EM & Vector calculus #2 Physical Systems, Tuesday 23 Jan 2007, EJZ Vector Calculus 1.2: Differential Calculus Ordinary derivatives Div, Grad, and Curl.

EM & Vector calculus #2Physical 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. 2.30 p.90)

• Electrostatic energy (Prob. 2.40 p.106), capacitors (Ex. 2.10 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 energyEM # 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. 2.30 p.90)

• Electrostatic energy (Prob. 2.40 p.106), capacitors (Ex. 2.10 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)

E.dA = q/0=, E = F/q

ldEdr

rV

')'(

4

1)(

VE

F = q E = m a

W = qV

C = q/V

Conservative fields admit potentials

0 ldE

0 E

b

a

ldEV

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

V

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. 2.40 p.106: Energy between parallel plates

Ex. 2.10 p.104: Find the capacitance between two metal plates of surface area A held a distance d apart.

Electrostatic potential energy