Hand-In Problem Set 6
(due Friday, 08/07/2015, 5p.m.)
1. A uniform magnetic field, B=−20𝑚𝑇𝑧̂ acts on a -5.0nC charge with mass 1.0mg travelling with
instantaneous velocity v=(-200,200,10)m/s
I should have increased the velocity for this problem!
a) What is the component of the velocity in the direction
parallel to the magnetic field?
b) What is the component of the velocity perpendicular to the
magnetic field?
a) and b) were designed to help you with the rest of the problem
c) Calculate the radius r of the trajectory.
Do you understand the motion, and what part of the velocity
contributes to the magnetic force? (Use only v_perp in this
calculation.)
d) The charge will travel in a spiral (also called a helix). Why? What is the direction of axis of
the spiral? How far along the axis will the charge progress in the time it takes to make one
loop of the spiral? (In other words, what is the pitch, p, of the spiral?)
Velocity is not always perpendicular to the B-field. What does the motion look like if it is
not?
2. A metal bar of mass m and length L is suspended from two
conducting wires, as shown in the figure. A uniform
magnetic field of magnitude B points vertically downward.
Find the angle θ the suspending wires make with the
vertical when the bar carries a current I.
Can you combine magnetic force with forces in
equilibrium? Can you visualize the 3D example?
3. (This one WILL be graded.) When at rest, a proton experiences a net electromagnetic force (also
called the Lorentz force) of magnitude 8.0 x 10-13 N pointing in the positive x direction. When the
proton moves with a speed of 1.5 x 106 m/s in the positive y direction, the net electromagnetic
force on it decreases in magnitude to 7.5 x 10-13 N, still pointing in the positive x direction. Find
the magnitude and direction of (a) the electric field and (b) the magnetic field. You may assume
that By=0T.
Introducing the term Lorentz force – total electromagnetic force is merely a combination of the
electric and magnetic force – superposition! Skill: Do you realize/draw that the force due to B
cancels soe of the force due to E.
4. The book describes how a cyclotron works, and there are also some mastering physics questions
and tutorials made available to you. Cyclotrons are one kind of particle accelerators that
somehow accelerate particles while these particles are on a more or less circular path. Here is an
image of a 12” diameter student-built cyclotron at a university on the east coast. A possible track
of a proton in the cyclotron is shown in dashed lines (spiral) on the drawing. All the air is
removed from the chamber of the cyclotron so the accelerated charges won't run into anything as
they circulate around. There is a very similar image in the book.
The purpose of this question was to get you thinking about how a cyclotron works. For constant
B, the frequency of the particle is independent of r and v!
Proton Source – injects
protons at center
The magnetic field of the cyclotron is produced by an electromagnet: coil surrounding the
cyclotron. That way the students could control the magnetic field by the current through the coil.
The graph on the next page shows the magnetic field inside the cyclotron with respect to distance
from the center, for three different coil currents.
The D-shaped sections (called D's because of the shape) are attached to electronics that result in a
potential difference of several thousand volts between them (don’t be confused you do not need
to know anything about the interior of this electronic). There is no electric field in the inside of
the D’s, they are made out of metal.
a) Sketch the B field in the D’s. The B field in a cyclotron is kept constant.
b) Sketch the E field between the D's. This is the region where the charge is accelerated.
c) Draw four free body diagrams:
a. The proton is in the interior of the left D
b. The proton is in the interior of the right D
c. The proton is between the two D’s going left to right (at the very beginning after leaving
the proton source
d. The proton is between the two D’s moving right to left
d) Every half-cycle of the charge’s motion, the E-field has to change direction in order to keep
accelerating the proton. Show that the time a charge takes to complete a half-cycle in a uniform
magnetic field is independent of its velocity and radius.
Hint: Express the radius of a proton with speed v in terms of the magnetic field, its mass and its
charge.
e) Clearly, this will only work if the magnetic field is constant. Use the provided graph to show
what is the maximum speed a proton has when it leaves the cyclotron?
Hint: the proton has to leave the cyclotron before the B-field starts to change.
f) How does that maximum speed depend on the voltage of the power supply creating the E field
between the D’s?
g) The distance between the two D’s is 2cm. The potential difference between the 2 D’s is 1000V.
The proton source is located in such a way that the protons will be accelerated for the complete
distance of 2cm. What is the velocity of the proton at location A, where it penetrates the right D
for the first time?
h) An AC (alternating) voltage source produces a voltage that switches sign every half cycle. The
frequency of this source will need to match how frequently the charge makes a full turn in the
cyclotron. What is the frequency needed in the voltage source?
Hint: How long (t) does it take the proton to complete a half circle and reach the gap again?
f=1/2t