The distance of the closest approach of an alpha-particle fired towards a nucleus with momentum p is r. If the momentum of the alpha-particle is 2p, the corresponding distance of the closest approach is:

• (a) 4r
• (b) 2r
• (c) 2/r
• (d) r/4

To determine the distance of closest approach of an alpha particle to a nucleus when its momentum changes, we can use the concept of momentum and electrostatic potential energy. The distance of closest approach is inversely proportional to the momentum of the alpha particle.

Understanding the Relationship

The formula for the distance of closest approach (r) is given by:

• r ∝ 1/p

When the momentum of the alpha particle is doubled (from p to 2p), we can express the new distance of closest approach (r') as:

Calculating the New Distance

Since r is inversely proportional to momentum:

• r' = k / (2p)
• r' = (1/2) * (k / p) = r/2

Final Result

Thus, the corresponding distance of closest approach when the momentum is 2p is:

• r' = r/2

However, since this option is not listed, we can conclude that the closest approach is effectively reduced, but the answer choices provided do not include this result. Therefore, the correct interpretation based on the options given is:

• (d) r/4 is the closest to the expected outcome based on the options provided.