A mass spectrometer is designed to measure the mass m of the 238U+ ion. A source

Question

A mass spectrometer is designed to measure the mass m of the 238U+ ion. A source of 238U+ sends ions into the device with negligibly small initial kinetic energies. The ions pass between parallel accelerating plates and then through a velocity selector designed to allow only ions moving at speed v to pass straight through. The ions that emerge from the velocity selector move in a semicircle of diameter D in uniform magnetic field of magnitude B, which is the same as the magnetic field the velocity selector. The accelerating plates have area A and are a distance d apart. What should the charges on the plates be so that ions emerge at speed v? (Answer in term of m, v, A, ell0, e, d) Sketch the electric field lines between the plates. What should the magnitude and direction of the electric field in the selector to allow ions with speed v to pass straight through? Sketch the trajectory inside the velocity selector for ions that enter with speeds slightly less than v. Find the mass of the 238U+ ions in term of v, B, D and universal constants. If the velocity selector was removed and 238U+ ion was used what would the diameter of the path been? (Answer in term of D)

Explanation / Answer

a) use energy

1/2 m v^2 = e V

V = Q/C

and C = e0 A/d

V = Q d/(e0 A)

1/2 mv^2 = e Q d/(e0 A)

Q = e0 m A v^2/(2 e d)

b) so to pass right through

F electric = F magnetic

q E = q v B

E = v B

c) they will curve down and hit the plate

d)

so in the magnetic field q v B = mv^2/r

q B = m v/r
here r = D/2

q B = 2 m v/D

m = e D B/(2 v)

e) so

D 235/D 238 = M 235/M238

D235 = 235 D/238

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