Buggé: Magnetism 4 Solutions
Magnetic Force Exerted by a Magnetic Field on a Single Moving Charged Particle
If a magnetic field exerts a force on current carrying wires, is it reasonable to believe that the magnetic
field also exerts a force on individual moving electrically charged particles (electrons)?
Hollow
Hot
4.1 Observe and find a
anode
cathode
pattern A cathode-ray tube
(CRT) is part of a
+
–
–
traditional television set or
–
electrons
of an oscilloscope.
Electrons “evaporate” from
a hot filament called the
cathode. They accelerate
across a potential difference
and then move at high speed toward a scintillating screen. The electrons
form a bright spot on the screen at the point at which they hit it. A magnet
held near the CRT sometimes causes the electron beam to deflect.
Scintillating
screen
a. Watch the video at
http://paer.rutgers.edu/pt3/experiment.php?topicid=10&exptid=114
or use the QR code at the right.
r
b. Devise a rule for the direction of the force Fm that the magnet exerts on the moving electrons
r
r
relative to the direction of their velocity v and the direction of the magnetic field B produced
by the magnet. Use the information provided in the table. (The red side of the magnet is the
north pole.)
€
€
Experiment
Point the north pole of a magnet at the front of the
scintillating screen—opposite the direction the electrons are
moving.
Point the north pole of the magnet from the right side (as
you face the coming beam) perpendicular to the direction the
electrons are moving.
Point the south pole of the magnet from the right side
perpendicular to the direction the electrons are moving.
Point the north pole of the magnet from the left side (as you
face the coming beam) perpendicular to the direction the
electrons are moving.
Point the south pole of the magnet from the left side
perpendicular to the direction the electrons are moving.
Point the north pole of the magnet down from the top of the
CRT, perpendicular to the direction the electrons are
moving.
Point the south pole of the magnet down from the top of the
CRT, perpendicular to the direction the electrons are
moving.
€
Observation
Nothing happens to the beam.
The beam deflects up.
The beam deflects down.
The beam deflects down.
The beam deflects up.
The beam deflects left.
The beam deflects right.
Buggé: Magnetism 4 Solutions
If the magnetic field is perpendicular to the velocity of the electrons, the force exerted on the electrons is
perpendicular to both the magnetic field and the velocity.
c. Your friend says that the beam of electrons is deflected by the magnet because the electrons
are charged particles and the magnet is made of iron. Because all conductors attract
electrically charged particles, the experiment above is not related to magnetism. How can you
convince your friend that she is mistaken?
Electrically charged particles exert forces that are either towards (attractive) or away (repulsive) from
it. In this case the force on the electron is perpendicular to both the magnetic field and the velocity.
4.2 Represent and reason For each situation below, decide if a non-zero magnetic force is exerted on
the moving electric charge (test object). If not zero, indicate the direction of the magnetic force.
Small force out of page
4.3 Derive Using the equation for the magnitude of force that a magnetic field exerts on a current
carrying wire, develop an expression for the magnitude of the force that the magnetic field exerts on
a single charged object with charge q moving at speed v. To help, begin by thinking of the current I
in the wire as a large number of positively charged particles, each with a charge q, that pass a cross
section of wire in a time interval Δt.
F = ILBsinθ
I = q/Δt
F = (q/Δt)LBsinθ = q(L/Δt)LBsinθ = qvBsinθ
Buggé: Magnetism 4 Solutions
Magnetic force exerted by magnetic field on a charged particle
The magnitude of the magnetic force that a magnetic field exerts on a particle with electric charge q
moving at speed v is:
FB on q = |q|vBsinθ
where θ is the angle between the direction of the velocity of the particle and the direction of the B-field.
The direction of this force is determined by the right hand rule for the magnetic
force. If the particle is negatively charged, then the force points in the opposite the direction.
What is the magnitude of force if the velocity and the B field vectors are parallel?
Zero
When is the magnitude of the force maximum?
When the direction of velocity of the particle and the direction of the B-field are perpendicular.
4.4 Particles in a magnetic field Each of the lettered dots shown in Figure below represents a small
object with electric charge of +2.0 x 10-6 C moving at the speed of 3.0 x 107 m/s in the directions
shown. Determine the magnetic force (magnitude and direction) that a 0.10-T B-field exerts on each
object. The B-field points in the positive y direction. Hint: First, use the right hand rule for the
magnetic force to determine the directions of the magnetic force exerted on each object.
First, use the right hand rule for the magnetic force to determine the directions of the magnetic force
exerted on each object.
Then use FB on q = |q|vBsinθ to determine the magnitude of each force.
FB on A = (2.0×10-6 C) (3.0×107 m/s) (0.10 T) sin(90º) = 6.0 N into the paper
FB on B = (2.0×10-6 C) (3.0×107 m/s) (0.10 T) sin(180º) = 0 N
FB on C = (2.0×10-6 C) (3.0×107 m/s) (0.10 T) sin(90º) = 6.0 N out of the paper
FB on D = (2.0×10-6 C) (3.0×107 m/s) (0.10 T) sin(53º) = 4.8 N out of the paper
Buggé: Magnetism 4 Solutions
4.5 Represent and reason The mass-detecting part of a mass spectrometer is described below in
multiple ways.
Words
Sketch
Physical
Mathematical
representation
representation
An ion with mass m and
Top view
charge +e leaves a velocity
selector moving at speed v.
∑Fradial = m ac
.
.
.
It then moves in a half
Velocity
circle in a magnetic field
Fm = m v2/ r
selector
.
.
.
that is perpendicular to the
radial
+
2r
plane of its motion. At the
e v B = m v2/ r
.
.
.
end of this trip, it is
detected. The radius of the
or
Detector
circle can be used to
.
.
.
determine the mass of the
m = eBr/v
ion.
.
Are the representations consistent with each other? Notice that this problem involves a charged particle
moving in an external magnetic field whose origin is unknown.
Yes, they are all consistent.
4.6 Motion of protons in Earth’s magnetic field What happens to a cosmic ray proton flying into
Earth’s atmosphere above the equator at a speed of about 107 m/s? The average magnitude of Earth’s
B-field in this region is approximately 5 x 10-5 T. The mass m of a proton is 1.67 x 10-27 kg. Consider
(i) a proton moving perpendicular to the B-field lines and (ii) a proton moving at an angle θ relative
to the B-field lines.
See example 17.5 in PTPA textbook (online)
4.7 If the magnetic force is always perpendicular to the velocity of a charged particle, does it do any
work on it? Explain your answer.
No; cos90 = 0 so W = F∆xcosθ = 0 J in all instances.