cincirob

08-23-2008, 08:26 PM

Analyzing the Michelson-Morley Experiment

In the middle of the 19th century, Maxwell developed the equations of electromagnetism. He found that the equations predicted a disturbance in an electromagnetic field would proceed at the speed close to the known value of c at that time. And so it was that it came to be be believe that light was an electromagnetic phenomenon. At that point in history the knowledge of wave mechanics indicated that waves were waves of some medium. So, it was theorized that ether was the fluid that waved to create light.

Maxwell lamented that it wasn’t possible to examine this ether with the technology of his day. But by the 1880s, Michelson thought he had figured out a way. The idea goes like this: wave mechanics demands that a wave travel at a particular speed based on the characteristics of its medium. You can test this the next time you’re near a smoothly flowing stream. Toss a stone in the water. Notice that the waves travel away for the point where the stone struck the water in a circle and the whole circular pattern moves doesn stream with the velocity of the stream.

If you were floating down the stream at the location of the stone strike, you would see the wave travelling away from you in all directions with the same velocity. If you were near the stone strike location but paddling upstream, you would measure the speed of the wave going upstream as the wave velocity minus your velocity. The part of the wave going downstream would appear to have the velocity of the wave plus your velocity.

There were some ideas that the ether might be carried along with the Earth but astronomical observations would have see aberrations where ether would be carried along with other bodies. An alternative would be that all the ether in the universe was carried along with the Earth but that would be a return to pre-Copernican, Earth centered thinking. So, it was thought that the Earth must have some velocity through the ether. Michelson determined to measure it. His paper can be found at: http://spiff.rit.edu/classes/phys314/refs/mm.html

If you are trying to follow along, you should look at figures 1 and 2 in that document since this format doesn’t permit figures (or at least, I don’t have the tools).

Michelson starts by explaining that if the ether were perfectly still, the a ray of light would move through his device as follows: Starting at point s(source) the light strikes a half-silvered diagonal mirror that lets part of the light pass through and part of it reflect at a 90 degree angle. Other mirrors are located to reflect the light back to the diagonal mirror where the two rays of light are recombined. If there is no velocity through the ether, rotating this device would have no effect. By using monochromatic light, Michelson reasoned he would see an interference pattern because the two rays of light would take slightly different length paths.

Since, as noted above, Michelson believed there probably is a velocity through the ether and that situation is determined by figure 2. The two paths for the light are now different. The ray reflected at 90 degrees (or near 90) travels the two sides of the isosceles triangle shown while the other ray simply goes out and back along the other arm. If the device is aligned as shown along the direction of motion through the ether, then rotating the device would interchange the two paths. If the paths are different the rotation would cause the interference pattern to shift.

Michelson predicts what that shift would be for the velocity of the Earth around the Sun to see if the effect would be large enough to see. His calculations how that he would be able to discern that kind of velocity. We now know he could have used a velocity 10 times that value.

Let’s look at the calculations. The near 90 degree ray traverse the hypotenuse of a triangle whose base is vt/2 and whose height is L, the length of the arm of the device. So the distance along one leg of this path is

D(vertical) = sqr[(vt/2)^2 + L^2)]

And if the velocity of light is c, then the time for this traverse of one leg is

cT(vert) = D(vert)/c = sqr[(vT(vert)/2)^2 + L^2)]

c^2(T(vert)/2)^2 = [ v^2(T(vert)/2)^ 2 +L^2]

T(vert)^2[c^2 – v^2] = L^2

T(vert) = L/sqr[c^2 – v^2] = L/c/sqr[1 – (v/c)^2]

And the total time is twice this so

Tv = 2L/c/sqr[1 – (v/c)^2)^.5

Now for the other path. Since the device I traveling at v through the ether, and light travels at v through the ether, the time for the light to get to the mirror is

T(out) = L/(c-v)

And the return trip is

T(back) = L/(c+v)

The total time for the round trip is Th,

Th = t(out) + T(back) = L/(c-v) + L/(c+v) = L(c+v+c-v)/(c^2 – v^2) = 2Lc/(c^2 – v^2)

Th = 2L/c/(1 – (v/c)^2)

Notice, Tv does not equal Th and this time difference is what Michelson expected to cause an interference pattern shift when he rotated his device. The null result he did get, that is no pattern shift, was not expected and could not be explained by any known theory. Years later Fitzgerald and Lorentz look at these two time equations and made a startling suggestion. What if the length L along the horizontal leg of the device contracted? Here are the two equations

Tv = 2L/c/sqr[1 – (v/c)^2)^.5

Th = 2L/c/(1 – (v/c)^2)

Notice if instead of L in the Th equation is replaced by L(1 – (v/c)^2)^.5, the two equations become identical and this would explain the null result. This is the first time the length contraction formula was suggested and it is:

L = Lo(1 – (v/c)^2)^.5

This is pretty long so I will save showing how this equation is derived from the Lorentz transformations until next time.

In the middle of the 19th century, Maxwell developed the equations of electromagnetism. He found that the equations predicted a disturbance in an electromagnetic field would proceed at the speed close to the known value of c at that time. And so it was that it came to be be believe that light was an electromagnetic phenomenon. At that point in history the knowledge of wave mechanics indicated that waves were waves of some medium. So, it was theorized that ether was the fluid that waved to create light.

Maxwell lamented that it wasn’t possible to examine this ether with the technology of his day. But by the 1880s, Michelson thought he had figured out a way. The idea goes like this: wave mechanics demands that a wave travel at a particular speed based on the characteristics of its medium. You can test this the next time you’re near a smoothly flowing stream. Toss a stone in the water. Notice that the waves travel away for the point where the stone struck the water in a circle and the whole circular pattern moves doesn stream with the velocity of the stream.

If you were floating down the stream at the location of the stone strike, you would see the wave travelling away from you in all directions with the same velocity. If you were near the stone strike location but paddling upstream, you would measure the speed of the wave going upstream as the wave velocity minus your velocity. The part of the wave going downstream would appear to have the velocity of the wave plus your velocity.

There were some ideas that the ether might be carried along with the Earth but astronomical observations would have see aberrations where ether would be carried along with other bodies. An alternative would be that all the ether in the universe was carried along with the Earth but that would be a return to pre-Copernican, Earth centered thinking. So, it was thought that the Earth must have some velocity through the ether. Michelson determined to measure it. His paper can be found at: http://spiff.rit.edu/classes/phys314/refs/mm.html

If you are trying to follow along, you should look at figures 1 and 2 in that document since this format doesn’t permit figures (or at least, I don’t have the tools).

Michelson starts by explaining that if the ether were perfectly still, the a ray of light would move through his device as follows: Starting at point s(source) the light strikes a half-silvered diagonal mirror that lets part of the light pass through and part of it reflect at a 90 degree angle. Other mirrors are located to reflect the light back to the diagonal mirror where the two rays of light are recombined. If there is no velocity through the ether, rotating this device would have no effect. By using monochromatic light, Michelson reasoned he would see an interference pattern because the two rays of light would take slightly different length paths.

Since, as noted above, Michelson believed there probably is a velocity through the ether and that situation is determined by figure 2. The two paths for the light are now different. The ray reflected at 90 degrees (or near 90) travels the two sides of the isosceles triangle shown while the other ray simply goes out and back along the other arm. If the device is aligned as shown along the direction of motion through the ether, then rotating the device would interchange the two paths. If the paths are different the rotation would cause the interference pattern to shift.

Michelson predicts what that shift would be for the velocity of the Earth around the Sun to see if the effect would be large enough to see. His calculations how that he would be able to discern that kind of velocity. We now know he could have used a velocity 10 times that value.

Let’s look at the calculations. The near 90 degree ray traverse the hypotenuse of a triangle whose base is vt/2 and whose height is L, the length of the arm of the device. So the distance along one leg of this path is

D(vertical) = sqr[(vt/2)^2 + L^2)]

And if the velocity of light is c, then the time for this traverse of one leg is

cT(vert) = D(vert)/c = sqr[(vT(vert)/2)^2 + L^2)]

c^2(T(vert)/2)^2 = [ v^2(T(vert)/2)^ 2 +L^2]

T(vert)^2[c^2 – v^2] = L^2

T(vert) = L/sqr[c^2 – v^2] = L/c/sqr[1 – (v/c)^2]

And the total time is twice this so

Tv = 2L/c/sqr[1 – (v/c)^2)^.5

Now for the other path. Since the device I traveling at v through the ether, and light travels at v through the ether, the time for the light to get to the mirror is

T(out) = L/(c-v)

And the return trip is

T(back) = L/(c+v)

The total time for the round trip is Th,

Th = t(out) + T(back) = L/(c-v) + L/(c+v) = L(c+v+c-v)/(c^2 – v^2) = 2Lc/(c^2 – v^2)

Th = 2L/c/(1 – (v/c)^2)

Notice, Tv does not equal Th and this time difference is what Michelson expected to cause an interference pattern shift when he rotated his device. The null result he did get, that is no pattern shift, was not expected and could not be explained by any known theory. Years later Fitzgerald and Lorentz look at these two time equations and made a startling suggestion. What if the length L along the horizontal leg of the device contracted? Here are the two equations

Tv = 2L/c/sqr[1 – (v/c)^2)^.5

Th = 2L/c/(1 – (v/c)^2)

Notice if instead of L in the Th equation is replaced by L(1 – (v/c)^2)^.5, the two equations become identical and this would explain the null result. This is the first time the length contraction formula was suggested and it is:

L = Lo(1 – (v/c)^2)^.5

This is pretty long so I will save showing how this equation is derived from the Lorentz transformations until next time.