Ampere’s Law:Symmetry

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. Consider three wires with current flowing in/out as shown • Consider three different loops surrounding the wires X 2 A Y 3 A Z 1 A Quiz: Ampere’s Law 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 • 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. Draw a path for Ampere’s law that crosses inside the solenoid length l 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 =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 • 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. 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.