1. The story so far… dB r dI
Magnetic field generated by current element: Biot-Savart I
Ampere’s law
closed path
surface bounded by path

2. Exam 2 results Grade cutoffs: Ave=69 A: 86 AB: 79 B: 66 BC: 58 C: 37
Mon. Mar. 31, 2008
Physics 208, Lecture 18

3. Ampere’s law Sum up component of B around path
Equals current through surface.
Component of B along path I
closed path
Ampere’s law
surface bounded by path
Mon. Mar. 31, 2008
Physics 208, Lecture 18

4. “Ampere’s law” in electrostatics
Work done by E-field = So
is work per unit charge to
bring charge back to where it started.
Why didn’t we have something like this in electrostatics?
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)
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?
Positive
Negative
Zero
Mon. Mar. 31, 2008
Physics 208, Lecture 18

8. Gauss’ law in magnetostatics
Net magnetic flux through any closed surface is always zero:
Compare to Gauss’ law for electric field
No magnetic ‘charge’, so right-hand side=0 for mag.
Basic magnetic element is the dipole
Mon. Mar. 31, 2008
Physics 208, Lecture 18

9. Comparison with electrostatics
Gauss’ law
Ampere’s law
Electrostatics
Magnetostatics
Use Phi_E and Phi_B in addition to integrals, or instead of integrals.
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
Demonstrate by moving magnetic around in air, pointing to where electric fields were created. How did Faraday measure these electric fields?
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:
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 EMF no longer zero around closed loop EMF around loop
Magnetic flux through surface bounded by path
Make comparison to battery. Show that this acts just like battery. Maybe use shaking flashlight to show that this works.
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 v
Constant v
Talk about magnetic fields generated by the induced currents, and what sense magnet that produces.
Constant I
Constant I
Mon. Mar. 31, 2008
Physics 208, Lecture 18

15. Faraday’s law Faraday’s law Biot-Savart law Result
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. Lenz’s law
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.
Do demo with magnet and copper plate. Copper plate in nitrogen.
Mon. Mar. 31, 2008
Physics 208, Lecture 18

17. Quick quiz
What direction force do I feel due to Lenz’ law when I push the magnet down? Up
Down
Left
Right
Strong magnet
Copper
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?
CCW CW
No induced current y x
Mon. Mar. 31, 2008
Physics 208, Lecture 18

19. Quick Quiz CCW CW No induced current
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?
CCW CW
No induced current I v
B-field from wire into page at loop Loop moves to region of smaller B, so flux decreases Induced 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. Motional EMF Conductor moving in uniform magnetic field
+ / - charges in conductor are moving.
Magnetic field exerts force.
Charges pile up at ends
Static equilibrium: E-field generated canceling magnetic force L -
Demo of moving magnet and copper sheet.
Solid conductor
Mon. Mar. 31, 2008
Physics 208, Lecture 18