Question

singly charged uranium-238 ions are accelerated through a potential difference of 2.00KV and wnter a uniform magnetic field of magnitude 1.20 T directed perpendicular to their velocities. Significance of this problem:

99% of all uranium that comes out of the ground (uranium mine) is U238. But it is U235 that is used for nuclear power plants (3~6% U235) and nuclear weapons (~90% U235). So the concentration of U235 must be increased (enriched). The concentration of U235 is what must be tested by inspectors when they go to a suspected location (in North Korea, or Iran, or …).

a) For U238 which has mass m = 238 amu (atomic mass units), what is the mass?

b) What is the work done to accelerate the U238 ion?

c) What is the speed of the U238 ion?

d) Find the radius and diameter of the path of the ion.

Answer 1

(a) 1 amu = 1.66 10^-27 kg

mass of ²³⁸U = 238 amu = 238 1.66 10^-27 = 3.951 10^-25 kg

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

(b) workdone , q V = 1.60210^-19 200 10³ = 3.204 10^-14 J

where q is charge on ²³⁸U ion and V is accelerating potential

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

(c) speed of ²³⁸U ion = (1/2)mv² = 3.204 10^-14 J

where m is mass of ²³⁸U ion

hence speed of ²³⁸U ion is obtained as,

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

(d) q B v = m v² / r ................(1)

where LHS is force experienced by ²³⁸U ion in magnetic field of flux density B.

RHS is centripetal force and r is radius of circular path

we get from eqn.(1), r = mv/ ( qB )

r = ( 3.951 10^-25 4.027 10⁵ ) / ( 1.602 10^-19 1.2 ) = 0.828 m