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Introduction:
Vectors and Integrals
Vectors
Vectors are characterized by two parameters:
a
• length (magnitude)
• direction
a
These vectors
are the same
a
Sum of the vectors:
b
a
a
ab b a
b
b
a
Vectors
Sum of the vectors: for a larger number of
vectors the procedure is straightforward
b
a
c
b
a
a
a
a  a  2a
Vector c a (where c is the
positive number) has the
same direction as a , but
its length is c times larger
Vector c a (where c is the
negative number) has the
direction opposite to a ,
and c times larger length
ab c
c
a
2a
a
2a
Vectors
The vectors can be also characterized
by a set of numbers (components), i.e.
a  (a1 , a2 ,...)
This means the following: if we
introduce some basic vectors, for
example x and y in the plane, then we
can write
a  a1 x  a2 y
a
a2 y
y
x
a1 x
x , y usually have unit magnitude
Then the sum of the vectors is the
sum of their components:
a  (a1 , a2 )
b  (b1 , b2 )
a  b  (a1  b1 , a2  b2 )
a  (a1 , a2 )
ca  (ca1 , ca2 )
Vectors: Scalar and Vector Product
Scalar Product
a
a b
is the scalar (not vector)
 ab cos( )

If the vectors are orthogonal then the scalar product is 0
b
a
a b  0
b
Vector Product
c
a
a b  c

is the VECTOR, the
magnitude of which is
ab sin( )
Vector c is orthogonal
to the plane formed by a
and b
b
If the vectors have the same direction then vector product is 0
b
a
a b  0
Vectors: Scalar Product
a
Scalar Product

a b
b
is the scalar (not vector)
 ab cos( )
If the vectors are orthogonal then the scalar product is 0
a
b
a b  0
It is straightforward to relate the scalar product of
two vectors to their components in orthogonal basis
a
a2 y
y

x
a1 x
a  a1 x  a2 y
If the basis vectors x , y are orthogonal and have unit
magnitude (length) then we can take the scalar product
of vector a  a1 x  a2 y and basis vectors x , y :
a cos( )  a  x  a1 x  x  a2 y  x  a1
from the definition of
the scalar product
=1 (unit magnitude)
=0 (orthogonal)
a cos( / 2   )  a sin( )  a  y  a1 x  y  a2 y  y  a2
a
a
ab
ab
y
x
b
a1  a  x
a2  a  y
a2 y
a1 x
a
a
2a
a
a  b  ab cos( )

2a
a
a b  0
b
b
c
a
a  (a1 , a2 )
a b  c
c  ab sin( )

b
b
a
a b  0
Vectors: Examples
The magnitude of
a
a
is 5
What is the direction and the magnitude of b  0.2a
b
The magnitude of b is b  0.2  5  1 , the direction is opposite to
a
The magnitude of
angle  is  / 3

a
is 5, the magnitude of
What is the scalar and vector product of
b
a  b  5  2cos( / 3)  5
c
a

b
a b  c
c  5  2sin( / 3)  5 3
a
b
is 2, the
and
b
a
Integrals
Basic integrals:
dx
1  1
1 


a x n n  1  a n1 bn1 
b
b
n
x
 dx 
a
1
b n 1  a n 1 

n1
You need to recognize these types of integrals.
Examples:
b
•
dx
a ( x  c )n
introduce new variable
y  xc
dy  dx
b
b c
dx
dy

a ( x  c )n a c y n
Important: Different
Limits in the Integrals
b
•
xdx
a ( x 2  c )n
introduce new variable
y  x c
dy  2 xdx
2
b
b2  c
xdx
1
dy

a ( x 2  c )n 2 2 y n
a c
Integrals
y
Integrals containing vector functions E ( t )
b
 E ( t )dt
t
a
a
E (t )
x
b
How can we find the values of such integrals?
b
 E ( t )dt
a
- this is the vector, so we can calculate each component of this vector
We can write E (t )  E1 (t ) x  E2 (t ) y , where only scalar functions E1 ( t ), E2 ( t )
depend on t, but not the basis vectors x , y then integral takes the form
Then the integral takes the form
b
b
b
 E (t )dt  x  E (t )dt  y  E (t )dt
1
a
a
2
a
so now there are two integrals which contain only scalar functions
Integrals
r ( )

Example:
2
 r ( )cos( )d
y
x
0
r ( ) - along the radius, then we can write the radial vector in terms of radius r
r ( )  r1 ( ) x  r2 ( ) y  r cos( ) x  r sin( ) y
Then we have the following expression for the integral
2

0
2
2
0
0
r ( )cos( )d  r x  cos 2 ( )d  r y  cos( )sin( )d  r  x
2
1
2
  [1  cos(2 )]d 

2 0
2
1

2
2
 sin(2 )d  0
0
1. Waves and Particles
2. Interference of Waves
Chapter 20
Traveling Waves
Waves and Particles
Wave – periodic oscillations in
space and in time of something
It is moving as a whole with
some velocity v
t0
vt
t0
x
Particle and Waves
Sinusoidal Wave
Particles
Sinusoidal Wave
Plane wave
c - Speed of wave
y
changes only along one direction
y

- wavelength
x

Period of “oscillation” – T 
c
(time to travel a distance
of wavelength)
Frequency of wave
f 
1 c

T 
x
maximum
minimum
Sin-function
E( x)
Amplitude
Phase (initial)
x


E ( x )  E0 sin  2   



Amplitude E0
 0

2
3
x
Sinusoidal (Basic) Wave
x
x



 Distribution of some Field in
E ( x, t )  E0 sin  2  2 ft   E0 sin  2   t 





 space and in time with
frequency f
1 c
Distribution of Electric Field
in space at different time
f 
T


x


E ( x )  E0 sin  2   t  - usual sin-function


 with initial phase,
depending on t as
Source of
the Wave
t   t
c
Time
x
Sinusoidal Wave
x
x



 Distribution of some Field in
E ( x, t )  E0 sin  2  2 ft   E0 sin  2   t 





 space and in time with
frequency f
x
At a given time t we have sin-function of x with “initial” phase,
depending on t
x


E ( x )  E0 sin  2   t 



 t  2 f t   t
t
At a given space point x we have sin-function of t with “initial”
phase, depending on x
E( x )  E0 sin  t   x 
 x  2
x

Particle and Waves
We can take the sum of many sinusoidal waves (with
different wavelengths, amplitudes) = wave pack
Sum
=
Any shape which is
moving as a whole
with constant velocity
Wave Pack
Wave pack can be
considered as a particle
Particle and Waves
How can we distinguish between particles and waves?
For waves we have interference, for particles – not!
Chapter 21
Interference of Waves
Sin-function
E( x)
Amplitude
Phase (initial)
E( x )  E0 sin  2 f t   
Amplitude E0
 0
E( x)
1/ f
2/ f
3/ f
t
Amplitude E0
 0
t
Interference: THE SUM OF TWO SIGNALS (WAVES)
Sin-function: Constructive Interference
E( x)
Amplitude
Phase (initial)
E( x )  E0 sin  2 f t   
Amplitude E0
 0
E( x)
1/ f
2/ f
3/ f
t
Amplitude E0
 0
t
Amplitude 2E0
 0
The phase difference
between two waves
should be 0 or integer
number of 2
t
Sin-function: Destructive Interference
E( x)
Amplitude
Phase (initial)
E( x )  E0 sin  2 f t   
Amplitude E0
 0
E( x)
Amplitude
1/ f
2/ f
3/ f
t
E0
  180o ( )
t
E( x)
Amplitude  0
(no signal)
t
The phase difference between two waves should
be  or  plus integer number of 2
Waves and Particles
Interference of waves: THE SUM OF TWO WAVES
Analog of Interference for particles: Collision of two particles
The difference between the interference of waves and collision of particles
is the following: THE INTERFERENCE AFFECTS MUCH LARGER
REGION OF SPACE THAN COLLISION DOES
Waves: Interference
x


E1 ( x , t )  E0 sin  2  2 ft 



Interference – sum of two waves
 x  2
1
x1


E1 ( x , t )  E0 sin 2 ft   x1
x


E2 ( x, t )  E0 sin  2  2 ft 




E2 ( x , t )  E0 sin 2 ft   x2
 x  2
2


x2

 In constructive interference the amplitude of the
resultant wave is greater than that of either individual
wave
 In destructive interference the amplitude of the
resultant wave is less than that of either individual wave
Waves: Interference
E1 ( t )
Amplitude
 x  2
1
 x  2
2
Phase (initial)
E(t )  E0 sin  2 ft   x 
E0
x1

E2 ( t )
Amplitude
Amplitude
1/ f
2/ f
t
3/ f
E0
x2

t
Constructive Interference: The phase difference between two waves should be 0
or integer number of 2
m  0, 1, 2...
 x   x  2 m
1
2
Destructive Interference: The phase difference between two waves should be
or  integer number of 2
 x   x    2 m
m  0, 1, 2...
1
2

Conditions for Interference
To observe interference the following two conditions must
be met:
1) The sources must be coherent
- They must maintain a constant phase with
respect to each other
2) The sources should be monochromatic
- Monochromatic means they have a single
(the same) wavelength
Conditions for Interference: Coherence
coherent
E( x )  E0 sin  t  1 
E( x )  E0 sin  t  2 
The sources should be monochromatic
(have the same frequency)
E( x )  E0 sin 1t   
E( x )  E0 sin 2t   
Waves and Particles
The difference between the interference of waves and collision of particles
is the following: THE INTERFERENCE AFFECTS MUCH LARGER
REGION OF SPACE THAN COLLISION AND FOR A MUCH LONGER
TIME
If we are looking at the region of space that is much larger than the
wavelength of wave (or the size of the wave) than the “wave” can be
considered as a particle