Building a cyclotron...Magnetism question

[SaintJude]

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A certain urban hospital wants to expand its radiooncology department and will need a source of shortlived radioisotopes. A 6.5 MeV proton cyclotron is to be built, but must fit into an existing 10 x 10 m room in the basement. Assuming that the maximum attainable field is 1 T, can the hospital go ahead with its plans ? (The mass of a proton is 1.67262158 × 10-27 kg & the charge of the proton is 1.6 x 10−19 C.)

To show math work, can we maybe use variables instead of the numbers?

A. Yes, because the magnetic field is smaller than the maximum magnetic field attainable.
B. No, because the cyclotron would be too big for the room in the basement.
C. It depends on the number of shortlived radioisotopes needed.
D. Cannot be determined

 

[MedPR]

My answer? D. 🙂

I have no idea where to even start. Sorry.

 

[SaintJude]

Ah, seriously this question is so annoying. The answer is A. And you've got to do calculations...

 

[chiddler]

A certain urban hospital wants to expand its radiooncology department and will need a source of shortlived radioisotopes. A 6.5 MeV proton cyclotron is to be built, but must fit into an existing 10 x 10 m room in the basement. Assuming that the maximum attainable field is 1 T, can the hospital go ahead with its plans ? (The mass of a proton is 1.67262158 × 10-27 kg & the charge of the proton is 1.6 x 10−19 C.)

To show math work, can we maybe use variables instead of the numbers?

A. Yes, because the magnetic field is smaller than the maximum magnetic field attainable.
B. No, because the cyclotron would be too big for the room in the basement.
C. It depends on the number of shortlived radioisotopes needed.
D. Cannot be determined

ouch. here's my attempt:

we have Voltage, maximum radius of rotation, max B field, mass of e-, and q.

so can this cyclotron produce a rotation that fits?

qV = 1/2mv^2 = (1.6e-19)(6.5e6) = 1/2 mv^2

find v

mv^2/r = qvB
mv/r = qB
r = mv/qB

we find r as we have velocity, mass of proton, and strength of B field.

the calculations are too complicated 😛

in the unlikely event that i'm correct, then r = 0.36 m which fits. but i had to use wolframalpha!

 

[SaintJude]

Hmm, I wonder if that works. EDIT: They say your approach works!! But they choose an alternative approach. They instead try to find the magnetic field that would be produced and then derive that the magnetic field is less than the possible maximum.

 

[SaintJude]

I was talking nonsense--so I erased this. Thanks, chiddler!

How did you know to do "qV" and what does this value represent?

 

[chiddler]

I was talking nonsense--so I erased this. Thanks, chiddler!

How did you know to do "qV" and what does this value represent?

very welcome!

W(ork) = qV = qΔφ, where φ is the electric potential (=kq/r)

 

[SaintJude]

But it's circular motion, so is there work? . Kaplan is suggesting it's "energy"--so are they referring to kinetic energy?

 

[chiddler]

it must gain the velocity to enter uniform circular motion. this velocity is the energy supplied by the voltage. After it enters UCM, then no work is done by the Bfield because the force is always perpendicular to its motion.

you should know: i don't have much confidence in my physics.

edited for clarity.

 

[milski]

it must gain the velocity to enter uniform circular motion. this velocity is the energy supplied by the voltage. After it enters UCM, then no work is done by the Bfield because the force is always perpendicular to its motion.

you should know: i don't have much confidence in my physics.

edited for clarity.

You're doing great!👍

It will be an accelerating circular motion at first - there will be work during that period, that's how it will gain the KE at the end.

 

[procrastinator2]

I was just working on this problem, its in the Kaplan review notes.... Answer is A.
F=qvB
(centripetal acceleration=v^2/r) F= mv^2/r
Cyclotron will orbit = circle

qvB=mv^2/r
Solve for B; B = mv/qr

E= (1/2) mv^2
v=sqrt 2E/m


B=mv/qr ; (v=sqrt 2E/m)
=(m/qr) (sqrt (2E/m))
= (sqrt 2mE) / qr <-- can someone show me the steps of simplifying to get to this?
= (sqrt 2(1.67*10^-27) (1.04*10^-12))/(1.6*10^-19)(5.0)
=0.074T

 

[gettheleadout]

So just to clarify, are we supposed to know what a cyclotron is? I haven't taken physics since AP a couple years ago but my buddy in engineering didn't know and he had just taken physics...

 

[milski]

So just to clarify, are we supposed to know what a cyclotron is? I haven't taken physics since AP a couple years ago but my buddy in engineering didn't know and he had just taken physics...

We did cover it during intro physics - a few of the problems that we had about magnetic field were about cyclotrons. I certainly don't remember the formulas that we derived for its radius. But that was the engineering/physics majors calculus based course - I don't know if the algebra based one discusses it.