In Rutherford's scattering experiments, alpha particles (charge = +2e) were fired at a gold foil. Consider an alpha particle, very far from the gold foil, with an initial kinetic energy of 3.3 MeV heading directly for a gold atom (charge +79e). The alpha particle will come to rest when all its initial kinetic energy has been converted to electrical potential energy. Find the distance of closest approach between the alpha particle and the gold nucleus.
Set
M V^2/2 = Z*2*e^2/d
where M is the mass of the alpha particle and Z is the atomic number of gold (79). Solve for the minimum separation, d.
V is the velocity associated with the 3.3 MeV energy of the alpha particle.
(Or just convert 3.3 MeV to Joules for the left side)
Thank you, but in the above equation what does the e in e^2 equal?
To find the distance of closest approach between the alpha particle and the gold nucleus, we can use the conservation of energy principle, which states that the initial kinetic energy of the alpha particle is equal to the final electrical potential energy when it comes to rest.
The initial kinetic energy of the alpha particle is given as 3.3 MeV. To convert it into joules, we can use the conversion factor: 1 MeV = 1.6 x 10^-19 J. Therefore, the initial kinetic energy is (3.3 x 1.6 x 10^-19) J.
When the alpha particle comes to rest, all of its initial kinetic energy is converted to electrical potential energy due to the repulsion between the positive charges of the alpha particle and the gold nucleus.
The electrical potential energy between the alpha particle and the gold nucleus can be calculated using the equation:
Electrical Potential Energy = (k x q1 x q2) / r
Where k is the electrostatic constant (k = 8.99 x 10^9 N.m^2/C^2), q1 and q2 are the charges of the particles (2e and 79e respectively), and r is the distance between the particles.
Since the alpha particle comes to rest, the final electrical potential energy becomes equal to its initial kinetic energy. Therefore:
(k x q1 x q2) / r = (3.3 x 1.6 x 10^-19) J
Simplifying the equation and solving for r gives us:
r = (k x q1 x q2) / (3.3 x 1.6 x 10^-19)
Plugging in the values for k, q1, and q2 will give us the distance of closest approach between the alpha particle and the gold nucleus.