Problem 8 - Fundamentals of Physics

Question

An electron of \(\beta=0.999987\) moves along the axis of an evacuated 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 \(S^{\prime}\) 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 contracted length measured by observer \( S^{\prime} \) is approximately \( 0.015297 \mathrm{~m} \).

Step 1: Understand the Lorentz Contraction

When an object is moving at relativistic speeds (close to the speed of light), its length in the direction of motion appears contracted for an observer at rest compared to when the object is at rest. The contracted length can be calculated using the Lorentz contraction formula.

Step 2: Write Down the Lorentz Contraction Formula

The formula for Lorentz contraction is: \( L' = L \sqrt{1 - \beta^2} \), where \( L' \) is the length measured by the observer in motion relative to the object, \( L \) is the original length as measured by the stationary observer, and \( \beta \) is the velocity of the moving observer as a fraction of the speed of light.

Step 3: Substitute the Given Values into the Formula

We are given \( L = 3.00 \mathrm{~m} \) and \( \beta = 0.999987 \). Substitute these values into the formula: \( L' = 3.00 \sqrt{1 - (0.999987)^2} \).

Step 4: Calculate \( \beta^2 \)

First calculate \( \beta^2 \): \( \beta^2 = 0.999987^2 = 0.999974000169 \).

Step 5: Calculate \( 1 - \beta^2 \)

Subtract \( \beta^2 \) from 1 to get: \( 1 - \beta^2 = 1 - 0.999974000169 = 0.000025999831 \).

Step 6: Calculate the Square Root

Find the square root of \( 1 - \beta^2 \): \( \sqrt{0.000025999831} \approx 0.005099 \).

Step 7: Calculate the Contracted Length

Multiply the original length by the square root to get the contracted length: \( L' = 3.00 \times 0.005099 \approx 0.015297 \mathrm{~m} \).

Step 8: Final Answer Verification

Double-check the calculations to ensure the accuracy of the contracted length. The calculations confirm that the contracted length is \( 0.015297 \) meters.

Key Concepts

Relativistic Speeds

When talking about relativistic speeds, we refer to velocities that are close to the speed of light in a vacuum, approximately \(3 imes 10^8\) meters per second. At these speeds, the effects predicted by Einstein's theory of relativity become significant.
These effects aren't noticeable at everyday speeds but become prominent when dealing with particles moving extremely fast, such as electrons.

• Relativistic speeds affect how we perceive time, length, and mass.
• For something like an electron moving at \(\beta = 0.999987\), where \(\beta\) is the velocity as a fraction of the speed of light, these changes are measurable.

Understanding these changes helps in studying high-speed particles like electrons using particle accelerators. This knowledge is essential to modern physics, illustrating how our universe operates at speeds incomprehensible in daily experiences.

Length Contraction

Length contraction is an intriguing concept in physics where an object in motion appears shorter along the direction of movement to an observer at rest relative to the object. This principle emerges from Einstein's theory of special relativity.The formula for calculating the contracted length \(L'\) is:

• \(L' = L \sqrt{1 - \beta^2}\)
• \(L'\) is the contracted length.
• \(L\) is the original length as viewed by a stationary observer.
• \(\beta\) represents the velocity of the object as a fraction of the speed of light.

In the given scenario, even though the tube's original length is 3 meters, a different length is observed by someone moving with the electron at such a high speed. This contraction effect confirms the prediction of special relativity that moving objects undergo measurable changes in dimension from the perspective of an observer at rest.

Special Relativity

Special relativity, formulated by Albert Einstein, is a cornerstone of modern physics. It describes the physical phenomena in the absence of gravitational fields. There are two key postulates:

• The laws of physics are identical for all observers in uniform motion relative to one another (principle of relativity).
• The speed of light in a vacuum is constant and will be the same for all observers, regardless of their motion relative to the light source.

Special relativity leads to effects like time dilation and length contraction. These concepts revolutionize our understanding of space and time, transforming how we grasp events at high velocities.
These principles come to life when examining high-speed particles, like electrons, which highlight the effects of length contraction as they move at relativistic speeds. Through calculations such as those in the Lorentz contraction example, special relativity provides precise predictions applicable in real-world high-velocity contexts such as in particle accelerators.

Observer Frames

In physics, the term "observer frames" refers to different points of reference from which measurements are made. When dealing with relativistic speeds, the perspective or frame of the observer greatly influences how events or objects are measured.

• An observer at rest relative to an object compares differently to one moving relative to the same object.
• This relativity of motion is essential for interpreting phenomena like length contraction.

In the situation with the electron and the laboratory tube, observer frame differences are critical.
The laboratory observer sees the electron tube at full length (3 meters), while the moving observer (with the electron) measures it contracted.
Observer frames emphasize the relativity and context-dependence in physical laws, showing that our perception of events and distances hinges on our state of motion relative to what we're observing.