Physics 8.03 Fall 2012
Homework 1 Solutions
1.1
1.1.a.
X (t) = Aej (t/2)
x(t) = A sin t
(1)
x(t) = A cos t
(2)
2
x(t) = A sin t
(3)
1.1.b.
X (t) = Bj ej (t+/6)
X (t) = B ej (t+/6)
3j
X (t) = B
+
(cos t + j sin t)
2
2
3
1
x(t) =
B cos t B sin t
2
2
1
3
x(t
Physics 8.03 Fall 2012
Homework 2 Solutions
2.1
2.1.a.
The block has three forces acting on it: the spring pulling it from the left, the
spring pulling it from the right, and friction:
mx = kx + k ( sin t x) bx
mx + bx + 2kx = k sin t
12
2
x + x + 0 x = 0
Massachusetts Institute of Technology
Physics 8.03 Fall 2012
Homework 3
due Friday, September 28, 2012 at 5 PM in Physics Homework Boxes, 3rd oor between Bldg 8
and 16
Information about Exam #1
Exam 1 will take place on Monday, October 1, 2012 during norm
Massachusetts Institute of Technology
Physics 8.03 Fall 2012
Homework 4
due Friday, October 12, 2012 at 5 PM in homework boxes
Reading Assignment
Note: from now on we will be jumping between chapters and pages in the books. French pages
161-178,223-230,25
Massachusetts Institute of Technology
Physics 8.03 Fall 2012
Homework 1
due Friday, September 14, 2012 at 5 PM, dropo boxes on 3rd oor between bldgs 8 and 16
Reading Assignment
French pages 3-16 and 41-70 (required). Beke & Barret 1-47 (very helpful).
Pro
Physics 8.03
Vibrations and Waves
Lecture 9
Wave equation in 2D and 3D
Time-independent Fourier analysis
Last time: Boundary Conditions
v2 v1
2v 2
Reflection and
r =
and =
v2 + v1
v2 + v1
transmission
Harmonic pulses y ( x, t ) = y cos(kx t + )
0
(traveli
Physics 8.03
Vibrations and Waves
Lecture 21
Diffraction + Interference
Diffraction gratings
Interference so far
Linear array of N sources
separated by d
Adjacent sources have
relative intrinsic phase
I is intensity at observation
angle
(
sin N
2
I = I
Physics 8.03
Vibrations and Waves
Lecture 17
EM waves meet dielectrics
Last time: waveguides
Single hollow conductor
TE or TM mode, but not TEM
mx j (t k z z )
E = y E0 sin
e
a
k x E0 mx j (t k z z )
k z E0 mx j (t k z z )
B = x
+ zj
cos
sin
e
Physics 8.03
Vibrations and Waves
Lecture 6
Driven Coupled Oscillators
Last time: Coupled oscillators
Normal modes of oscillation
Harmonic motion at fixed (eigen)frequencies
Amplitude ratios for each mode (constant)
Any old motion
All allowed motions are
Physics 8.03
Vibrations and Waves
Lecture 14
Dipole Radiation
Last time: polarization
Components of E0
E0x = E0y e j
E0x E0y and = n linearly polarized
E0x = E0y and = n/2 circularly polarized
E0x E0y and n/2 elliptically polarized
Energy carried by EM
Physics 8.03
Vibrations and Waves
Lecture 10
Fourier Analysis
Last time:
Wave equation in 2-D
1 2
2
2
( x, y , t ) +
( x, y , t ) = 2 2 ( x, y , t )
v t
x 2
y 2
n
k 2 = k x2 + k y2 = x
Lx
Arbitrary motion
Superposition of
normal modes
2
n y
+
L
Physics 8.03
Vibrations and Waves
Lecture 15
EM waves meet conductors
Transmission Lines
Last time
Radiation from accelerating charges
Dipole approximation
q a n (t r / c)
E rad (r , t ) =
4 0 rc 2
1
B rad (r , t ) = r E rad
c
1
S rad (r , t ) =
E rad B
Physics 8.03
Vibrations and Waves
Lecture 7
The Wave Equation
Solutions to the Wave Equation
Last time:
External driving force
Applied an external driving force to a coupled
oscillator system
In steady-state coupled system takes on frequency of
the drivin
Physics 8.03
Vibrations and Waves
Lecture 8
Boundary Conditions Applied to
Pulses and Waves
Last time:
Wave Equation and its Solutions
Waves oscillations in space and time
y (x, t )
Transverse or longitudinal waves
Traveling or standing waves
Solutions
Massachusetts Institute of Technology
Physics 8.03 Fall 2012
Homework 7
due Friday, November 9, 2012 at 5 PM in Physics Dropo Boxes
Reading Assignment
Chapter 3 in B&B pages 221-250 on polarization and radiation pressure. Read Chapter 4 251-267
about radi
Massachusetts Institute of Technology
Physics 8.03 Fall 2012
Homework 2
due Friday, September 21, 2012 at 5 PM in Physics boxes, 3rd oor between Bldg 8 and 16
Reading Assignment
French pages 77-101 required, 102-112 suggested, Beke & Barret 48-69 suggeste
Physics 8.03 Fall 2012
Homework 3 Solutions
3.1
3.1.a.
The tension in the strings acts along the direction of the string, so we can just
use Newtons Laws to nd the equations of motion for this problem. A torque
approach may also work just as well for the
Physics 8.03 Fall 2012
Homework 5 Solutions
5.1
5.1.a.
The displacement of the nth normal mode on a string of length L xed at both
ends can be written as
y (x, t) = A sin
v=
nvt
L
nx
sin
L
TL
M
where x = 0 represents the left end of the string. The energy
Massachusetts Institute of Technology
Physics 8.03 Fall 2012
Homework 5
due Friday, October 19, 2012 at 5 PM in Physics Homework Boxes
Quiz 2
Quiz 2 will take place on Monday October 22, 2012 in 50-340 during normal lecture hours. The
exam will last 1 hou
Physics 8.03 Fall 2012
Homework 4 Solutions
4.1
4.1.a.
v=
T
22 m/s
(1)
4.1.b.
= vT 2.2 m
(2)
4.1.c.
We know the waves traveling in the +x direction will take the form
y (x, t) = A cos
2
x
t
+
T
v
(3)
with A = 0.02 m. We know that when x = t = 0, the str
Massachusetts Institute of Technology
Physics 8.03 Fall 2012
Homework 10
due Friday, 7 December, 2012 at 5 PM in Physics Boxes
Reading Assignment
Due 4:00 PM, Friday, December 5, 2003
Read B&B Chapter 8, pages 519-553 on interference, pages 559-584 on di
Physics 8.03 Fall 2012
Homework 9 Solutions
9.1
Propagation in a dielectric is the same as in vacuum, except with the replace
ment 0 e 0 (and thus c v c/ e ). Note then that k = v/ , with
= 2f .
1
Ex = E0 cos(kz t + )
2
1
Ey = E0 cos(kz t + )
2
1 E0
Bx =
Physics 8.03 Fall 2012
Homework 6 Solutions
6.1
6.1.a.
6.1.b.
Plugging the ansatz into this modied wave equation, the exponential form
clearly works assuming a dispersion relation is satised.
2
2
2
= a 2 c
2
t
z
2
2 Aejtjkz = ak 2 Aejtjkz c Aejtjkz
2
2
Massachusetts Institute of Technology
Physics 8.03 Fall 2012
Homework 6
due Friday, November 2, 2012 at 5 PM in Physics Dropo Boxes
Reading Assignment
With this homework we nish the dispersion (last lecture before Quiz #2) and we start detailed
studies of
Physics 8.03
Vibrations and Waves
Lecture 11
Fourier Analysis with traveling waves
Dispersion
Last time:
Arbitrary motion Superposition of ALL
possible normal modes
m =1
n =0
y ( x, t ) = Am sin (k m x ) cos(mt + m ) + Bn cos(k n x ) cos(nt + n )
Orthogon