Download ∇ Homework Assignment #8 due Friday 10/24 1. Problem 16.1 2. Problem 16.2

ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ
∇∙E=0
Homework
Assignment #8 due Friday 10/24
αβγδεζηθικλμνξοπρςστυφχψω
∇ × E = −∂B /∂t
1. Problem 16.1
+<=>|~±×÷′″⁄⁒←↑→↓⇒⇔
∂Δ∇∈∏∑
2. Problem 16.2
∇∙B=0
(ε1 3.
e1 + Problem
ε 2 e2 )
16.3
∇ × B = + μ0ε0 ∂E /∂t
4. Problem 16.10
∓∔⁄∗∘∙√∞∫∮∴
5. Problem 16.13
≂≃≄≅≆≠≡≪≫≤≥
6. (a) Explain briefly (1 paragraph with one figure) how to produce linearly polarized
electromagnetic waves. (b) Explain briefly (1 paragraph with one figure) how to
e−iωt produce circularly polarized waves.
ε
Recitation
Session Tuesday at 6:30.
−
Chapter 16 : Waves in vacuum
MAXWELL’S EQUATIONS IN VACUUM
A linearly polarized electromagnetic plane
wave; polarized in the x direction and
propagating in the z direction.
∇∙E=0
∇ × E = −∂B /∂t
∇∙B=0
∇ × B = + μ0ε0 ∂E /∂t
These are infinite planes;
the fields E(r,t) and B(r,t) do
not depend on x or y on these
planes.
There are neither charges nor
currents in vacuum.
x
y
z
Harmonic solutions (i.e., harmonic in time).
The problem is to find a complete set of
solutions that are harmonic in time.
“Harmonic” means that there is a fixed
angular frequency ω.
We’ll use complex fields to solve the field
equations. Then the physical fields are the
real parts of the complex fields
complex fields
physical fields
E
= Re E
B
= Re B
Do not confuse the complex fields (E,B) and
the physical fields (E,B) although we’ll use
the same symbols for both (E, B).
E(r,t) = E(r) e−iωt
B(r,t) = B(r) e−iωt
∂ /∂t is the same as
multiplication by −iω
so
∇ ∙ E(r) = 0
∇ × E(r) = +iω B(r)
∇ ∙ B(r) = 0
∇ × B(r) = −iω μ0ε0 E(r)
The equations for E(r) and B(r) are
linear. Therefore the natural solutions
are exponentials.
{{ dy /dx = y implies y(x) = C exp(x) }}
∇ × (∇ × E ) = − ∇ 2 E
= i ω ∇ × B = ω 2 μ 0ε 0 E
− ∇ 2 E = (ω2 /c2) E
− ∇ 2 E = (ω2 /c2) E
Solve by separation of variables
E(r) = E0 e i k∙r and B(r) = B0 e i k∙r
Solving the Maxwell equations ...
∇∙E=0
Note:
∇ is the same as multiplication by ik.
Requires k∙E0 = 0
∇∙B=0
Requires k∙B0 = 0
∇ × E = −∂B /∂t
Requires k × E0 = ω B0
∇ × B = + μ0ε0 ∂E /∂t
Requires k × B0 = −ω/c2 E0
Result: E(r) = E0 e i k∙r
Also, recall Faraday’s law, iω B(r) = i k × E(r) ;
thus B(r) = B0 e i k∙r
where B0 = k/ω × E0
The transverse plane
Both E0 and B0 are perpendicular to k;
i.e., they lie in the transverse plane.
Polarization
What is the direction of E(r,t) (the
physical field) as a function of r and t ?
We have this complex field
E(r,t) = ( ε1 e1 + ε2 e2 ) eiφ(r,t)
,
where φ(r,t) = k∙r − ωt.
Let e1 and e2 be orthogonal basis vectors for the
transverse plane. (These are not uniquely
determined; but once they have been chosen we’
ll keep them fixed.)
Then E0 = ε1 e1 + ε2 e2
B0 = k/ω × (ε1 e1 + ε2 e2 ) = (−ε2 e1 + ε1 e2 )/c
Take k to be real (for some applications, k
might be complex); also, ω = ck.
Now write
ε1 = A exp(i δ1) and
ε2 = B exp(i δ2)
where A, B, δ1 and δ2 are real.
The electric field is
E(r,t) = Re [ ( ε1 e1 + ε2 e2 ) eiφ(r,t) ]
● Linear Polarization
If δ = 0,
then this is a linearly polarized wave.
The direction of the electric field
alternates between +a and −a :
●
Circular Polarization
If δ = π/2 or −π/2, and A = B,
then this is a circularly polarized wave. The direction
of the electric field rotates around the wave vector (k),
with constant magnitude in the transverse plane.
●
Elliptical Polarization
This is the general case, with no conditions on A, B, δ1
and δ2 . (Both linear and circular polarization are
special cases of elliptical polarization.) The direction of
the electric field rotates around k, and the magnitude
varies according to the position on an ellipse.
ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ
∇∙E=0
Homework
Assignment #8 due Friday 10/24
αβγδεζηθικλμνξοπρςστυφχψω
∇ × E = −∂B /∂t
1. Problem 16.1
+<=>|~±×÷′″⁄⁒←↑→↓⇒⇔
∂Δ∇∈∏∑
2. Problem 16.2
∇∙B=0
(ε1 3.
e1 + Problem
ε 2 e2 )
16.3
∇ × B = + μ0ε0 ∂E /∂t
4. Problem 16.10
∓∔⁄∗∘∙√∞∫∮∴
5. Problem 16.13
≂≃≄≅≆≠≡≪≫≤≥
6. (a) Explain briefly (1 paragraph with one figure) how to produce linearly polarized
electromagnetic waves. (b) Explain briefly (1 paragraph with one figure) how to
e−iωt produce circularly polarized waves.
ε
Recitation
Session Tuesday at 6:30.
−