Understanding the Distance of Closest Approach
The distance of closest approach (r0) of an alpha particle to a nucleus is influenced by its kinetic energy (E). As the kinetic energy increases, the dynamics of the interaction change.
Concept of Distance of Closest Approach
- The distance of closest approach can be derived from the balance between kinetic energy and potential energy during the interaction between the alpha particle and the nucleus.
- The kinetic energy E is converted to potential energy at the point of closest approach.
Relation Between Kinetic Energy and Distance
- The potential energy between charged particles is inversely proportional to the distance.
- When an alpha particle approaches a positively charged nucleus, the kinetic energy is given by the equation: E = k * (Z1 * Z2) / r, where Z1 and Z2 are the charges of the alpha particle and nucleus, and k is a constant.
Doubling the Kinetic Energy
- If the kinetic energy is doubled (2E), the relationship can be rewritten: 2E = k * (Z1 * Z2) / r.
- Rearranging this gives: r = k * (Z1 * Z2) / 2E.
New Distance of Closest Approach
- From this equation, it is clear that if the kinetic energy is doubled, the distance of closest approach (r) is halved.
- Therefore, if the initial distance of closest approach is r0, the new distance (r') becomes: r' = r0 / 2.
Conclusion
- When the kinetic energy of an alpha particle is doubled, the distance of closest approach is halved.
- This illustrates the inverse relationship between kinetic energy and distance in the context of nuclear interactions.