Problem in integrating to find Rutherford's formula

In summary, the challenge in integrating to derive Rutherford's formula lies in the complexity of the mathematical concepts involved, particularly in accurately accounting for the forces and interactions at play in atomic structures. This process requires a deep understanding of classical physics, electrostatics, and the nuances of differential equations, making it a non-trivial task for many.
  • #1
MatinSAR
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Homework Statement
Figure below.
Relevant Equations
Figure below.
Could someone guide me on what change of variable was used to obtain equation 9.138 from equation 9.137?
1714250757096.png

Book : Classical Dynamics of Particles and Systems 5th Edition by Stephen T. Thornton (Author), Jerry B. Marion (Author)

They told us to check equation 8.38 and in that page they had ##1/r^2## in the numerator so they used ##u=1/r## then they get ##du=-dr/r^2##.
But I cannot use that here because I have ##1/r## in the numerator ...
 
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  • #2
If you take ##r## out from the square root in the denominator you get ##1/r^2## in the numerator.
 
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  • #3
Hill said:
If you take ##r## out from the square root in the denominator you get ##1/r^2## in the numerator.
I was oblivious. Thanks for your clever idea! I will retry to solve.
 

FAQ: Problem in integrating to find Rutherford's formula

What is Rutherford's formula and why is it important?

Rutherford's formula describes the scattering of alpha particles by a thin foil of gold, leading to the discovery of the atomic nucleus. It is important because it provided a model for understanding atomic structure and laid the groundwork for modern nuclear physics.

What are the main components needed to integrate for Rutherford's formula?

The main components include the differential cross-section, which describes the scattering probability, and the impact parameter, which is the perpendicular distance from the center of the nucleus to the initial trajectory of the alpha particle. Integrating these components over the appropriate limits allows for the calculation of the total scattering cross-section.

What challenges do scientists face when integrating to find Rutherford's formula?

One challenge is dealing with the mathematical complexity of the integrals involved, particularly when considering different scattering angles and the potential energy between the alpha particles and the nucleus. Additionally, ensuring accurate limits of integration and handling singularities can complicate the process.

How do you handle the singularities that arise in the integration process?

To handle singularities, scientists often employ techniques such as regularization or the introduction of a small cutoff parameter to avoid undefined behavior at certain points. This allows for a more manageable integration process and helps to yield finite results.

What are the practical applications of Rutherford's formula in modern science?

Rutherford's formula has practical applications in fields such as nuclear physics, particle physics, and medical imaging. It helps in understanding nuclear interactions, designing particle accelerators, and improving techniques in radiation therapy for cancer treatment.

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