One way in which Uranium 235 was separated from Uranium 238 in World War 2 was t
ID: 1383581 • Letter: O
Question
One way in which Uranium 235 was separated from Uranium 238 in World War 2 was to use a mass spectrograph to separate the 2 isotopes. Suppose the uranium atoms are ionized so that they have a net charge of +2 elemental charge units. These ions are then accelerated through a constant electric field of 204 N/C for 53 centimeters in a constant magnetic field. The constant magnetic field is produced by a solenoid with 155 coils per meter wrapped around a cylinder with 6.3 Amps of current running through it. The two ions have different paths through the magnetic field into 2 different containers as in the diagram below. How far apart, d in centimeters, do the centers of the containers have to be to maximize collecting both isotopes?
Explanation / Answer
for m1 = U235
in the electric field
0.5*m1*v1^2 = dV*q = E*L*q1
v1 = sqrt(2*E*L*q1/m1) = sqrt(2*204*0.53*2*1.6e-19)/(235*1.67e-27)
v1 = 1.33e4 m/s
in the magnetic field it describes a circular path of radis r1
B = uo*n*I = 4*3.14e-7*155*6.3 = 1.23e-3 T
Fb = q1*v1*B = m*v1^2/r1
r1 = m1*v1/(q1*B) = (235*1.67e-27*1.33e4)/(2*1.6e-19*1.23e-3 = 13.26 m
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similarly for m2 = U238
for m2 = U238
in the electric field
0.5*m2*v2^2 = dV*q = E*L*q2
v2 = sqrt(2*E*L*q2/m2) = sqrt(2*204*0.53*2*1.6e-19)/(238*1.67e-27) = 1.32e4 m/s
v2 = 1.32e4 m/s
in the magnetic field it describes a circular path of radis r1
B = uo*n*I = 4*3.14e-7*155*6.3 = 1.23e-3 T
Fb = q2*v2*B = m*v1^2/r1
r2 = m2*v2/(q2*B) = (238*1.67e-27*1.32e4)/(2*1.6e-19*1.23e-3) = 13.32 m
d = r2 - r1 = 13.32 - 13.26 = 0.06 m = 6 cm <-answer
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