Electricity and Magnetism

1. Hw: All Chapter 5 problems and exercises Electricity and Magnetism Physics 208 Dr. Tatiana Erukhimova Lectures 13-15

2. Average 75 Median 78 Test 1 results 521 – 79 522 – 76 523 – 73.3 524 – 76.5 525 – 76 526 – 69.7 >90 >80 >70 >60 >50 <50

3. Outline • Applications of Gauss’s Law - The single Fixed Charge • Field of a sphere of charge • Field of a spherical shell • A Line of Charge • Conductors and Insulators • The electric field of a conductor • The field in the cavity of a conductor; Faraday’s Cage

4. Solid conducting sphere with charge Q A E V r A r

5. Electric field of a ball of charge Q Electric field outside of a charged sphere is exactly the same as the electric field produced by a point charge, located at the center of the sphere, with charge equal to the total charge on the sphere.

6. Electric field of a spherical shell Q The field outside the shell is like that of a point charge, while the field everywhere inside the shell is zero.

7. Electric field of a line of charge

8. A Charged, Thin Sheet of Insulating Material + + + + + + + + + + +

9. Conductors and insulators Charges reside at the surface of the conductor + + + + + + + + Conductor + + E=0 + + + + +

10. What have we learned about conductors? • There is no electric field inside a conductor • Net charge can only reside on the surface of a conductor • Any external electric field lines are perpendicular to the surface (there is no component of electric field that is tangent to the surface). • The electric potential within a conductor is constant

11. Electric field near a surface of a conductor a

12. Two parallel conducting plates - + + + - + a - + + d (the total field at any point between the plates)

13. An Apparent Contradiction - + + + - + - + +

14. An Apparent Contradiction - + + + - + - + + Near the surface of any conductor in electrostatics

15. 1) There is a conducting spherical shell, inner radius A and outer radius B. If you put a charge Q on it, find the charge density everywhere. 2) There is a conducting spherical shell, inner radius A and outer radius B. A charge Q is put at the center. If you put a charge Q2 on the shell, find the charge density everywhere.

16. A sphere of radius A has a charge Q uniformly spread throughout its volume. Find the difference in the electric potential, in other words, the voltage difference, between the center and a point 2A from the center.

17. There is a conducting spherical shell, inner radius A and outer radius B. A charge Q1 is put at the center. If you now put charge -2Q1 on the shell, find the charge density at r=A and r=B.

21. since inside the conductor. For any two points and inside the conductor The conductor’s surface is an equipotential.

22. Equipotential Surfaces An equipotential surface is a surface on which the electric potential V is the same at every point. Because potential energy does not change as a test charge moves over an equipotential surface, the electric field can do no work on such a charge. So, electric field must be perpendicular to the surface at every point so that the electric force is always perpendicular to the displacement of a charge moving on the surface. Field lines and equipotential surfaces are always mutually perpendicular.

24. - - - - - - - - - - - Method of images: What is a force on the point charge near a conducting plate? Equipotential surface - -

25. The force acting on the positive charge is exactly the same as it would be with the negative image charge instead of the plate. The point charge feels a force towards the plate with a magnitude:

26. Equipotential surface - - - - - - - - - - - Method of images: A point charge near a conducting plane. - -

28. Equilibrium in electrostatic field: Earnshaw’s theorem There are NO points of stable equilibrium in any electrostatic field! How to prove it? Gauss’s Law will help! Imaginary surface surrounding P P If the equilibrium is to be a stable one, we require that if we move the charge away from P in any direction, there should be a restoring force directed opposite to the displacement. The electric field at all nearby points must be pointing inward – toward the point P. But that is in violation of Gauss’ law if there is no charge at P.

29. Thomson’s atom 1899 If charges cannot be held stably, there cannot be matter made up of static point charges (electrons and protons) governed only by the laws of electrostatics. Such a static configuration would collapse!

30. Capacitors Consider two large metal plates which are parallel to each other and separated by a distance small compared with their width. Area A L The field between plates is

31. The capacitance is:

32. Have a great day!