has a magnetic dipole moment !
3
Ampere’s Law r
r
Directions of Closed Paths
r
!
∫ B ⋅ d l = ∫ B ⋅ ˆldl = ∫ B cosθdl = ∫ Bdl = B ∫ dl
I
!
!
µI = 2πrB = 2πr 0 = µ 0 I 2πr
r r ∫ B ⋅ ˆldl = ∫ B ⋅ ˆldl +
r2
+
r
top
bottom
top
=
r1
!
r r µ I B( r ) = 0 ϕˆ 2π r B-field is tangential Path: dot product ! cosθ=1 !
!
side
!
B-field: RHR
Tangential B-field Path: dot product !
side
!
bottom
µ0 I (r2dθ ) − µ 0 I (r1dθ ) 2πr2 2πr1
!
=0
Ampere’s Law
Line integral outside wire ! ! !
!
!
nˆ
cosθ=1 along top cosθ=-1 along bottom cosθ=0 perp. To Bfield ! “Sides” of box
Ampere’s Law
I
nˆ
!Line integral outside wire r ∫ B ⋅ ˆldl ! Annular box
∫ Bdl
!
!For every closed path, we must define a direction in which to circulate about it !There is an associated “normal direction” or “circulation vector” which is related to the path direction by RHR
Circle
∫ B ⋅ ˆldl + ∫ B ⋅ ˆldl
= ∫ Bdl −
I
r
Line integral around wire
Arbitrary shape Can be constructed of annular boxes Inner lines cancel (dot products of +1 and –1) Total contribution is contribution of each box ! Integral=0
!
!
I
Arbitrary shape ! Imagine an shape infinitesimally different, connected to a circle !
! !
!
Does not enclosed current
Sum has to be 0 Circle calculated already: -µ0I ! Sign: rotation sense (RHR) Arbitrary shape equal/opposite: µ0I
Over arbitrary line integral ! Outside current !
I
!
!
I
=0
Around current !
Line integral containing wire !
Put together
µ0I
Ampere’s Law
!
!
∫
r r B d l = µ0 I
(This is half a Maxwell’s equation!) Signs: Line integral taken in right-hand sense about current I ! If you took it in left-hand sense you’d get a minus sign
4
Solenoid
Solenoid !
Solenoid !
!
!
Outside solenoid, field must be weak For each current loop ! Both sides of loop nearly same distance ! Each contribution to B from one bit of loop cancelled by a contribution from other side of loop
!
!
!
!
B=µ0NI
!
Calculate line-integral
!
τ
!
Ampere’s Law Torque
Wanted force law in terms of field B !Found we needed !
!
r
Cross-product in field (Biot-Savart) Cross-product and velocity in force (Lorenz Force)
Found experimentally that there are no confirmed “magnetic charges” (e.g. a single North pole) to be source of magnetic field B !Used Biot-Savart to calculate fields in a few cases
r
r B
Force law on loop
!
!
!
Calculating field of small current loop found current loops are sources of magnetic dipole fields Calculating field of straight wire found Ampere’s Law ! Ampere’s Law and Biot-Savart equivalent ! Could have started with Ampere and derived Biot-Savart
Current-current force
Current force in B-field
Experimental fact: uncharged current-carrying wires feel forces || force: µ0I1I2/2πr2 ! Attractive if currents parallel, repulsive if anti-parallel ⊥ force: 0
Should not depend on longitudinal position, but might on radial position Our box stays at a fixed radius Line integral ! B.dl=0 on sides ! B=0 outside r r ∫ B ⋅ d l = Bl
Magnetism Summary
How We Got Here !
N=number of turns / length
Line integral must equal ! µ I 0 encl=µ0NlI
Let B be the field inside solenoid
Equal
Draw boxlike path through coil at any point
!
Total current enclosed ! NlI !
!
!
!
For box of length l
ΦB =
∫∫
r r B ⋅ da = 0
Closed Surface
Cl
∫
r r B ⋅ d l = µ 0 I encl
dP h
r r F ( x) r r = I×B l
r r
Fpath ( ) Integrate along
Dipole
l
Field lines cannot end
No “magnetic charge” (no monopoles)
=
µ 0 I1 I 2
Force (Newtons)
Magnetic-field
µ
(Tesla)
r µ 0 I dˆl × rˆlx 2 dB = l 4 Small current loop Electron in atomic “orbit” Spinning electron charge
F
r r r r r E( ) B( )
(A-m2 or J/T)
IA
r r F ( x ) ||
B
Sum contributions from all currents (Biot-Savart)
Electric currents (Amps) I
5
Electrostatics U
Potential Energy
What We Know About E and B
Integrate F.x over path
(Joules or N.m)
∆U = q∆V Φ
(Volts; J/Coulomb)
V AB = Φ A − Φ B
Voltage
q ∑ i ch arg es 4πε 0 ri
(Volts; J / C)
V
(Newtons)
Q=CV
Closed Surface
Integrate E.x over path
r r ∫ E ⋅ dl = 0
(N/Coulomb or Volts/m)
Gradient
p Dipole
r E=
(Coulomb. m)
p = qd Charges (Coulomb)
Closed Path
Coulomb’s Law
r r Q Φ ( flux ) = ∫ E ⋅ da = encl ε0
!
! !
!
!
!
!
r
r r B da
ΦB Now we want to consider non-closed surfaces
∫∫
But our “fundamental
Small circular surface in plane of large current loop
∫
µI =− 2R 0
!
Flux !
µ 0 I ( zˆ ) nˆ da − ⋅
∫ 2R
coil
2 0I da = − µ πr 2 R ∫
Biot-Savart
r µ 0 Idl r 2 dB = ! DirectionB = ! Into page !
r r B ⋅ da =
Coil loop is boundary of surface (circle)
Calculate B-field at center !
Ismall=0
Rcoil
Take each patch of area, dot B into normal, multiply by area, sum over the surface
∫∫
!
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
!
No monopoles No magnetic “charge” No magnetic flux over closed surface ! Enclosed charge is always 0
r B nˆ da
!
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
Open-Surface Flux of a Loop
Calculate a magnetic flux the same way as electric flux !
!
Energy conservation (Kirchoff)
r r r r r r F ( x ) = q (E ( x ) + v × B( x ))
We didn’t spend too much time on magnetic flux !
!
Closed rPath
Magnetic Flux, Redux
We know there is some connection between E and B
!
No monopoles
r r ∫ B ⋅ d l = µ0 I encl
q ∑ i 2 rˆi ch arg es 4πε 0 ri Gauss’ Law
q
Gauss’ Law
Closed Surface
E
E-Field
r r Q E ⋅ da = encl ε0
r r ∫∫ B ⋅ da = 0
r r F = qE
Gradient
Electric Potential
∫∫
F
Force
!
!
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
No current in small coil
6
Changing Current
Time-Dependent Magnetic Flux
!Now change the current in the loop !
If Φ changes as function of t, find that there is a current induced in the small coil !
I=I(t)
!
Flux also a function of time µ 0πr 2 !
r
Φ(t ) = −
Rcoil
2 Rcoil
Once Φ is settled, no more current !
I (t )
!
Suppose we examine what happens on the boundary of the surface
!
!
If we put resistor on coil !
Current induced is inversely proportional to resistor
I smallα I
small
1 dΦ
Rα
dΦ
dΦ Vα
In fact, the voltage here is equal to the change in flux (with a minus sign)
!
! !
This is voltage across resistor What about Kirchoff’s rule?
dΦ dt
Coil
Current induced is inversely proportional to resistor
I smallα
1 dΦ R dt
Faraday’s Law There is an “electromotive force” around the boundary of the surface
!
I(t)
Rcoil
If we put resistor on coil !
Time-Dependent Magnetic Flux
Ismall0 r
Current is due to change of flux
I smallα
!
!
Small coil is on boundary of the surface
I(t)
!
Ismall0 r
!
R Rcoil
!
Actually, what is induced is an electric field, not a force Line integral of E-field (a voltage) on the boundary is the “EMF”
Ismall0 r R Rcoil
r r dΦ B E ⋅ dl = − closed dt ∫ path
Sign: direction of voltage with respect to “sense” of flux calculation
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 !
I(t)
2
Φ(t ) = −
µ 0πr I ( t ) 2R 2
E =
µ 0πr I '(t ) 2 R il
7
What We Know About E and B r r Q E ⋅ da = encl ε0 ClosedSurface r r ∫∫ B ⋅ da = 0
∫∫
ClosedSurf ace
r
r
∫ E ⋅ dl = 0 −
ClosedPath
r
dΦ B dt
r
∫ B ⋅ dl = µ I
0 encl
ClosedPath
r r r r r r r F ( x ) = q (E ( x ) + v × B( x ))
Gauss’ Law
Summary !
!
No monopoles Energy conservation (Kirchoff) + Faraday’s Law Ampere’s Law Lorenz Force Law
How fields E and B behave (Maxwell eq’ns)
!
!
=0 on closed surface ≠0 on open surface
Magnetic Induction !
!
encl
r r Φ B = ∫∫ B ⋅ da = 0 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 =
How particles respond to fields
r
0
Restatement of Biot-Savart
Magnetic Flux Redux !
!
r
∫ B ⋅ dl = µ I
Ampere’s Law
r
r
∫ E ⋅ dl = −
Closed Path
dΦ B dt
8
