1. Ampere’s Law:Symmetry
Ampere’s law because easy (just as gauss’s law did) when we have symmetry and a uniform field
From a infinitely long wire:
Radial symmetric, so
B(2pr)=m0I
(same expression as from Biot-Savart law)

2. Quiz: Ampere’s Law
2 A
3 A
1 A
Consider three wires with current flowing in/out as shown X Y Z
Consider three different loops surrounding the wires
Which of the loops has the largest and smallest integrals of the magnetic field around the loops drawn?
A) X > Y > Z C) Y > Z > X
B) X > Z > Y D) Y > X > Z

3. Ampere’s Law: Wire B(2pr)=m0I(pr2)/(pR2) B=m0I(r)/(2pR2)
We can use ampere’s law on problems difficult to solve with the Biot-Savart law
We know that the current is uniformly distributed within the wire, so Ienc=I(pr2)/(pR2)
B(2pr)=m0I(pr2)/(pR2)
B=m0I(r)/(2pR2)
Inside the field is proportional to the distance from the center
Outside the field falls over off

4. Magnetic Field From a Plane of Current
What is the magnetic field?
By symmetry, magnetic field will be the same on both sides
N wires in length L
Draw a loop crossing the boundary to use Ampere’s Law L
Current I in each wire

5. Magnetic Field of an Ideal Solenoid
If the length is much greater than the radius,
Magnetic field far away is zero
Magnetic field everywhere outside is small
Length L
N turns
Draw a path for Ampere’s law that crosses inside the solenoid
length l
=Bl R
Current I

6. Current Coil A single circular loop has an axial field
Use Biot-Savart law to derive
Far away from the loop

7. Current Coil as a Dipole
Far away from the loop
Rewrite in different form
So we can yet again consider a current-loop as a magnetic dipole

8. Magnetic Flux
Magnetic Flux is the amount of magnetic field flowing through a surface
It is the magnetic field times the area, when the field is perpendicular to the surface
It is zero if the magnetic field is parallel to the surface
Normally denoted by symbol B.
Units are T·m2, also known as a Weber (Wb)
Magnetic Field
Magnetic Field
B = BA
B = 0

9. Magnetic Flux B = BnA= BA cos 
When magnetic field is at an angle, only the part perpendicular to the surface counts (does this ring a bell? Remember Gauss’s law? ) B
Multiply by cos  Bn 
For a non-constant magnetic field, or a curvy surface, you have to integrate over the surface
B = BnA= BA cos 

10. Quiz
A sphere of radius R is placed near a very long straight wire that carries a steady current I. The total magnetic flux passing through the sphere is:
moI
moI/(4pR2)
4pR2moI
D) zero

11. Gauss’s Law
Magnetic field lines are always loops, never start or end on anything (no magnetic monopoles)
Net flux in or out of a region is zero
Gauss’s Law for electic fields

12. Four closed surfaces are shown
Four closed surfaces are shown. The areas Atop and Abot of the top and bottom faces and the magnitudes Btop and Bbot of the uniform magnetic fields through the top and bottom faces are given. The fields are perpendicular to the faces and are either inward or outward. Which surface has the largest magnitude of flux through the curved faces?

13. Ampere’s Law I
Ampere’s law says that if we take the dot product of the field and the length element and sum up (i.e. integrate) over a closed loop, the result is proportional to the current through the surface
This is not quite the same as gauss’s law

14. A Problem with Ampere’s Law
Consider a parallel plate capacitor that is being charged
Try Ampere’s Law on two nearly identical surfaces/loops I I
Magnetic fields cannot be different just because surfaces are chosen slightly different
When current flows into a region, but not out, Ampere’s law must be modified

15. Ampere’s Law Generalized
When there is a net current flowing into a region, the charge in the region must be changing, as must the electric field.
By Gauss’s Law, the electric flux must be changing as well
Change in electric flux creates magnetic fields, just like currents do
Displacement current – proportional to time derivative of electric flux
Magnetic fields are created both
by currents carried by conductors
(conduction currents) and by
time-varying electric fields.

16. Generalized Ampere’s Law Tested
Consider a parallel plate capacitor that is being charged
Try Ampere’s modified Law on two nearly identical surfaces/loops I I

17. Comparision of Induction
No magnetic monopole, hence no magnetic current
Electric fields and magnetic fields induce in opposite fashions

18. Magnetism in Matter: Types
Depending on details of the electronic structure on can have permanent or induced magnetic moments – similar to the case with dipoles.
Diamagnetic substances: induced magnetic moments counter to the
applied field.
All substances have a diamagnetic response, but it is weak.
Paramagnetic substances: the molecules have permanent moments, but are
weakly coupled and require a field to line them up, and the net moments
are in the direction of the field.
Ferromagnetic substances: the atoms have permanent moments and are
strongly coupled. The atoms can remain aligned in the absence of a field.