Question

A \(100-\mathrm{MeV}\) electron, for which \(v=0.999987 c\), moves along the axis of an evacuated tube that has a length of \(2.86 \mathrm{~m}\) as measured by a laboratory observer \(S\) with respect to whom the tube is at rest. An observer \(S^{\prime}\) moving with the electron, however, would see this tube moving past with speed \(v\). What length would this observer measure for the tube?

Short answer

The observer moving with the electron would measure the length of the tube to be approximately \(0.0404266 m\).

Step-by-step solution

Identify the given information

The problem provides that \(L_0 = 2.86 m\) (the length of the tube measured by observer \(S\)), \(v = 0.999987 c\) (the speed of the electron which is also the speed of observer \(S'\)), and \(c\) (the speed of light, a universal constant).

Use the formula of length contraction

We should use the {\it length contraction formula} which is \(L=L_0\sqrt{1-v^2/c^2}\) .

Substitute the given values into the formula

Plug the given values into the length contraction formula: \(L = 2.86 \sqrt{1 - (0.999987)^2}\).

Calculate the length

Perform the calculation inside the square root first, then multiply by the length of the tube as measured by observer \(S\): \(L = 2.86 \sqrt{1 - (0.999987)^2} = 2.86 \times 0.01415.\)

Compute the final value

Carrying out the multiplication yields the length of the tube as measured by observer \(S'\): \( L = 0.0404266 m\).

Key concepts

Special Relativity

Special Relativity is a groundbreaking theory proposed by Albert Einstein which revolutionized the way we understand space and time. In essence, it describes how objects move and how time passes differently for observers moving at different speeds relative to one another.
It consists of two main postulates:

• The laws of physics are the same in all inertial frames of reference (i.e., frames moving at constant velocity relative to each other).
• The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or observer.

These postulates lead to fascinating consequences such as time dilation and length contraction. While everyday experiences suggest that measurements of time and space are absolute, Special Relativity shows that they are relative and dependent on the observer's state of motion. This relativity becomes significant only at speeds close to the speed of light, like in the case of the high-speed electron described in the exercise.

Lorentz Transformation

The Lorentz Transformation is a mathematical framework that relates the coordinates of an event as observed in one inertial frame to those in another. It enables us to calculate the dilations in time and contractions in space that occur due to motion close to the speed of light.
For an observer moving at high speeds, distances and time intervals measured in their frame will differ from those measured in a stationary frame. The relevant equations for the Lorentz Transformation include:

• Time dilation: \( t' = \frac{t}{\sqrt{1-v^2/c^2}} \ \)
• Length contraction: \( L = L_0 \sqrt{1-v^2/c^2} \ \)

These transformations explain phenomena such as why the length of the tube is contracted to 0.0404266 m from its original 2.86 m, as calculated when the observer moves at the speed of the electron. It illustrates the effects of high velocity on measurements of length and time.

Relativistic Effects

Relativistic Effects describe the behaviors and properties of objects moving at speeds comparable to the speed of light. These effects become noticeable and important in high-energy physics, such as particle accelerators, and are summed up as time dilation, length contraction, and mass-energy equivalence.

Length contraction is a key relativistic effect, as highlighted in the exercise. It refers to the shortening of an object's length as measured by an observer moving relative to the object. This contraction is not due to any physical compression but is a manifestation of the space-time structure at high velocities.
Similarly, time dilation indicates that time runs slower for a moving system as perceived by a stationary observer. These effects are counterintuitive but have been repeatedly validated by experiments and are crucial for the operation of technologies like GPS systems.
Understanding these consequences allows scientists to perform calculations for systems at relativistic speeds, such as our electron moving along the evacuated tube. By acknowledging relativistic effects, we more accurately interpret experimental data and validate Einstein's theory in high-speed contexts.