Chapter 22

1. Chapter 22 Patterns of Fields in Space • Electric flux • Gauss’s law • Ampere’s law • Maxwell equations

2. Patterns of Fields in Space What is in the box? vertical charged plate? no charges?

3. Patterns of Fields in Space Box versus open surface …no clue… Seem to be able to tell if there are charges inside Gauss’s law: If we know the field distribution on closed surface we can tell what is inside.

4. Electric Flux: Surface Area Definition of electric flux on a surface: 3. Surface area flux through small area:

5. The Electric Field of a Large Plate Symmetry: Field must be perpendicular to surface Eleft=Eright

6. The Electric Field of a Uniform Spherical Shell of Charge • Symmetry: • Field should be radial • The same at every location • on spherical surface A. Outer sphere: B. Inner sphere:

7. Gauss’s Law: Properties of Metal Can we have excess charge inside a metal that is in static equilibrium? Proof by contradiction: =0

8. Gauss’s Law: Hole in a Metal =0 What is electric field inside? = • No charges on the surface of an empty hole • E is zero inside a hole

9. Gauss’s Law: Charges Inside a Hole =0 +5nC

10. Gauss’s Law: Screening Is the field zero inside the box because the metal blocks the field?

11. Gauss’s Law: Circuits Can we have excess charge inside in steady state?

12. Gauss’s Law: Junction Between Two Different Metal Wires n2<n1 u2<u1 i1=i2 n1Au1E1 = n2Au2E2 There is negative charge along the interface!

13. Magnet Cut in Half & Pulled Apart No magnetic monopole! Try to cut a magnet down to a single pole, just get smaller magnets No magnetic Charge!

14. Gauss’s Law for Magnetism Gauss’s law for magnetism or Dipoles: Electric field: ‘+’ and ‘–’ charges can be separated Magnetic field: no monopoles Suppose magnetic dipole consists of two magnetic monopoles, each producing a magnetic field similar to the electric field. One cannot separate them total magnetic ‘charge’ is zero.

15. Patterns of Magnetic Field in Space Is there current passing through these regions? There must be a relationship between the measurements of the magnetic field along a closed path and current flowing through the enclosed area. Ampere’s law

16. Quantifying the Magnetic Field Pattern Curly character – introduce: Similar to Gauss’s law (Q/0)

17. Ampère’s Law All the currents in the universe contribute to B but only ones inside the path result in nonzero path integral Ampere’s law is almost equivalent to the Biot-Savart law: but Ampere’s law is relativistically correct

18. Ampere’s Law: A Long Thick Wire for thick wire: (the same as for thin wire) Can B have an out of plane component? Is it always parallel to the path? Would be hard to derive using Biot-Savart law

19. Ampere’s Law: An Infinite Sheet What is on sides? Number of wires inside:(N/L) Uniform: Does not depend on distance from sheet. Opposite directions above and below sheet.

20. Ampere’s Law: A Toroid Symmetry:B || path Is magnetic field constant across the toroid?

21. Maxwell’s Equations (incomplete) Three equations: Gauss’s law for electricity Gauss’s law for magnetism Ampere’s law for magnetism Is anything missing? ‘Ampere’s law for electricity’

22. Maxwell’s Equations (incomplete) Gauss’s law for electricity Gauss’s law for magnetism Incomplete version of Faraday’s law Ampere’s law (Incomplete Ampere-Maxwell law) First two: integrals over a surface Second two: integrals along a path Incomplete: no time dependence

23. Motional EMF Revisited Heat Ring and it expands What is the direction of the electric field on the ring? Curly electric fields! Ampere’s law What is changing inside the ring? B