Problem Statement: With what velocity should an alpha particle travel towards the nucleus of a Cu atom so as to arrive at a distance of 10^-13 m?

Solution:

To solve the problem, we can use the principle of conservation of energy. The initial energy of the alpha particle is given by its kinetic energy, which can be calculated using the formula:

Kinetic Energy = (1/2)mv^2

where m is the mass of the alpha particle, and v is its velocity. The final energy of the alpha particle is given by the potential energy at a distance of 10^-13 m from the nucleus, which can be calculated using the formula:

Potential Energy = (kQq)/r

where k is the Coulomb constant, Q is the charge of the nucleus, q is the charge of the alpha particle, and r is the distance between the nucleus and the alpha particle.

Step 1: Calculate the potential energy of the alpha particle at a distance of 10^-13 m from the nucleus of a Cu atom.

The charge of the nucleus of a Cu atom is +29e, where e is the elementary charge. The charge of an alpha particle is +2e. The distance between the nucleus and the alpha particle is 10^-13 m. Therefore, the potential energy of the alpha particle is:

Potential Energy = (kQq)/r
= (9x10^9 Nm^2/C^2) x (+29e) x (+2e) / (10^-13 m)
= 1.74x10^-12 J

Step 2: Calculate the kinetic energy of the alpha particle at a distance of infinity from the nucleus.

When the alpha particle is at infinity from the nucleus, its potential energy is zero. Therefore, its total energy is equal to its kinetic energy, which can be calculated using the formula:

Total Energy = Kinetic Energy = (1/2)mv^2

where m is the mass of the alpha particle, and v is its velocity. The mass of an alpha particle is 6.64x10^-27 kg.

Total Energy = Kinetic Energy
= (1/2)mv^2
= (1/2) x 6.64x10^-27 kg x v^2
= 3.32x10^-27 v^2

Step 3: Use the principle of conservation of energy to calculate the velocity of the alpha particle.

According to the principle of conservation of energy, the initial energy of the alpha particle (its kinetic energy) is equal to its total energy (the sum of its kinetic and potential energies) when it is at a distance of 10^-13 m from the nucleus. Therefore, we can equate the two energies and solve for v:

Kinetic Energy = Total Energy - Potential Energy

(1/2)mv^2 = 3.32x10^-27 v^2 - 1.74x10^-12

v^2 = 5.24x10^15

v = 2.29x10^8 m/s

Conclusion: The velocity of the alpha particle should be 2.29x10^8 m/s to arrive at a distance of 10^-13 m from the nucleus of a Cu