Question

At the 10.0 -km-long Stanford Linear Accelerator, electrons with rest energy of 0.511 MeV have been accelerated to a total energy of \(46 \mathrm{GeV}\). How long is the accelerator as measured in the reference frame of the electrons?

Short answer

The accelerator length in the electrons' frame is approximately 0.111 m.

Step-by-step solution

Understand the Concept of Length Contraction

In the reference frame of an object moving at a high velocity, distances along the direction of motion appear contracted. This phenomenon is described by the theory of special relativity. We can express it using the length contraction formula.

Write the Length Contraction Formula

The length contraction formula is given by: \[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]where \(L\) is the length in the moving frame (electrons' frame), \(L_0\) is the proper length (in the lab frame), \(v\) is the velocity of the object, and \(c\) is the speed of light.

Express Total Energy in Terms of Rest and Kinetic Energy

The total energy \(E\) of an electron can be expressed as:\[ E = \gamma m_0 c^2 \]where \(\gamma\) is the Lorentz factor, \(m_0\) is the rest mass energy, and \(c\) is the speed of light. Given the rest energy \(0.511 \text{ MeV}\) and total energy \(46 \text{ GeV}\), \(\gamma\) can be determined.

Calculate the Lorentz Factor (\(\gamma\))

The Lorentz factor \(\gamma\) is defined as:\[ \gamma = \frac{E}{m_0 c^2} \]Convert all energies to the same units (1 GeV = 1000 MeV):\(\gamma = \frac{46,000 \text{ MeV}}{0.511 \text{ MeV}} \approx 89921.72 \)

Find the Length in the Electrons' Reference Frame

Since \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\), we can derive:\[ L = \frac{L_0}{\gamma} \]where \(L_0 = 10 \text{ km}\).Calculate \(L\):\[ L = \frac{10,000 \text{ m}}{89921.72} \approx 0.111 \text{ m} \]

Key concepts

Length Contraction

Length contraction is a fascinating phenomenon that emerges from Einstein's theory of special relativity. It tells us that when an object moves close to the speed of light, its length in the direction of motion will appear shorter. This contracted length is what we observe in the reference frame that the object is moving relative to. This was exactly what happened in the exercise involving the Stanford Linear Accelerator.

In simpler terms, if there's a highly speedy object, like an electron traveling through the accelerator, its perceived length shrinks. Why does this happen? It's because distances along the direction of motion seem compressed when you're observing them from an outside stationary frame of reference. This special relativity effect was vividly noted in the exercise problem, using the length contraction formula to compare the accelerator's length in different frames.

Lorentz Factor

The Lorentz Factor () is a crucial part of understanding special relativity. It's a number that helps us understand how time, length, and relativistic mass change for an object moving at a significant fraction of the speed of light. In equations, it often emerges when calculating effects like time dilation and length contraction.

For the Stanford Linear Accelerator problem, calculating  was essential. We used the formula: dacEcm_0c^2e, where total energy (E) and rest mass energy (m_0) help show how the energy and movement interplay. The Lorentz factor we found was quite large, implying significant relativistic effects. By using this  value, we extrapolated information about the electron's perspective, including how short the accelerator appears.

Rest Energy

When discussing special relativity, rest energy is a fundamental concept. It refers to the amount of energy that an object possesses due to its mass, even when it is not moving. The famous equation: \[ E = mc^2 \] expresses this intrinsic energy.

For the electron in the Stanford Linear Accelerator, the rest energy is given as 0.511 MeV. This represents the unchanging energy tied to the electron's mass, regardless of its speed. Understanding rest energy is crucial as it serves as a baseline when considering how much more energy is added to an object as it moves faster and gains kinetic energy.

Total Energy

The concept of total energy in special relativity combines rest energy with the additional energy an object gets as it moves. Total energy (f) is computed as:\[ E =  m_0 c^2 \] where  stands for the Lorentz factor. In our exercise, electrons reached a total energy of 46 GeV within the accelerator.

This contrasts with their rest energy of 0.511 MeV. The gap between these values highlights how much energy is used to accelerate the electrons to significant fractions of light speed. Recognizing the connection between rest energy and the total energy gives insights into relativistic motion's complexity and the substantial energy requirements of modern accelerators.

Stanford Linear Accelerator

The Stanford Linear Accelerator, often called SLAC, is a monumental piece of scientific equipment. It's 10 km long and is vital for particle physics research, particularly for accelerating electrons to near-light speeds.

In examining the exercise, the accelerator plays a role in showing how relativity alters our understanding of space. As electrons zoom through this extensive track, we witness relativistic effects like length contraction. From the electrons' point of view, the massive structure appears dramatically shorter than its actual length. In practical terms, SLAC helps scientists explore the boundaries of particle physics, testing theories like those in special relativity and helping advance our understanding of the universe at the smallest scales.