Methods of Math. PhysicsDr. 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

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 =

ˆ ˆx y zx y z

ˆ ˆf f f

f x y zx y z

ˆ ˆyx zVV V

x y zx y z

V

V

Practice: 1.15: Calculate the divergence and

curl of v = x2 x + 3xz2 y - 2xz z2 2(3 ) ( 2 )

ˆ ˆ ...x xz xz

x y zx y z

V

zyx

xzxzxzyx

zyx

ˆˆ

222

V

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

2 V V 2 f f 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)

volume surface

d d flux v v a

surface boundary

d d circulation v a v l

Derive Gauss’ Theorem

volume surface

d d flux v v a

Apply Gauss’ thm. to Electrostatics

0

0

0

charge density therefore

volume surface

volume

volume volume

qE d E d

dqd q

d

E d d

E

a

Derive Stokes’ Theorem

surface boundary

d d circulation v a v l

Apply Stokes’ Thm. to Magnetostatics

0

0

0

current density , so

surface boundary

surface

surface surface

B d B d I

dIJ I J d

da

B d J d

B J

a l

a

a a

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).

222ˆˆˆ zyxrzzyyxx r

2 2 2

ˆ ˆ ˆ' ( ') ( ') ( ')

' ( ') ( ') ( ')

x x x y y y z z z

x x y y z z

r r

r r

r

r

Origin

Source (e.g. a charge or current element)

Point of interest, orField point

See Griffiths Figs. 1.13, 1.14, p.9

(separation vector)rr’

r

Dirac Delta Function

2

ˆ

r

rf

0 0( )

0

if xx

if x

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

( ) ( ) ( )f x x a dx f a

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)

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

1 ( ')( ) '

4

rV r d E dl

r

VE

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? (RE=6.4 x 106 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

zyx

VVVzyx

zyx

zyx

ˆˆV