A certain commercial mass spectrometer is used to separate uranium ions of mass

Question

A certain commercial mass spectrometer is used to separate uranium ions of mass 3.92 10-25 kg and charge 3.20 10-19 C from related species. The ions are accelerated through a potential difference of 90 kV and then pass into a uniform magnetic field, where they are bent in a path of radius 0.70 m. After traveling through 180° and passing through a slit of width 1.00 mm and height 1.00 cm, they are collected in a cup.

(a) What is the magnitude of the (perpendicular) magnetic field in the separator?
= _____ T

(b) If the machine is used to separate out 90 mg of material per hour, calculate the current of the desired ions in the machine.
= _____ A

(c) In that case, calculate the thermal energy produced in the cup in 1.00 h.
= _____ J

Explanation / Answer

a) We can used the energy to find velocity and then compute the field

0.5mv^2 = qV =====> v =sqrt[2qV/m] = sqrt[(2*3.2*10^-19*90000)/3.92*10^-25] = 3.83*10^5 m/s

Now B = mv/qr = (3.92*10^-25*3.83*10^5)/(3.2*10^-19*0.70) = 0.67 T

b) We can use the rate of accumulation of mass to find the current

90*10^-6 g = (#ions / sec)*( 3600 sec)(3.92*10^-25 kg/ion)

  (#ions / sec) = 90*10^-6/(3600*3.92*10^-25) = 6.38*10^16 ions/sec

Now we can find the current = i =  (6.38 ×10^16 ions / sec) (3.20 ×1019C /ion) = 0.020 A

c) We can also find the energy since we know the number of ions and the energy carried by each ion

U =qV = 3.2*10^-19*90000 = 2.88*10^-14 J/ion

Energy = (6.38*10^16 ions/sec)*(3600s/hr)*(2.88*10^-14 J/ion) = 6614784 J

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