Question

Suppose a cyclotron is operated at an oscillator frequency of 10.0 MHz and has a dee radius 42.9 cm. Estimate the total path length traveled by a deuteron in the cyclotron during the (entire) acceleration process. Assume that the accelerating potential between the dees is 94.9 kV. The deuteron mass is m = 3.34x10^-27 kg.

Answer 1

first find magnetic field B , using f = qB/2pim

B = 2pimf/q

B = 2*3.14* 3.34*10^-27 * 10*10^6/(2*1.6*10^-19)

B = 0.655T

-----------------------------------------------------------

apply centripetal force = magnetic force

i.e mv^2/r = qvB,

where m = mass o the charged particle

v = velocity,   r = radius ,

q = charge = 1.6*10^-19 C

B = magnetic field

so now , r = mv/qB

-----------------------------------------------------------------------------------

also KE = PE

0.5 mv^2 = eV

or v = sqrt(2eV/m)

v = sqrt(2* 2*1.6*10^-19 * 95900/3.34*10^-27)

V = 4.28 *10^6 m/s

-------------------------------------------------------------------------------------------

so r = mv/qB

r = 3.34*10^-27 * 4.28 *10^6/(1.6*10^-19 * 0.655T)

r = 0.1364 m

--------------------------------------------------------------------------------

finally

total path length = 2piR/T

= 2* 3.14 * 0.1364*10*10^6

= 7.336 *10^6 m

-----------------------------------------------------------