Question

# You have landed a great summer job in the medical school assisting in a research group...

You have landed a great summer job in the medical school assisting in a research group investigating short-lived radioactive isotopes which might be useful in fighting cancer. Your group is working on a way of transporting α-particles (Helium nuclei) from where they are made to another room where they will collide with other material to form the isotopes. Since the radioactive isotopes are not expected to live very long, it is important to know precisely how much time it will take to transport the α-particles. Your job is to design that part of the transport system which will deflect the beam of α-particles (mα = 6.64 × 10−27 kg, q = +2e = +3.2 × 10−19 C) through an angle of 90° by using a magnetic field. The alpha-particle beam will be traveling horizontally in an evacuated tube. At the place the tube is to make a 90° turn you decide to put a dipole magnet which provides a uniform vertical magnetic field of 0.030 T. Your design has a tube of the appropriate shape between the poles of the magnet. Before you submit your design for consideration, you must determine how long the α-particles will spend in the uniform magnetic field in order to make the 90°-turn. In the following, each question should help you to answer the next one.

1. In order to make the beam, initially heading north, bend to end up heading west, in which direction must the uniform magnetic field point?

2. Because the magnetic force cannot do work on moving charged particles, the α-particles will follow a circular path at a constant speed. Set the centripetal force equal to the magnetic force and solve for the radius of the path in terms of mass, speed, charge, and magnetic field.

3. Because of the uniform magnetic field, the α-particle beam will travel along one quarter of a circular path of radius r. How much distance d does the beam then cover while in the magnetic field (in terms of r)? 24

4. The time t that the particles take to go through the 90° bend is what we need to know. What is the relationship between the time t, the speed v, and the distance d?

5. We do not know the speed at which the particles enter our beam deflector. Combine answers from questions 2, 3, and 4, what relationship can you find for the time it takes to travel through the bend?

6. Using the values provided for the mass and charge of an α-particle, as well as the strength of your magnetic field, how much time does a particle take to be deflected by 90°?

given
m = 6.64*10^-27 kg
q = 3.2*10^-19 C
B = 0.030 T

1) direction of B: downward

because, F = q*(v cross B)

2)

F_centrieptal = q*v*B*sin(theta)

m*a = q*v*B*sin(90)

m*v^2/r = q*v*B

==> r = m*v/(B*q)

3) distance tracelled, d = 2*pi*r/4

4) time taken to travel d distance, t = d/v

5) t = (2*pi*r/4)/v

= (2*pi/(4*v))*r

= (2*pi/(4*v))*(m*v/(B*q))

= pi*m/(2*B*q)

6) t = pi*6.64*10^-27/(2*0.03*3.2*10^-19)

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