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In the extreme relativistic regime, velocities of particles, such as electrons, approach the speed of light. This significant speed alters the normal relationships between energy and momentum. The extreme relativistic relation is summarized by the equation:

• \( p = \frac{E}{c} \)
• where \( p \) is the momentum, \( E \) is the energy, and \( c \) is the speed of light.

This formula is particularly useful for high-energy particles. In these scenarios, the particle's kinetic energy is much greater than its rest energy. This condition allows us to simplify the calculations by relying on the extreme relativistic relation.
In the given problem, the electron has a huge energy of 50 GeV, sufficient for using this relation. This energy is much higher than the electron's rest energy, which is approximately 0.511 MeV.
When energy is this significant, traditional equations such as \( E = mc^2 \) are less effective, as they do not consider the kinetic contributions effectively. Thus, the extreme relativistic formula helps in computing values like momentum, crucial for understanding dynamics at near-light velocities.

Momentum is a fundamental concept when studying particles in motion, especially at high energies. For electrons, momentum is key in determining their behavior and interactions.
In this context, we utilize the extreme relativistic formula to determine electron momentum:

• Given the energy \( E = 50 \, \text{GeV} \), converted into joules to facilitate calculations \( E = 8.0 \times 10^{-9} \, \text{J} \).
• The speed of light \( c = 3.0 \times 10^8 \, \text{m/s} \) is a constant crucial for obtaining momentum.
• The momentum is calculated as \( p = \frac{E}{c} = 2.67 \times 10^{-17} \, \text{kg m/s} \).

Calculating the electron momentum requires energy conversion to joules, enabling the proper use of physical constants.
The result gives a practical understanding of how the electron interacts under extreme conditions at a subatomic level.
Furthermore, mastering momentum in relativistic contexts supports exploring phenomena like scattering, which reveals internal structures of particles, such as atomic nuclei.

The nuclear radius is an important measurement in understanding the structure and size of atomic nuclei. It gives insight into the space within which the nuclear particles, often protons and neutrons, exist.
For the given problem:

• The nuclear radius is specified as \( R = 5.0 \, \text{fm} \), where 1 femtometer (fm) equals \( 10^{-15} \, \text{m} \).
• This converts to \( R = 5.0 \times 10^{-15} \, \text{m} \).

This tiny measurement reflects the compact nature of atomic nuclei.
The de Broglie wavelength \( \lambda \) of an electron helps in probing the nuclear structure by scattering, effectively acting as a measuring "tape."
The computed ratio \( \frac{R}{\lambda} \) provides a comparison of the nucleus size to the wavelength. Here, \( \frac{R}{\lambda} \approx 201.6 \), denotes that the electron鈥檚 wavelength is much shorter compared to the nuclear radius.
This short wavelength is essential for high-resolution investigations, as it means the electron can detail the nucleus with precision during interactions.