Halliday, Resnick, Walker 9E problem 27 Ch. 28 principles of physics A mass spec

Question

Halliday, Resnick, Walker 9E problem 27 Ch. 28
principles of physics

A mass spectrometer is used to separate  ions of mass 3.92*10^-25 kg. and charge 3.2*10^-19C. Ions are accelerated through a potential difference of 100kV. Then pass into a uniform magnetic field where they are bent in a path of radius 1 m. after traveling 180 degrees and passing through a slit width 1mm and h 1cm they are collected in a cup.

A) what is the magnitude of perpendicular magneticc field in separator? if the machine is used to separate out 100 mg of material per hour calculate B) current of desired ions in the machine, C) thermal energy produced in the cup in 1hr

Explanation / Answer

Part A)

To find the magnitude of the magnetic field, we use the formula for the radius of a charged particle in a magnetic field.

r = mv/qB

The velocity is found by conservation of energy. The ions are given an energy when accelerated through the voltage. The energy is found by qV. That must equal the KE given to the ions, so

qV = .5mv2

v = (2qV/m)

v = [(2)(3.2 X 10-19)(1 X 105)/(3.92 X 10-25)]

v = 1.56 X 103 m/s

Now we can find B

B = mv/qr

B = (3.92 X 10-25)(1.56 X 103)/(3.2 X 10-19)(1)

B = 1.92 X 10-3 T

Part B)

The current is found by charge over time. We are given 100 mg per hour. Converting that to kg/s we have

2.78 X 10-8 kg/s

Since each ion has a mass of 3.92 X 10-25 kg, we get (2.78 X 10-8)/(3.92 X 10-25) = 7.09 X 1016 Ions/sec

Since each ion has a charge of 3.2 X 10-19 C, we finally get

(7.09 X 1016)(3.2 X 10-19) = 2.27 X 10-2 C/s

Since charge over time is current, that is our current. 2.27 X 10-2 A or you can say 22.7 mA

Part C)

If all of the kinetic energy of the particles is converted into thermal energy, we can find the total KE from all of the charges in one hour.

Total mass in one hour is 1 X 10-4 kg. We already have the velocity

KE = .5mv2 = (.5)(1 X 10-4)(1.56 X 103)2

KE = 122 J

So thermal energy is 122 J

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