Acertain 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 112 kv and then pass into a uniform magnetic field, where they are bent in a path of radius 0.904 m. after traveling through 180° and passing through a slit of width 0.846 mm and height 1.14 cm, they are collected in a cup. (a) what is the magnitude of the (perpendicular) magnetic field in the separator? if the machine is used to separate out 1.41 mg of material per hour, calculate (b) the current of the desired ions in the machine and (c) the thermal energy produced in the cup in 0.796 h.

(a) 0.579 T

When the ions are accelerated through the potential difference, the kinetic energy gained is equal to the change in electric potential energy:

where m is the mass of the ions, v their final speed, q their charge, and the potential difference.

We can re-write the equation as

(1)

When the ions enter the region with magnetic field, the magnetic force acts as centripetal force. So we can write

(2)

where B is the magnitude of the magnetic field and r is the radius of the path. Combining (1) with (2), we find an expression for the magnetic field:

And using:

m = 3.92 × 10-25 kg

q = 3.20 × 10-19 C

r = 0.904 m

We find:

(b)

The total mass separated per hour is

Which corresponds to a number of ions of

So the total charge of these ions is

This is the total charge passing through the slit in one hour, which corresponds to a time of

t = 1 h = 3600 s

Therefore, the current is

(c)

First of all, we need to calculate the number of ions collected in the cup in 0.796 h.

We already know that the number of ions collected in 1 hour is (part b)

, therefore

The kinetic energy carried by each ion is (part a)

Therefore, the thermal energy produced in the cup in 0.796 h will be equal to the energy carried by a single ion times the number of ions:

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