Understanding Distance of Closest Approach
The distance of closest approach is a concept in nuclear physics that describes how close a charged particle, such as a proton, can get to a nucleus before being repelled by the Coulomb force.
Key Concepts
- Energy of the Proton
The energy of the proton is given as 5.0 MeV. This energy determines how much potential energy the proton has as it approaches the nucleus.
- Coulomb's Law
The repulsive force between the proton and the positively charged gold nucleus can be described using Coulomb's Law, which states that the force between two charges decreases with the square of the distance between them.
Calculating Distance of Closest Approach
- Total Energy and Potential Energy
As the proton approaches the nucleus, its kinetic energy is converted into potential energy due to the electrostatic repulsion. At the distance of closest approach, all kinetic energy is converted into potential energy.
- Formula
The potential energy (U) at the distance of closest approach (r) can be expressed as:
U = k * (Z₁ * Z₂ * e²) / r
Where:
- k is Coulomb's constant
- Z₁ and Z₂ are the atomic numbers of the proton and gold nucleus
- e is the elementary charge
- Setting Energies Equal
At the closest approach, the kinetic energy (KE) of the proton is equal to the potential energy (U):
KE = U
Thus, you can rearrange the equation to solve for r.
Conclusion
The distance of closest approach for a 5.0 MeV proton to a gold nucleus can be calculated using the principles of energy conservation and Coulomb's law. The resulting value will indicate the minimum distance that the proton can reach before the repulsive force from the nucleus prevents any closer approach.