EM & Vector calculus #3Physical Systems, Tuesday 30 Jan 2007, EJZ

Vector Calculus 1.3: Integral Calculus

• Line, surface, volume integrals

• Fundamental theorems

• Integration by parts

Ch.3a: Special Techniques (Electrostatics)

• Quick homework review

• Poisson’s and Laplace’s equations (Prob. 3.3 p.116)

• Uniqueness

• Method of images (Prob. 3.9 p.126)

1.2.6 Product rules

1.2.7 Second derivatives

1.3.1 Line, Surface, Volume integrals

0

qW F dl E da d q

1.3.2 Fundamental theorem of calculus

1.3.3 Fundamental theorem for Gradients

1.3.4 Fundamental theorem for Divergences

1.3.5 Fundamental theorem for Curls

Consequences

0d I and I d

d d

B l J a

B l B a

Gauss’s law and fundamental theorem for divergences:

Ampere’s Law and fundamental theorem for curls:

0

qd and q d

d d

E a

E a E

E&M Ch.3: Techniques for finding V

Why?

• Easy to find E from V

• Scalar V superpose easily

How?

• Poisson’s and Laplace’s equations (Prob. 3.3 p.116)

• Guess if possible: unique solution for given BC

• Method of images (Prob. 3.9 p.126)

• Separation of variables (next week)

Poisson’s equation

Gauss: Potential:

combine to get Poisson’s eqn:

Laplace equation holds in charge-free regions:

Prob.3.3 (p.116): Find the general solution to Laplace’s eqn. In spherical coordinates, for the case where V depends only on r.

Do the same for cylindrical coordinates, assuming V(s).

(See Laplacian on p.42 and 44)

0

E

VE

Method of images

A charge distribution induces on a nearby conductor.

The total field results from combination of and .

+ -

• Guess an image charge that is equivalent to .

• Satisfy Poisson and BC, and you have THE solution.

Prob.3.9 p.126 (cf 2.2 p.82)