Lecture 11 Examples of Induction
There are an enormous number of different ways in which induction is important. Here
are simplified examples of induction.
1
A linearly moving wire segment
Consider a very simple conducting loop that lies in the X Y plane.
Lecture 14 LC Circuits
We can write for a simple circuit with an inductor of inductance, L, and a capacitor with
capacitance, C, and nothing else that:
L
dI
Q
+
= 0
dt
C
We can write that
dQ
dt
I =
(1)
(2)
Therefore, for our simple circuit:
L
Q
d2 Q
+
= 0
Lecture 13 Complex Numbers
To solve for the flow of electrons through a circuit with various devices, we can very
conveniently use complex
numbers. A brief introduction is provided here.
We define i = 1 as the imaginary number. A complex number, z can hav
Lecture 23 Light Scattering: the Blue Sky
We now consider a simple model to describe the interaction of light with matter. If a
particle of charge q is accelerated by the amount x
, then the power radiated is:
P =
q2 x
2
6 0 c3
(1)
The derivation of this
Lecture 12 Solenoids as Inductors
An extremely important electrical device is an inductor. A typical system is a coil of wires
that looks like a solenoid.
With a solenoid of length, l, total number of coils, N , and current I, then the magnetic
field is g
Lecture 28 Interference: The Double Slit
Interference is a phenomenon which depends upon light acting as a wave. In computing
interference, we add electric fields of different waves. We can often describe the addition
by combining two different sine waves
Lecture 20 Classical and Quantum Properties of Light
The demonstration that light is an electromagnetic wave is one of the most important
consequences of Maxwells Equations. It opens a door for a much better understanding of
the natural world.
A solution
Lecture 18 Maxwells Equations
In differential form, we can write for the electric and magnetic fields that:
~ B
~ = 0
~ E
~ =
0
(1)
(2)
~
~ E
~ = B
(3)
t
These equations are fine. However, the differential form of Amperes Law cannot be correct.
That is:
Lecture 27 Reflecting Dishes
Armed with the principles of refraction and reflection, we can now consider the various devices
that have been invented to focus and control light. There are, of course, a very large number of
applications, and we can only con
Lecture 21 Momentum Carried by Light
For a particle of mass m and speed v, we can write that the relationship between the kinetic
energy, E, and the momentum, p, is:
p =
2E
v
(1)
We would like to find the analogous relationship between energy and momentum
Lecture 15 RC and LRC Circuits
For a circuit with only a resistor and a capacitor, the equation governing its behavior
is:
or:
IR +
Q
= 0
C
(1)
dQ
=
dt
1
Q
RC
(2)
The solution can be written as:
Q = Re Q0 ei(! t +
0)
If so, then because this is a first or
Lecture 17 Fourier Analysis (Enrichment)
Just as we can write a function as a Taylor Series an infinite polynomial,
we can also write any function as the sum of sin and cosine functions. Here,
we will consider the simplified case of fitting any function w
Lecture 22 Einsteins Famous Equation
Consider a thought experiment where two light emitters/receivers are placed on opposite ends of a flat railroad car of mass M and length L. Assume that the rail car lies
on an essentially frictionless railroad. Assume
Lecture 16 Driven Harmonic Oscillators
We can write for a simple circuit with an inductor of inductance, L, a capacitor with
capacitance, C, and resistor with resistance, R, and a sinusoidally varying applied EMF
that:
d2 Q
dQ
Q
L 2 + R
+
= E0 ei t
(1)
dt
Lecture 19 Light
We now solve Maxwells equations for the special and important case of a vacuum where
= 0 and ~j = 0. In this case, the equations become:
~ B
~ = 0
(1)
~ E
~ = 0
(2)
~
~ E
~ = B
t
(3)
~
~ B
~ = 0 0 E
(4)
t
These 4 equations can be complic
Lecture 29 Angular Resolution
We want to describe the minimum angular size that can be observed with any optical
system. Consider first a 1-dimensional hole of width D. We approximate it as a system
with two slits separated by distance D. If we measure fr
Lecture 33 Relativistic Dynamics
Because objects cannot move faster than the speed of light, we need a new formulation of
dynamics. We write that
d~
p
F~ =
(1)
dt
where p~ is the relativistic momentum. A fully self-consistent formulation of the dynamics
t
Lecture 26 Fermats Principle and Geometrical Optics
Fermats Principle is a fascinating re-statement of the laws of geometric optics. It
basically states that the path that light takes the least time compared to other nearby
paths. Mathematically, Fermats
Lecture 32 Addition of Velocities
Consider the addition of velocities. We assume that frame O0 is moving at speed V
along the X axis relative to frame O. We wish to compute the velocity ux0 of an object
measured in the O0 frame which is moving with speed
Lecture 30 Special Relativity I
In inertial frames of references frames which are unaccelerated the laws of physics
are always the same.
Consider two co-ordinate systems, O and also O0 which is moving along the +X direction with speed V . Assume no charge
Lecture 31 The Lorentz Transformation
Consider two co-ordinate systems, O and also O0 which is moving along the +X direction with speed V . We define
V
=
(1)
c
where c is the speed of light or 2.9979 108 m s1 and
1
= p
1 2
(2)
Note that > 1.
With these
Lecture 25 Earths Average Surface Temperature
The flux from a blackbody, F (Watt m2 Hz1 ), is given by:
F d =
2 h 3
1
d
2
h/k
BT 1
c
e
(1)
where T is the temperature of the substance, and h, kB and c (the speed of light) are
constants. The integrated flux
Lecture 24 Scattering and Attenuation of Light
We are highly motivated to understand the propagation of light through materials. For a
monodirectional beam of light that carries flux, F , we can write that:
dF
= F n
dx
(1)
where n is the number density of
Discussion 3: Magnetism III: Faradays Law
Relevant Formul:
Faradays Law
)
(
=
cos( )
cos( ) + sin( )
=
Magnetic Field of a long straight Wire
=
0
2
Magnetic Field inside of a long Solenoid
= 0 =
Self-Inductance of a Solenoid
0 2
1 2
=
2
=
Energy
Discussion 2: Magnetism II
Relevant Formul:
Force on a current
=
Force on a straight wire in a uniform magnetic field
Force on a point Charge
Ampres Law
Biot-Savarts Law
Faradays Law
Magnetic Field of a long straight Wire
Magnetic Field inside of a lo
Project 3
Boulder Blast
For questions about this project, first consult your TA.
If your TA cant help, ask Professor Nachenberg.
Time due:
Part 1: 9 PM, Sunday, February 22
Part 2: 9 PM, Friday, February 27
WHEN IN DOUBT ABOUT A REQUIREMENT, YOU WILL NEVE
Project 3
Boulder Blast
For questions about this project, first consult your TA.
If your TA cant help, ask Professor Nachenberg.
Time due:
Part 1: 9 PM, Sunday, February 22
Part 2: 9 PM, Saturday, February 28
WHEN IN DOUBT ABOUT A REQUIREMENT, YOU WILL NE
Project #3
NachMan
For questions about this project, first consult your TA.
If your TA cant help, ask Professor Nachenberg.
Time due:
Part 1: Thursday, February 24 at 9 pm
Part 2: Tuesday, March 1 at 9 pm
Table of Contents
Introduction. 3
Game Details. 5