1. When a ball is thrown horizontally, it falls a distance of 4.9 m in the first second. By the principle of equivalence, a horizontal light beam also drops 4.9 m in 1 s.
A. Why isn’t this drop observed?
Light travels so fast (300,000,000 m/s to be exact) that it has gone too far away in that one second to notice the drop. In addition, it is going so far so fast that it could be far away from the Earth and its gravitational force after one second, so the whole notion that it is falling for 1 second isn't even right. If you could keep gravity acting on the light, however, you would be able to measure that for every 300,000,000 meters it travels, it falls about 5 meters. That, for all practical purposes, is essentially a straight line.
B. Describe an example in which gravity noticeably bends light. What is required for this to happen?
If you want to see light bend, put it in a gravitational field that is stronger than the Earth's. In other words, make it fall faster than a measly 4.9 meters every second. Where would you find such gravity? The Sun is one place; and when the light of the Sun is blocked (as in during a solar eclipse), the stars behind it can be seen. In fact, starlight coming towards the Sun actually is bent in such a way that stars appear to be in different places than normal. This effect is known as gravitational lensing. Gravitational lensing has also been recently found to occur around distant galaxies and apparently seems to a good feature of a black hole.
2. Consider the following two scenarios. In picture A, you are sitting inside a box and observing a beam of light as it travels from the bottom of your box and to the top. In picture B, you are observing the same box and the same beam of light from the outside. Think about the following:
A. Compare the amount of distance that the beam of light is observed to travel from each perspective. Which observed distance is longer? Why?
If you are inside the box (A), you see the laser and its light beam travel with you. In other words, you simply see it go straight up to the top of the box. If you are outside of the box (B), you see the beam not only go upwards, but you also see it travel away from you with the box. So, from B's perspective, you see the beam of light go on a diagonal path upward and away from you. This path is longer than the "straight-up" path observed by A.
B. Compare the time each beam of light is observed to travel. Which is longer? Why?
Each of you measures the same speed of light, c (300,000,000 m/s). This feature of light is known as the "constancy of the speed of light." You each measure the same speed, but you measure different distances for the same event. Ugh. This means that the person who measures the longer distance (B) also measures a longer time for the same event, in order for the speed of light to be measured the same from each reference frame.
C. How do your answers support the idea of time dilation in special relativity?
This is time dilation in action. The person in the box is measuring the proper time, while the person outside of the box is measuring a dilated time for the same event. What person A says takes 5 seconds, person B might say takes 6 seconds. Strange, but if the above observations and conclusions are correct, then there is no other choice. In other words, if the speed of light is always measured to be the same value, regardless of reference frame, then some other measurement has be vary between reference frames.
Ugh.
3. In the spaceship diagramed below, you shoot two laser pulses (at the speed of light) from the center of your craft to the ends. Your spaceship is moving from left to right.
A. From your perspective (inside the spaceship), compare the time it takes for each beam of light to hit a wall. Remember that the speed of light is the same in all directions and from all perspectives.
You and the laser and the beams of light are all in the same reference frame. In other words, you are all experiencing the same perspective. You see the laser in the middle of the room, equal distances away from each wall. Both beams of light hit a wall at the same exact time.
B. From an outsider’s perspective, compare the time it takes for each beam to hit a wall. Consider how much distance each pulse of light must travel to reach a wall (as seen from this perspective).
The outsider sees a strange thing: She sees an idiot flying by in a transparent box with a dual-sided laser! Amazing! In addition, because the box is moving from left-to-right, the right side of the box actually moves away from the source of the light, while the left side of the box moves towards the source of the light. As a result, the distance that the one laser beam travels to the left is shorter than the distance that the other laser beam travels to the right! Each beam moves with the same speed of light, so the left wall gets hit before the right wall gets hit (from the outsider's perspective).
C. How do your answers support the idea of time dilation in special relativity?
As in a previous situation, this shows that two different observers from different reference frames see events occur at different times. This case is especially weird, since the difference between two events for the insider is zero time, the difference between the same two events for the outsider is a measurable amount of time. Egads.
In a strict sense, this isn't an example of "time dilation" specifically; but it does show how the whole notion of time is not absolute, but relative, to its observers. The idea that simultaneous event in one reference frame can be non-simultaneous from the perspective of another reference frame is known as "simultaneity." (Now why do you suppose they call it such???)
What if the "outsider" had the laser beam? How would things work in that case? Oh no . . .
4. According to the Bohr model of the atom, explain why an atom’s electrons only emit specific wavelengths of light, rather than a smooth rainbow of all wavelengths.
The Bohr model of the atom shows electrons only existing at specific energy levels, with no existence in between energy levels. Electrons, behaving as waves, must "fit" the atom just like a sound wave and its harmonics fit an open-ended organ tube. Thus, only a certain "fundamental" wavelength electron and its higher "harmonics" can "fit" in the atom. Different wavelengths mean different energies, so only specific energies are allowed.
Fine (really?!), but what about the light? If an electron can only fall from one specific energy level to another, then it will release a very specific amount of energy. This comes in the form of light. And, light's energy is determined by its wavelength. Thus, you only allow certain energies of electrons, resulting in specific energies of light getting dumped off the atom, resulting in specific wavelengths of that light.
5. The photoelectric effect demonstrated that light can behave as a particle. We also show evidence that "particles" can behave as waves. (In fact, everything is both a particle and a wave!) Using the Heisenberg uncertainty principle, explain why a constant velocity stream of electrons would have many different velocities (directions) after traveling through a very narrow opening (in which there was little uncertainty in their position).
One thousand curses to you, Dr. Johnston! My head hurts!!! (While not the right answer, this seems like a completely valid response.)
By forcing the electrons to go through a narrow opening, you are forcing them to be in a very exact position (the narrow opening). This exact position has a very very small uncertainty in position. Heisenberg's uncertainty principle states that as uncertainty in position decreases, uncertainty in momentum (velocity) increases. So, as soon as you force the electrons to go through the opening, you no longer have any certainty in where they are traveling (velocity or momentum). As a result, electrons diffract, shooting every which way out of the narrow opening. The narrower you make the opening, the more the electrons scatter in all directions. In fact, this is exactly what any self-respecting wave does; so we see that electrons behave as waves in this case. This has actually been done experimentally.