The two-mile linear accelerator at Stanford University in California "appears" to be less than a meter long to the electrons that travel in it. Explain.
For the moving electron, length contraction
reduces the apparent length of the 2-mile long
tube. Since electrons move with nearly the
speed of light, the contraction is very signifi-
cant.
Length contraction is the relativistic physical phenomenon of a decrease in length detected by an observer of objects that travel at any non-zero velocity relative to that observer.
L = 2 miles = 3218.7 m is the length observed by an observer in relative motion with respect to the object,
Lₒ =3219.7 m is the proper length (the length of the object in its rest frame),
L=Lₒ•sqrt(1 -β²),
1- β² =(L/ Lₒ)²,
β = sqrt{1 –(L/ Lₒ)²}=
= sqrt{1 – (3218.7/ 3219.7)²} =0.025
β = v/c,
v = β•c = 0.025 •3•10^8 =7.5•10^6 m/s.
The phenomenon that makes the two-mile linear accelerator at Stanford University appear shorter to the travelling electrons is known as "length contraction" or "Lorentz contraction." It is a consequence of the theory of special relativity proposed by Albert Einstein.
According to special relativity, as an object moves closer to the speed of light, its length in the direction of motion appears to contract from the perspective of a stationary observer. This contraction occurs in the direction of motion and is relative to the observer's frame of reference.
In the case of the linear accelerator at Stanford University, the electrons are accelerated close to the speed of light. As their speed increases, the length contraction effect comes into play. From the perspective of the stationary observer, the electrons' contracted length is observed. Thus, to the electrons traveling in the accelerator, which are moving at a significant fraction of the speed of light, the length of the accelerator appears shortened. In other words, the distance they need to travel seems to decrease due to their high velocities.
This phenomenon is a consequence of the relativistic equations, and it has been experimentally observed and verified.
The phenomenon you are referring to is known as time dilation, which is a consequence of Einstein's theory of relativity. According to this theory, the perception of time can change depending on the relative motion between observers.
In the case of the two-mile linear accelerator at Stanford University, electrons are accelerated to speeds close to the speed of light. At such high speeds, time dilation comes into play. As the electrons approach the speed of light, their perception of time slows down compared to a stationary observer's perception.
To understand this concept, let's consider a simplified example: Imagine two observers, one stationary and one on board the electron as it travels through the accelerator.
From the perspective of the stationary observer, time appears to flow normally. As the electron travels along the two-mile accelerator, the observer sees it cover the entire distance in a certain amount of time, let's say 1 second.
However, for the observer on board the electron, time is different. Due to the high velocity, the electron experiences time dilation, causing its perception of time to slow down relative to the stationary observer. This means that, from the electron's perspective, time is dilated, or stretched out.
As a result, the electron experiences the journey through the two-mile accelerator as if it were shorter in length. The actual distance of two miles appears significantly compressed or contracted, making it appear less than a meter long to the electrons.
This illusionary contraction of space is a consequence of time dilation, allowing the electrons to effectively traverse the accelerator in a much shorter perceived distance than what is observed by the stationary observer.
To calculate the exact amount of compression, one would need to apply the equations of special relativity, taking into account the velocity of the electrons and their associated time dilation effects. However, it is important to note that this compression refers to the perceived distance by the electrons, not the actual physical length of the accelerator.