Chapter 12

1. Chapter 12 Special Relativity and Elementary Particles

2. Special Relativity • Imagine sitting in the back of a truck as a friend throws a ball to you. • You are moving away from your friend at 20 km/h. • The ball is thrown at 50 km/h relative to the ground.

3. Special Relativity, cont’d • The speed of the ball, relative to you, depends on how you are moving relative to the ball. • If you were standing, the ball would approach you at 50 km/h. • Since you are moving away, the ball appears to approach you at 30 km/h.

4. Special Relativity, cont’d • Now imagine traveling away from you friend in a spaceship as he shines a light at you. • You travel at 200,000 km/s away from your friend. • The light travels away from your friend at 300,000 km/s.

5. Special Relativity, cont’d • The speed of the light, relative to you, does not depend on you are moving relative to your light. • You measure the light at 300,000 km/s from the spaceship. • If standing still, you still measure 300,000 km/s.

6. Special Relativity, cont’d • So, the speed of light does not depend upon the motion of the observer. • Light, behaving as Maxwell predicted, does not act as we would expect. • Based on the physics we’ve studied so far.

7. The postulates of special relativity • Einstein recognized the contradiction between the predictions of classical mechanics and electromagnetism. • Classical mechanics says the speed of light depends on the motion of the observer. • Electromagnetism says the speed of light does not depend on the motion of the observer. • He put forth two postulates to overcome this discrepancy.

8. The postulates of special relativity, cont’d • The 1st postulate of relativity states: • The speed of light, c = 300,000 km/s, is the same for all observers, regardless of their motion. • The speed of light is a fundamental constant of nature. • Much like the gravitational constant, G. • Prior to Einstein, several experiments were unsuccessful at demonstrating different values of the speed of light based on the observer’s motion.

9. The postulates of special relativity, cont’d • The 2nd postulate of relativity states: • The laws of physics are the same for all observers moving uniformly. • Uniformly means at a constant velocity. • This is known as the principle of relativity. • This means if two observers traveling toward one another at a constant acceleration perform the same experiment, they obtain identical results.

10. The postulates of special relativity, cont’d • These two postulates constitute Einstein’s special theory of relativity. • It is “special” because it is restricted to uniform motion.

11. Predications of special relativity • These two postulates have some unusual implications. • Time dilation— you observe the clock of someone moving, relative to you, to run slower than your clock. • Length contraction— you observe the length of an object moving, relative to you, to be less than an identical object at rest, relative to you. • In the direction of the object’s motion.

12. Time dilation • Consider an experiment. • Create a clock out of a flashbulb and a photo-detector. • The bulb sends out a flash of light. • The flash reflects off the mirror and triggers the detector.

13. Time dilation, cont’d • You synchronize your clock with a friend’s clock in a spaceship traveling at a speed v. • The question is: will the clocks keep the same time?

14. Time dilation, cont’d • For your clock: • The distance the light must travel from the bulb to the detector is d. • The speed at which light travels is c. • The amount of time between clicks is:

15. Time dilation, cont’d • For your friend’s clock (as seen by you): • Since the spaceship is traveling with speed v, you think the light must travel farther than the distance 2d. • From your observation, let the time interval between “ticks” of your friend’s clock is Dt′.

16. Time dilation, cont’d • The distance the light must travel during a “click” is • Since this is the distance light travels during a “click,” we also have

17. Time dilation, cont’d • So we can equate these two: • After some algebra, we can solve for Dt′.

18. Time dilation, cont’d • Recall that from your clock, you know: • So you can relate the two time intervals:

19. Time dilation, cont’d • So what does this formula tell us? • Dt is the interval between clicks of your clock. • Dt′ is the interval between clicks of your friend’s clock as observed by you. • You see your friend’s clock run more slowly than your click. • Even though they are synchronized to “click” at the same rate when side-by-side at rest.

20. Time dilation, cont’d • How significant is this effect? • For “everyday” speeds (v < 0.5c), the effect is negligible. • For higher speeds, the effect can become very significant. • Even doubling or tripling how slow his clock appears to run.

21. Time dilation, cont’d • Is this real? • Muons are subatomic particles created high in the atmosphere through collision with atmospheric molecules and energetic particles from space. • They are created about 10 km or more above the Earth’s surface. • They have an average lifetime of 0.000 002 s. • Since they die so quickly, they should not reach the surface in any great quantity. • So, why do we detected such a large number at the Earth’s surface?

22. ExampleExample 12.1 What is the mean lifetime of a muon as measured in the laboratory if it is traveling at 0.90c with respect to the laboratory? The mean lifetime of a muon at rest is 2.2×10-6 seconds.

23. ExampleExample 12.1 ANSWER: The problem gives us: So the lifetime as seen from the lab is

24. ExampleExample 12.1 DISCUSSION: So the muon “thinks” it only exists for 2.2 ms (as measured by its moving clock). But we on the ground think the muon exists for twice as long (as measured by our stationary clocks): 5.0 ms.

25. Length contraction • Another phenomenon predicted by special relativity is length contraction. • This is the apparent shortening of moving objects in the direction of their motion. • So how does this happen? • One method to measure the length of an object by timing how long it takes light to traverse the object.

26. Length contraction, cont’d • But we just discovered that moving clocks appear to run more slowly than stationary clocks. • Since the moving clock appears to take less time to measure the traversal of light, it predicts that the length should be shorter. • Recall that • So a shorter time interval indicates a shorter length.

27. Length contraction, cont’d • As for time dilation, length contraction is not observed at “normal” speeds. • But it is noticeable in particle accelerators. • These are scientific devices used to accelerate particles to very high speed.

28. Length contraction, cont’d • Our muon story also demonstrates length contraction. • Even though the muon is 10 km or more above the surface, they reach the surface. • From the muon’s perspective, their lifetime is too short to reach the Earth’s surface. • But since they do, the muon’s must measure the distance to be less than 10 km. • The speed and lifetime are fixed. • The distance is the only thing that can be changed in order for them to accomplish this.

29. Rest energy • Our definitions of kinetic energy and momentum no longer work in the realm of special relativity. • In order for conservation of kinetic energy and momentum to hold, we must include a rest energy, E0, in our calculations. • The rest energy is

30. Rest energy, cont’d • The total relativistic energy of a particle has two terms. • the rest energy, and • the kinetic energy.

31. Rest energy, cont’d • The relativistic kinetic energy is then • This reduces to our usual formula for very low speeds:

32. ExampleExample 12.2 In an x-ray tube, an electron with mass m = 9.1×10-31 kg is accelerated to a speed of 1.8×108 m/s. How much energy does the electron possess? Give the answer in joules and MeVs (million electron-Volts).

33. ExampleExample 12.2 ANSWER: The problem gives us: The energy is The speed ratio is

34. ExampleExample 12.2 ANSWER: The relativistic energy is then

35. ExampleExample 12.2 ANSWER: To express this in MeV

36. ExampleExample 12.2 DISCUSSION: Notice that the rest energy of an electron is

37. ExampleExample 12.2 DISCUSSION: The classical result for the energy is Too small by almost 30%.

38. The four forces • There are four fundamental forces.

39. The four forces, cont’d • Every interaction in our environment is due to these forces. • We initially defined a force as a push or pull. • This is acceptable in classical physics. • It needs to be expanded to deal with subatomic particles. • It has to include every process a particle can undergo. • disintegration, annihilation, reaction, creation, …

40. The four forces, cont’d • So we typically use the term four basic interactions instead of four basic forces. • An interaction means the mutual action or influence of one or more particles on another.

41. The four forces — gravity • Gravity is the most familiar of the four basic interactions. • We investigated it in Chapter 2. • In the world of particle physics, it is the least important. • Its strength is so feeble compared to the other interactions so it can be ignored. • It is 10-38 times weaker than the electromagnetic interaction.

42. The four forces — gravitycont’d • Gravity is probably the most important for large-scale interactions. • Since most objects are electrically neutral, the gravitational interaction is more important than the electromagnetic interaction. • Realize that both are “infinite-range” forces. • It is responsible for the distribution of matter in the Universe. • Newton’s law of gravity has been replaced by Einstein’s general theory of relativity.

43. The four forces — electromagnetic • The next most-familiar interaction is the electromagnetic. • We discussed it in Chapter 8. • The electromagnetic force can be attractive, repulsive or irrelevant. • Gravity is always attractive. • It is also an “infinite-range” interaction. • But if objects are electrically neutral, there is no significant interaction.

44. The four forces — electromagneticcont’d • It can affect certain, non-charged particles. • The photon is the “force carrier” for the electromagnetic interaction. • The photon has no electrically charge. • But it does involve an electric and magnetic field. • So the electromagnetic interaction influences the fields associated with a photon.

45. The four forces — weak • The interaction intermediate to the EM and gravitational interaction is the weak nuclearinteraction. • It is a nuclear interaction. • It only influences nuclear processes. • So we never encounter it in daily life. • It is “weak” since it is 10-5 times weaker than the EM interaction. • But still 1031 times stronger than gravity. • It only acts over ranges of 10-18 meters.

46. The four forces —weakcont’d • It is also “weak” because it is unlikely that processes involving this interaction occur. • It plays a critical role in the generation of energy in the Sun and in building-up heavy elements. • Another aspect of it being “weak” is that it occurs on longer time scales than the electromagnetic force. • About a million times “slower.”

47. The four forces —weakcont’d • There is a very close relationship between the electromagnetic and weak interactions. • Each can be responsible for the same types of interactions. • But the EM interaction is much more likely to be responsible. • These two theories have been joined into an electroweak interaction. • This is similar to electricity and magnetism being joined into electromagnetism.

48. The four forces — strong • There is another nuclear force, called the strong interaction. • It is about 100 times stronger than the EM interaction. • But it operates within very small ranges. • Although the EM interaction is an “infinite-range” interaction, the strong exerts influence only over a range of 10-15 meters. • It is always attractive and does not depend upon electric charge.

49. The four forces — strongcont’d • The strong force is responsible for overcoming the electric repulsion of protons in the nucleus.

50. Force carriers • Since we have to change our definition of force, we have a new impression of how forces cause particles to interact. • Each force has its own type of force carrier. • These are particles that are exchanged due a certain interaction and the type of particle depends on the interaction.