Methods of Math. PhysicsDr. E.J. Zita, The Evergreen State College, 6 Jan.2011

Lab II Rm 2272, zita@evergreen.edu

Winter wk 1 Thursday: Electromagnetism

* Overview of E&M

* Review of basic E&M: prep for charge/mass ratio workshop

* Griffiths Ch.1: Div, Grad, Curl, and

Introduction to Electromagnetism

• 4 realms of physics• 4 fundamental forces• 4 laws of EM• statics and dynamics• conservation laws• EM waves• potentials• Ch.1: Vector analysis• Ch.2: Electrostatics

4 realms of physics, 4 fundamental forces

Classical Mechanics(big and slow:

everyday experience)

Quantum Mechanics(small: particles, waves)

Special relativity(fast: light, fast particles)

Quantum field theory(small and fast: quarks)

Four laws of electromagnetism

Electric Magnetic

Gauss' Law

Charges → E fields

Gauss' Law

No magnetic monopoles

Ampere's Law

Currents → B fields (and changing E→ B fields)

Faraday's Law

Changing B → E fields

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

Ch.1: Vector Analysis

Dot product: A.B = Ax Bx + Ay By + Az Bz = A B cos

Cross product: |AxB| = A B sin

zyx

zyx

BBB

AAA

zyx

zB y B x B ,zA yA xA zyxzyx BA

Examples of vector products

Dot product: work done by variable force

Cross product:

angular momentum

L = r x mv

cosW F dl F dl

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

yz

yy

xx

ˆˆ

y

fz

y

fy

x

fxf

ˆˆ

y

Vz

y

Vy

x

Vx zyx

ˆˆV

zyx

VVVzyx

zyx

zyx

ˆˆV

Practice: 1.15: Calculate the divergence and

curl of v = x2 x + 3xz2 y - 2xz z

...)2(

ˆ)3(

ˆ22

y

xzz

y

xzy

x

xx

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

Develop intuition about fields

Look at fields on p.17 and 18.

Which diverge?

Which curl?

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

Fundamental theorems

For divergence: Gauss’s Theorem

For curl: Stokes’ Theorem

volume surface

d d flux v v a

surface boundary

d d circulation v a v l

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 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)

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 .