1. Methods of Math. Physics Dr. E.J. Zita, The Evergreen State College Lab II Rm 2272, zita@evergreen.edu Winter wk 3, Thursday 20 Jan. 2011 Electrostatics & overview Div, Grad, and Curl Dirac Delta Modern Physics Logistics: PIQs, e/m lab writeup

2. Electrostatics Charges → E fields and forces charges → scalar potential differences dV E can be found from V Electric forces move charges Electric fields store energy (capacitance)

3. Magnetostatics Currents → B fields currents make magnetic vector potential A B can be found from A Magnetic forces move charges and currents Magnetic fields store energy (inductance)

4. Electrodynamics Changing E(t) → B(x) Changing B(t) → E(x) Wave equations for E and B Electromagnetic waves Motors and generators Dynamic Sun

5. Some advanced topics Conservation laws Radiation waves in plasmas, magnetohydrodynamics Potentials and Fields Special relativity

6. Differential operator “del” Del differentiates each component of a vector. Gradient of a scalar function = slope in each direction Divergence of vector = dot product = outflow Curl of vector = cross product = circulation =

7. Practice: 1.15: Calculate the divergence and curl of v = x 2 x + 3xz 2 y - 2xz z Ex: If v = E, then div E ≈ charge. If v = B, then curl B ≈ current. Prob.1.16 p.18

8. 1.22 Gradient

9. 1.23 The  operator

10. 1.2.4 Divergence

11. 1.2.5 Curl

12. 1.2.6 Product rules

13. 1.2.7 Second derivatives Laplacian of scalar Lapacian of vector

14. Fundamental theorems For divergence: Gauss’s Theorem (Boas 6.9 Ex.3) For curl: Stokes’ Theorem (Boas 6.9 Ex.4)

15. Derive Gauss’ Theorem

16. Apply Gauss’ thm. to Electrostatics

17. Derive Stokes’ Theorem

18. Apply Stokes’ Thm. to Magnetostatics

19. Separation vector vs. position vector: Position vector = location of a point with respect to the origin. Separation vector: from SOURCE (e.g. a charge at position r’) TO POINT of interest (e.g. the place where you want to find the field, at r).

20. Origin Source (e.g. a charge or current element) Point of interest, or Field point See Griffiths Figs. 1.13, 1.14, p.9 (separation vector) r’ r

22. Dirac Delta Function This should diverge. Calculate it using (1.71), or refer to Prob.1.16. How can div(f)=0? Apply Stokes: different results on L ≠ R sides! How to deal with the singularity at r = 0? Consider and show (p.47) that

23. Ch.2: Electrostatics: charges make electric fields Charges → E fields and forces charges → scalar potential differences E can be found from V Electrodynamics: forces move charges Electric fields store energy (capacitance) F = q E = m a W = qV C = q/V

24. charges ↔ electric fields ↔ potentials

25. Gauss’ Law practice: 2.21 (p.82) Find the potential V(r) inside and outside this sphere with total radius R and total charge q. Use infinity as your reference point. Compute the gradient of V in each region, and check that it yields the correct field. Sketch V(r). What surface charge density does it take to make Earth’s field of 100V/m? (R E =6.4 x 10 6 m) 2.12 (p.75) Find (and sketch) the electric field E(r) inside a uniformly charged sphere of charge density .

26. Curl Curl of vector = cross product = circulation