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.

### 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.

### 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.

### 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