Problem 8 - Fundamentals of Physics

Question

An clectron of \(\beta=0.999987\) moves along the axis of an cvacuated tube that has a length of \(3.00 \mathrm{~m}\) as measured by a laboratory observer \(S\) at rest relative to the tube. An observer \(S^{\prime \prime}\) who is at rest relative to the electron, however, would see this tube moving with speed \(v(=\beta c) .\) What length would observer \(S^{\prime}\) measure for the tube?

Step-by-Step Solution

Answer

The observed length of the tube is approximately 0.0147 meters.

Step 1: Identify the Known Values

We are given that the velocity of the electron is expressed in terms of the speed of light as \( \beta = 0.999987 \). The rest length of the tube as measured by observer \( S \) is \( 3.00 \ \mathrm{m} \). We want to find the length of the tube as measured by an observer \( S' \) moving with the electron.

Step 2: Understand Lorentz Contraction

When an object moves at a high velocity relative to an observer, its length in the direction of motion appears contracted. This effect is described by the Lorentz contraction formula: \( L' = L \sqrt{1-\beta^2} \), where \( L \) is the rest length and \( L' \) is the length as measured by the moving observer.

Step 3: Calculate Lorentz Factor

First, we need to calculate the Lorentz factor, \( \gamma \), which is given by \( \gamma = \frac{1}{\sqrt{1 - \beta^2}} \). Substituting \( \beta = 0.999987 \), calculate \( \gamma \).

Step 4: Simplify the Lorentz Factor Calculation

Calculate \( 1 - \beta^2 = 1 - (0.999987)^2 \). This simplifies to a very small value as \( \beta \) is very close to 1.

Step 5: Compute the Length in Observer's Frame

Substitute the values into the Lorentz contraction formula. We know \( \beta = 0.999987 \) and the rest length \( L = 3.00 \ \mathrm{m} \). Compute \( L' = 3.00 \ \mathrm{m} \times \sqrt{1-(0.999987)^2} \).

Step 6: Final Calculation and Result

After computing, \( \sqrt{1-(0.999987)^2} \approx 0.0049 \). Hence, \( L' = 3.00 \ \mathrm{m} \times 0.0049 \approx 0.0147 \ \mathrm{m} \). Therefore, the observed length of the tube by observer \( S' \) is approximately \( 0.0147 \ \mathrm{m} \).

Key Concepts

Special Relativity

Special relativity is a fundamental theory in physics that was developed by Albert Einstein in 1905. It revolutionized our understanding of space and time. The central idea of special relativity is that the laws of physics are the same for all observers, regardless of their constant relative speed.

One of the key postulates of special relativity is the constancy of the speed of light. No matter how fast an observer is moving, they will always measure the speed of light in a vacuum to be the same value, approximately 299,792 kilometers per second. This striking idea challenged the classical notions of absolute space and time, leading to new concepts like time dilation and length contraction.

These effects become pronounced as objects approach the speed of light. At such high speeds, time appears to slow down for the moving object relative to a stationary observer, and lengths appear contracted. Special relativity also paved the way for the famous energy-matter equivalence formula: \( E = mc^2 \), indicating how mass can be converted to energy and vice versa.

Relativistic Physics

Relativistic physics is the study of physical phenomena when significant speeds near the speed of light are involved. It extends the principles introduced by special relativity and lays the groundwork for understanding the behavior of objects under such conditions.

Relativistic physics features crucial concepts such as:

• Time Dilation: Moving clocks run slower compared to stationary ones, as observed by someone not moving with the clock.
• Mass-Energy Equivalence: Energy and mass are interchangeable, leading to applications in nuclear energy and particle physics.
• Length Contraction: Objects in motion are measured to be shorter in the direction of motion, as seen by a stationary observer.

These principles are pivotal in modern technology and scientific research, including GPS systems and our understanding of the universe. Understanding relativistic physics enables scientists to explore extreme environments like black holes and conditions in high-energy particle accelerators.

Length Contraction

Length contraction is one of the intriguing phenomena predicted by the theory of special relativity. It describes how an object's length appears shorter to an observer in a different inertial frame where the object is moving rapidly relative to the observer.

Mathematically, length contraction is expressed by the equation:
\[ L' = L \sqrt{1 - \beta^2} \]Where:

• \( L' \) is the contracted length observed when the object is moving.
• \( L \) is the proper length of the object at rest.
• \( \beta \) is the ratio of the object's velocity \( v \) to the speed of light \( c \), i.e., \( \beta = \frac{v}{c} \).

Length contraction is only significant at speeds close to the speed of light. In everyday life, this effect goes unnoticed because the speeds are much lower than the speed of light. However, in the realm of high-energy physics, such as in particle accelerators, length contraction is an essential consideration.

Through the example given, an electron moving at nearly the speed of light significantly contracts the observed length of the tube from about 3.00 meters to approximately 0.0147 meters, showcasing the dramatic effect of special relativity.