Lecture 1 — Prelude to Optics: Basic Electromagnetic Theory — January 13, 1999
Physics 221 — Lecture 1 — “Review 1: Electromagnetic Fields and Waves”
January 13, 1999
Reading
Demonstrations
Problems
Reminders
Lipson, Sections 2.1-2.6, Klein and Furtak, pp. 44 ff.
Danny’s Mathematica Manual
See course schedule for Unit 1
See WWW for general course information; office hours today.
1.
Historical introduction: Faraday, Maxwell, Planck, Einstein, Bohr, Schrödinger
a. Big issues: classical electromagnetic fields, quantum properties of matter
b. Electromagnetic waves (X-rays to radio waves)
c. Atomistic origins of electrical and optical properties of matter
2.
Maxwell’s equations govern electromagnetic interactions with matter.
a. There are two kinds of charge, called (after Franklin) positive and negative;
their interactions obey the set of four Maxwell equations.
b. In ordinary matter, these are respectively ions and electrons; the ions are
the “fixed” atoms minus their outermost (chemically active) electron shells.
c. Charge is conserved and quantized in units of 1.6·10-19 C. The absence of an
electron at the normal site of an atomic core creates a positively charged
“hole” which is electronically and optically active.
d. Chemistry, condensed-matter physics, and electronic and optical physics is
due just to the spatial distribution of charge in atoms and molecules.
3.
Mathematical preliminaries: vector algebra and calculus
a. Electromagnetic fields are vector quantities, and must be understood with
reference to those properties. Normally, in most optical physics problems,
we deal only with symmetrical configurations of fields which can be reduced
to scalar quantities.
b. The vector differential operator “del” is the principal actor in the equations
governing light and optical phenomena. It has the following properties:
eˆx eˆ y eˆz
r
r r ∂Ax ∂Ay ∂Az
r r
∂
∂
∂
∇ ≡ eˆ x + eˆ y
+ eˆ
, ∇oA=
+
+
, ∇ × A ≡ ∂ x ∂y ∂ z
∂x
∂y z ∂z
∂x
∂y
∂z
Ax Ay Az (1)
c.
r
r r
∂ψ
∂ψ
∂ψ
∂ 2ψ ∂ 2ψ ∂ 2ψ
∇ ψ ≡ gradψ = eˆx
+ eˆ y
+ eˆz
, ∇ o ∇ ψ ≡ ∇2ψ = 2 + 2 + 2
∂x
∂y
∂z
∂x
∂y
∂z
A most remarkable vector
utility
in electromagnetism is
r
r identity
r
r ofr special
r
r
2
∇ × ∇ × A = ∇ ∇ o A −∇ A
(2)
(
)
(
)
We shall presently use of this identity to prove that electromagnetic waves
must exist and that they must be transverse waves, that is, what is oscillating is transverse to the propagation direction of the wave.
Lecture 1 — Prelude to Optics: Basic Electromagnetic Theory — January 13, 1999
4.
Maxwell’s equations reflect centuries of physics experiments.
a. Statement of the equations in terms of electric field E, magnetic induction
B, electric displacement D, magnetic field H, polarization P, magnetization
M, and current density
J.
r
r
r r
r
r r ∂D
r r
r r
∂B
∇× E= − , ∇×H = J+
, ∇ o D = −4πρ, ∇ o B = 0
∂t
∂t
(3)
r
r r r
r r
D = ε o E + P, B = µo H + M
(
5.
)
b. In most of optical physics, we are dealing with nonmetallic, nonmagnetic
materials where no currents are flowing. So how are these equations simplified in that case?
c. What does not go away necessarily is the polarization P, the dipole moment
per unit volume. This turns out to play a major role in optical phenomena.
From Maxwell’s equations, one can derive the wave equation obeyed by E and B.
a. If one applies the vector identity (2) to the equations in (3) for free space, one
can show that
r
r
∂ 2E r
∇ E − εo µo 2 = 0 and
∂t
2
r
r
∂2 B r
∇ B − ε oµo 2 = 0
∂t
2
(4)
This is a wave equation because wave functions, that is, functions of the form
f(r, t)=f(r-vt) will satisfy it. This equation is also true for each scalar component of the fields E and B. Why?
b. Among the many possible solutions to this wave equation are the complex
functions
r
E( x,y,z,t ) = eˆx Eo exp[i(kz − ωt )], remember that exp [ iϕ ] = cosϕ + i ⋅sinϕ
(5)
c.
which represents an electromagnetic wave travelling in the z-direction. The
number k is called the propagation constant or wave vector, while ω is the
angular frequency 2πv.
Thought question: What does it mean if the wave solution to this equation is
complex? Must not the fields be real if they are physically meaningful?