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

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)

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)

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

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

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 =

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

1.22 Gradient

1.23 The  operator

1.2.4 Divergence

1.2.5 Curl

1.2.6 Product rules

1.2.7 Second derivatives Laplacian of scalar Lapacian of vector

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

Derive Gauss’ Theorem

Apply Gauss’ thm. to Electrostatics

Derive Stokes’ Theorem

Apply Stokes’ Thm. to Magnetostatics

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).

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

Dirac Delta Function This should diverge. Calculate it using (1.71), or refer to Prob 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

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

charges ↔ electric fields ↔ potentials

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 .

Curl Curl of vector = cross product = circulation