2. 2D case:

3. If or then 2 or 3D cases:

4. Several dimensions: U(x,y,z) Compact notation using vector del, or nabla: Another notation: Partial derivative is taken assuming all other arguments fixed

5. Geometric meaning of the gradient Direction of the steepest ascent; Magnitude : the slope in that direction Direction of the steepest descent Magnitude : the slope in that direction http://reynolds.asu.edu/topo_gallery/topo_gallery.htm

6. 1)The electric potential V in a region of space is given by where A is a constant. Derive an expression for the electric field at any point in this region. 2)The electric potential V in a region of space is given by where c is a constant. The source of the field is at the origin. Derive an expression for the electric field at any point in this region.

7. Exercise 5 p. 52 An electron moves from one point to another where the second point has a larger value of the electric potential by 5 volts. If the initial velocity was zero, how fast will the electron be going at the second point?

8. Problem 3 p. 45

9. Electric potential V is a scalar!

10. An old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hour

11. Outline Area vector Vector flux More problems Solid angle Proof of Gauss’s Law

12. Electric field lines These are fictitious lines we sketch which point in the direction of the electric field. 1) The direction of at any point is tangent to the line of force at that point. 2) The density of lines of force in any region is proportional to the magnitude of in that region Lines never cross.

13. Density is the number of lines going through an area (N) divided by the size of the area For a charge q located at the origin It is important that the force is proportional to

14. Gauss’s Law The total flux of electric field out of any closed surface is equal to the charge contained inside the surface divided by.

15. What is or flux of any vector, e.g. velocity of a water flow? Consider a flow with a velocity vector. Let S be a small area perpendicular to. a) The volume of water flowing through S per unit time is S a) b) Now S is tilted with respect to. The volume of water flowing through S per unit time is n S b) Area vector is the angle between velocity vector and unit vector normal to the surface S. Flux:

16. S S Flux of electric field

17. The flux of

18. Have a great day! Hw: All Chapter 4 problems and exercises Reading: Chapter 4