a) The formula for a cyclotron's frequency:
f = (qB)/(2πm)
f is the frequency, q is the particle's charge, B is the magnetic field strength, and m is the particle's mass.
Given values:
q = 1.6×10⁻¹⁹C (proton charge)
B = 1.30T
m = 1.67×10⁻²⁷kg (proton mass)
Plug in these values and solve for f:
f = (1.6×10⁻¹⁹×1.30)/(2π×1.67×10⁻²⁷)
f = 1.98×10⁷Hz
b) The kinetic energy of a particle in the cyclotron is:
KE = 0.5q²B²r²/m
q is the particle's charge, B is the magnetic field strength, r is the radius of the particle's path, and m is the particle's mass.
Given values:
q = 1.6×10⁻¹⁹C (proton charge)
B = 1.30T
r = 0.670m
m = 1.67×10⁻²⁷kg (proton mass)
Plug in the values and solve for KE:
KE = (0.5)(1.6×10⁻¹⁹)²(1.30)²(0.670)²/1.67×10⁻²⁷
KE = 5.81×10⁻¹²J