The story so far…
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Transcript of The story so far…
The story so far…
dIMagnetic field generated by current element: Biot-Savart
Ampere’s law
dB
r
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dB = μo
4πIds × ˆ r
r2
€
B • ds∫ = μoIclosed path
surface bounded by path
I
Mon. Mar. 31, 2008 Physics 208, Lecture 18 2
Exam 2 resultsGrade cutoffs:A: 86AB: 79B: 66BC: 58C: 37D: 230
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Phy208 Exam 2
SCORE
Ave=69
Mon. Mar. 31, 2008 Physics 208, Lecture 18 3
Ampere’s lawSum up component of B around pathEquals current through surface.
Ampere’s law
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B • ds∫ = μoIclosed path
surface bounded by path
I
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r B
Component of B along path
Mon. Mar. 31, 2008 Physics 208, Lecture 18 4
“Ampere’s law” in electrostatics
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r E • d
r s
path∫ = ?
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WAB =r F Coulomb • ds
A
B
∫ = qr E • ds
A
B
∫Work done by E-field =
So
is work per unit charge to bring charge back to where it started.
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r E • d
r s
path∫
This is zero.
Mon. Mar. 31, 2008 Physics 208, Lecture 18 5
Gauss’ law in electrostatics Electric flux through surface charge enclosed
What about magnetic flux?
Mon. Mar. 31, 2008 Physics 208, Lecture 18 6
Magnetic flux Magnetic flux is defined
exactly as electric flux (Component of B surface) x (Area element)
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ΦB = B • dA∫
zero flux Maximum flux
SI unit of magnetic flux is the Weber ( = 1 T-m2 )
Mon. Mar. 31, 2008 Physics 208, Lecture 18 7
Magnetic flux
What is that magnetic flux through this surface?
A. PositiveB. NegativeC. Zero
Mon. Mar. 31, 2008 Physics 208, Lecture 18 8
Gauss’ law in magnetostatics Net magnetic flux through any closed
surface is always zero:
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Φmagnetic = 0
No magnetic ‘charge’, so right-hand side=0 for mag.Basic magnetic element is the dipole
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Φelectric = Qenclosed
εo
Compare to Gauss’ law for electric field
Mon. Mar. 31, 2008 Physics 208, Lecture 18 9
Comparison with electrostatics
Gauss’ law Ampere’s law
Magnetostatics
Electrostatics
Mon. Mar. 31, 2008 Physics 208, Lecture 18 10
Time-dependent fields Up to this point, have discussed only magnetic
and electric fields constant in time. E-fields arise from charges B-fields arise from moving charges (currents)
Faraday’s discovery
Another source of electric field Time-varying magnetic field creates electric field
Mon. Mar. 31, 2008 Physics 208, Lecture 18 11
Measuring the induced field A changing magnetic flux produces an EMF
around the closed path. How to measure this? Use a real loop of wire for the closed path.
The EMF corresponds to a current flow:
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ε=IR
Mon. Mar. 31, 2008 Physics 208, Lecture 18 12
Current but no battery?
Electric currents require a battery (EMF) Faraday:
Time-varying magnetic field creates EMF
Faraday’s law:EMF around loop = - rate of change of mag. flux
Mon. Mar. 31, 2008 Physics 208, Lecture 18 13
Faraday’s law
€
ε = E • ds∫ = − ddt
ΦB = − ddt
B∫ • dA
Magnetic flux through surface bounded by path
EMF around loop
EMF no longer zero around closed loop
Mon. Mar. 31, 2008 Physics 208, Lecture 18 14
Quick quiz Which of these conducting loops will have currents flowing in them?
Constant I I(t) increases
Constant IConstant v
Constant IConstant v
Mon. Mar. 31, 2008 Physics 208, Lecture 18 15
Faraday’s law Faraday’s law
Time-varying B-field creates E-field Conductor: E-field creates electric current
Biot-Savart law Electric current creates magnetic field
Result Another magnetic field created
Mon. Mar. 31, 2008 Physics 208, Lecture 18 16
Lenz’s law Induced current produces a magnetic field.
Interacts with bar magnet just as another bar magnet
Lenz’s law Induced current generates a magnetic field
that tries to cancel the change in the flux. Here flux through loop due to bar magnet is increasing.
Induced current produces flux to left. Force on bar magnet is to left.
Mon. Mar. 31, 2008 Physics 208, Lecture 18 17
Quick quizWhat direction force do I feel due to Lenz’ law when I
push the magnet down?
A. UpB. DownC. LeftD. Right
Copper
Strong magnet
Mon. Mar. 31, 2008 Physics 208, Lecture 18 18
Quick Quiz
A conducting rectangular loop moves with constant velocity v in the +x direction through a region of constant magnetic field B in the -z direction as shown.
What is the direction of the induced loop current?
X X X X X X X X X X X XX X X X X X X X X X X XX X X X X X X X X X X XX X X X X X X X X X X X
v
x
yA. CCWB. CWC. No induced current
Mon. Mar. 31, 2008 Physics 208, Lecture 18 19
Quick Quiz•Conducting rectangular loop moves with constant velocity v in the -y direction away from a wire with a constant current I as shown. What is the direction of the induced loop current?
A. CCWB. CWC. No induced current
I
v
B-field from wire into page at loopLoop moves to region of smaller B, so flux decreasesInduced loop current opposes this change, so must create a field in same direction as field from wire -> CW current.
Mon. Mar. 31, 2008 Physics 208, Lecture 18 20
L
Motional EMF Conductor moving in uniform magnetic field + / - charges in conductor are moving. Magnetic field exerts force.
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r v
-
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r v
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r F B
Charges pile up at ends
Static equilibrium: E-field generated canceling magnetic force
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qE = qvB
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EMF = vBLSolid conductor