Physics 9
WS M2 (rev. 1.0)
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M-2. The Lorentz force law
Questions for discussion
1.
The diagram below shows a proton moving through an external magnetic field at a particular instant of time.
a)
At the instant shown, would the magnetic field be exerting any force on the proton? If so, in which direction?
Sketch the force vector on the diagram above.
b) Would your answers change if the particle were an electron?
2.
A positively charged particle is moving with the indicated velocity through an external magnetic field, which is not
shown. This external magnetic field exerts a force on the moving charge, which is shown.
a)
What is the direction of the external magnetic field at the location of the particle?
b) Answer part (a) again, this time assuming the particle is negatively charged.
Physics 9
3.
WS M2 (rev. 1.0)
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In the figure below, a uniform magnetic field points into the page. (The magnetic field vectors are indicated by
⊗’s.) Four particles with the same mass follow the paths shown as they pass through this magnetic field with
identical, constant speeds.
What can you conclude about the charge on each particle?
4.
An electron moves horizontally from the left with speed v and enters a uniform vertical electric field of magnitude
E0 pointing upwards. In the absence of any other forces, the electron would be deflected vertically by the F on q =
qEext force. Sketch the direction of a magnetic field that could cancel this force and allow the electron to maintain
its horizontal path.
5.
When a particle moves under the influence of a magnetic field, the speed of the particle remains constant. How does
this come about?
Physics 9
WS M2 (rev. 1.0)
Page 3
Problems
1.
causing them to become singly ionized (which means
that they have lost one of their outer electrons).
When the ions emerge from the ionizing chamber,
they are accelerated through a potential difference V0,
at which point they enter a uniform magnetic field B 0
as shown. The magnetic field deflects the ions into a
circular path, and they eventually collide with a
detector plate.
The mass spectrometer.
In a sample of ordinary carbon, most of the carbon
atoms consist of six electrons bound to a nucleus of
six protons and six neutrons. But a small fraction of
the carbon atoms in the sample will be slightly
heavier, consisting of six electrons, six protons, and
eight neutrons.
The reason this whole scheme works is that ions of
different isotopes will end up in different places on
the detector plate. This allows identification of the
relative abundances of the isotopes.
In this problem we will determine the separation of
12
C and 14C on the detector plate.
a)
As you can see, the masses of these isotopes are very
nearly M12 = 12mp and M14 = 14 mp, respectively,
where mp = 1.67×10-27 kg is the mass of a proton.
Because these isotopes are the same as far as their
charged constituents, their chemical properties are for
all practical purposes identical. So we cannot
separate these isotopes from one another via chemical
procedures.
A mass spectrometer uses a magnetic field to separate
the isotopes as follows.
In the ionizing chamber, an electric discharge runs
through the atoms in the carbon gas,
How fast is a 12C ion going when it enters the
magnetic field? Answer in terms of V0, mp, and e.
(For simplicity, assume that the ions emerging from
the ionizing chamber are at rest.)
b) Draw a force diagram for a 12C ion when it is
traveling inside the magnetic field in a circular path
with the speed calculated in part (a). (Neglect
gravity.)
c)
Use Newton’s Second Law F net = ma to determine
the radius of the 12C ion’s circular path. Answer in
terms of V0, B0, mp, and e.
d) Looking at your answer for part (c), write down an
expression for the radius of a 14C ion’s circular path
within the magnetic field.
e)
What is the separation of the ions on the detector
plate? Answer in terms of V0, B0, mp, and e.
f)
If we want the separation to be at least 1 cm, and the
magnetic field we are using is 0.1 Tesla, then how
large is the accelerator voltage V0 going to have to
be?
g)
With the voltage set at this level (and with B 0 = 0.1
T), where will the 12C ions strike the detector plate?

Physics 9
2.
WS M2 (rev. 1.0)
In the figure below, a rectangular loop of wire is
immersed in an external magnetic field B 0 pointing
to the right.
c)
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If the loop were not held fixed in place, what would
be the net effect of the forces on each segment of the
loop?
d) What torque does segment 1 of the loop experience?
(Consider the magnetic force on the segment to act at
the midpoint of the segment, and take the torque
about the center of the loop.)
The loop carries current i0 in the clockwise direction,
and is held fixed in place.
a)
e)
Answer the same question for segments 2, 3, and 4.
In each case, consider the magnetic force on the
segment to act at the midpoint of the segment, and
take torques about the center of the loop.
f)
What is the net torque on the loop about its center?
Give the magnitude and direction.
g)
Now compute the torque on the loop using the
standard formula τon loop = µ×B, where µ = iA is the
magnetic moment of the loop. Does your answer
from part (f) agree, in both magnitude and direction?

How large is the force exerted on segment 1 of the
loop by the external magnetic field? In which
direction does it point?
b) Answer the same questions for segments 2, 3, and 4.
3.
A metal wire of mass m slides without friction on
two horizontal rails spaced a distance d apart, as
shown.
The track lies in a uniform external magnetic field
B 0, pointing perpendicular to the plane of the rails.
A constant current i0 flows from the generator G
along one rail, across the wire, and back along the
other rail.
a)
At any given time, what is the force exerted on the
sliding wire by the external magnetic field? Give
both the magnitude and direction.
b) Find the velocity of the sliding wire as a function of
time, assuming it to be at rest at time t=0. 