1. Introduction: Vectors and Integrals

2. Vectors Vectors are characterized by two parameters: length (magnitude) direction These vectors are the same Sum of the vectors:

3. Vectors Sum of the vectors: for a larger number of vectors the procedure is straightforward Vector (where is the positive number) has the same direction as, but its length is times larger Vector (where is the negative number) has the direction opposite to, and times larger length

4. usually have unit magnitude Vectors The vectors can be also characterized by a set of numbers (components), i.e. This means the following: if we introduce some basic vectors, for example x and y in the plane, then we can write Then the sum of the vectors is the sum of their components:

5. Vectors: Scalar and Vector Product is the scalar (not vector) is the VECTOR, the magnitude of which is Vector is orthogonal to the plane formed by and If the vectors are orthogonal then the scalar product is 0 Scalar Product Vector Product If the vectors have the same direction then vector product is 0

6. from the definition of the scalar product Vectors: Scalar Product is the scalar (not vector) If the vectors are orthogonal then the scalar product is 0 Scalar Product It is straightforward to relate the scalar product of two vectors to their components in orthogonal basis If the basis vectors are orthogonal and have unit magnitude (length) then we can take the scalar product of vector and basis vectors : =0 (orthogonal) =1 (unit magnitude)

8. Vectors: Examples The magnitude of is 5 What is the direction and the magnitude of The magnitude of is, the direction is opposite to The magnitude of is 5, the magnitude of is 2, the angle is What is the scalar and vector product of and

9. Integrals You need to recognize these types of integrals. Basic integrals: Examples: introduce new variable Important: Different Limits in the Integrals introduce new variable

10. Integrals How can we find the values of such integrals? Integrals containing vector functions - this is the vector, so we can calculate each component of this vector We can write, where only scalar functions depend on t, but not the basis vectors then integral takes the form Then the integral takes the form so now there are two integrals which contain only scalar functions

11. Integrals - along the radius, then we can write the radial vector in terms of radius Then we have the following expression for the integral Example:

12. 1. Waves and Particles 2. Interference of Waves

13. Traveling Waves Chapter 20

14. Waves and Particles Wave – periodic oscillations in space and in time of something It is moving as a whole with some velocity

15. Particle and Waves Sinusoidal Wave Particles

16. Sinusoidal Wave - Speed of wave Plane wave changes only along one direction maximumminimum - wavelength Period of “oscillation” – (time to travel a distance of wavelength) Frequency of wave

17. Sin-function Amplitude Phase (initial)

18. Sinusoidal (Basic) Wave Source of the Wave Time Distribution of Electric Field in space at different time Distribution of some Field in space and in time with frequency f - usual sin-function with initial phase, depending on t as

19. Sinusoidal Wave Distribution of some Field in space and in time with frequency f At a given time t we have sin-function of x with “initial” phase, depending on t At a given space point x we have sin-function of t with “initial” phase, depending on x

20. 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

21. Wave Pack Wave pack can be considered as a particle

22. Particle and Waves How can we distinguish between particles and waves? For waves we have interference, for particles – not!

23. Interference of Waves Chapter 21

24. Sin-function Amplitude Phase (initial) Amplitude Interference: THE SUM OF TWO SIGNALS (WAVES)

25. Sin-function: Constructive Interference Amplitude Phase (initial) Amplitude The phase difference between two waves should be 0 or integer number of

26. Sin-function: Destructive Interference Amplitude Phase (initial) Amplitude (no signal) The phase difference between two waves should be or plus integer number of

27. 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

28. Waves: Interference  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 Interference – sum of two waves

29. Waves: Interference Amplitude Phase (initial) Amplitude Constructive Interference: The phase difference between two waves should be 0 or integer number of Destructive Interference: The phase difference between two waves should be or integer number of

30. 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

31. Conditions for Interference: Coherence coherent The sources should be monochromatic (have the same frequency)

32. 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