EM & Vector calculus #2 Physical Systems, Tuesday 23 Jan 2007, EJZ Vector Calculus 1.2: Differential Calculus Ordinary derivatives Div, Grad, and Curl Product rules, Second derivatives Ch.2 Electrostatic potential and energy Quick homework review Review electrostatics, Gauss’ Law: charges  E field Conservative fields and path independence  potential V Boundary conditions (Ex. 2.5 p.74, Prob p.90) Electrostatic energy (Prob p.106), capacitors (Ex p.104)

1.2.1 Ordinary derivatives

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

Electrostatic potential and energy EM # 2, Physical Systems, Tuesday 23 Jan 2007, EJZ Quick homework review Review electrostatics, Gauss’ Law: charges  E field Conservative fields and path independence  potential V Boundary conditions (Ex. 2.5 p.74, Prob p.90) Electrostatic energy (Prob p.106), capacitors (Ex p.104)

Ch.2: Electrostatics (d/dt=0): charges  fields  forces, energy Charges make E fields and forces charges make scalar potential differences dV E can be found from V Electric forces move charges Electric fields store energy (capacitance) F = q E = m a W = qV C = q/V

Conservative fields admit potentials  depends only on endpoints.Therefore Easy to find E from V is independent of choice of reference point V=0 V is uniquely determined by boundary conditions Every central force (curl F = 0) is conservative (prob 2.25) Ex.2.5 p.74: parallel plates

Parallel plates

Electrostatic boundary conditions: E  is discontinuous across a charge layer:  E =  /  0 E || and V are continuous Prob 2.30 (a) p.90: check BC for parallel plates

Electrostatic potential: units, energy Prob p.106: Energy between parallel plates Ex p.104: Find the capacitance between two metal plates of surface area A held a distance d apart.

Electrostatic potential energy