Exercise 27.20
Cyclotrons are widely used in nuclear medicine for producing
short-lived radioactive isotopes. These cyclotrons typically
accelerate H− (the hydride ion, which has one proton and
two electrons) to an energy of 5MeV to 20MeV. This ion has a mass
very close to that of a proton because the electron mass is
negligible−about 1/2000 of the protons mass. A typical magnetic
field in such cyclotrons is 1.7 T .
Part A
What is the speed of a 5.0-MeV H−?
Express your answer with the appropriate units.
Part B
If the H− has energy 5.0MeV and B= 1.7 T , what is the
radius of this ions circular orbit?
Express your answer with the appropriate units.
Answer
General guidance
Initially, the velocity can be calculated by using the formula from the energy conservation. Later the forces in a magnetic field for a circular motion have to equate to calculate the expression for the radius of the circular path. Finally, the radius of the circular path can be calculated by substituting the given numerical values in the problem.
From energy conservation, the velocity of the Hydrogen is,
Here, 
is the energy of the hydrogen, 
is the speed of the hydrogen, and 
is the mass of proton.
At equilibrium, the forces in a magnetic field is,
Here, 
is the magnetic field, 
is the charge of proton and 
is the radius of the circular orbit.
Step-by-step
Step 1 of 3
(A)
The expression for the speed is,
Here, is the energy of the hydrogen, is the speed of the hydrogen, and is the mass of proton.
The expression for the speed of the hydrogen atom is related to the square root of twice of the energy of the hydrogen and then divided by the mass of the proton.
. The correct expression for calculating the speed of the hydrogen atom is Step 2 of 3
Substitute 
for
, 
for
.
The speed of the hydrogen is
.
The speed of the hydrogen in a magnetic field is calculated by using the expression for energy.
to calculate the speed of the hydrogen. It is incorrect because the correct relation for the speed is
.Step 3 of 3
(B)
For a charged object to move in a circular path under uniform magnetic field, the magnetic force and centripetal force acting on the charged particle must be balanced.
The expression for the radius of the circular path is,
Substitute 
for
, 
for
, 
for 
and 
for
The radius of the circular orbit is
The radius of the circular orbit is calculated by equating the forces acting on the hydride ion moving in a magnetic field.
to calculate the radius the circular orbit. It is incorrect because the correct relation for the radius of the orbit is
.Answer
The speed of the hydrogen is.
The radius of the circular orbit is
Answer only
The speed of the hydrogen is.
The radius of the circular orbit is