Final answer:
The speed of the alpha particle is 2.6x10^6 m/s. The period of revolution is 0.109 microseconds. The kinetic energy is 140 MeV and the potential difference required is 70.0 kV.
Step-by-step explanation:
To calculate the speed of the alpha particle, we can use the formula for the centripetal force exerted on a charged particle in a magnetic field. In this case, the centripetal force is provided by the magnetic force. We have the formula F = qvB, where F is the force, q is the charge, v is the velocity, and B is the magnetic field strength.
To find the speed of the alpha particle, we need to rearrange the formula to solve for v. We have qvB = mv^2/r, where m is the mass of the alpha particle and r is the radius of the circular path.
(a) v = (qBr/m)^0.5
Substituting the given values q = +2e, B = 1.20 T, r = 4.50 cm = 0.045 m, and m = 4.00 u = 4.00 x 1.661x10^-27 kg:
v = [(2e)(1.20 T)(0.045 m)]^0.5 / (4.00 x 1.661x10^-27 kg) = 2.6x10^6 m/s
Therefore, the speed of the alpha particle is 2.6x10^6 m/s.
(b) The period of revolution of the alpha particle can be calculated using the formula T = 2πr/v, where T is the period, r is the radius, and v is the velocity.
T = 2π(0.045 m) / (2.6x10^6 m/s) = 0.109 micro s
Therefore, the period of revolution of the alpha particle is 0.109 micro s.
(c) The kinetic energy of the alpha particle can be calculated using the formula KE = 0.5mv^2, where KE is the kinetic energy, m is the mass, and v is the velocity.
KE = 0.5(4.00 x 1.661x10^-27 kg) (2.6x10^6 m/s)^2 = 140 MeV
Therefore, the kinetic energy of the alpha particle is 140 MeV.
(d) The potential difference needed to accelerate the alpha particle can be calculated using the formula KE = qV, where KE is the kinetic energy, q is the charge, and V is the potential difference.
V = KE/q = (140 MeV) / (2e) = 70.0 kV
Therefore, the potential difference through which the alpha particle would have to be accelerated to achieve this energy is 70.0 kV.