Physics I
Class 24
Magnetic Force
on Moving Charges
Rev. 11-Apr-03 GB
24-1
Hendrick Antoon Lorentz
Hendrick A. Lorentz was a Dutch
physicist who refined certain aspects of
electromagnetic theory. He, along with
Irish mathematical physicist George F.
FitzGerald, deduced fundamental
properties of the electromagnetic
behavior of moving bodies that formed
the basis of Einstein’s Special Theory
of Relativity.
The force of a magnetic field on a
moving charge is sometimes called the
Lorentz Force.
H.A. Lorentz
1853-1928
24-2
Important Facts About Velocity
and Acceleration Vectors (Review)
v
a
v
Same direction: speeding up.
a
v
a
Opposite directions: slowing down.
Right angles: changing direction, same speed.
24-3
Vector Cross Product (Review)
 
   
c  a  b ; | c |  | a || b | sin( )
The direction comes from the
right-hand rule. It is at a right
angle to the plane formed by a
and b . In other words, the cross
productis at right angles to both

a and b . (3D thinking required!)
24-4
Drawing 3D Vectors in 2D
Y
-Y
+Y
+X
-X
-Z
X
Z
+Z
+Z is out of page
24-5
Magnetic Force on a
Moving Charge

 
F  q vB
charge of the particle (C; + or –)
v : velocity of the particle (m/s)
B : magnetic field (T)
q:

 Force is at a right angle to velocity.
 Force is at a right angle to magnetic field.
Important: If q is negative, that reverses the direction of force.
24-6
An Example
An Electron in a Magnetic Field
Y
X
Z
24-7
Analysis of the Magnetic Force
Y
X

 
F  q vB
Z
F
We will evaluate this expression before
the electron starts turning.
 
First, evaluate v  B . In this case,
 they are 90° apart, so all we

need is the direction. v is +X, B is –Z, so v  B is +Y.
Next, we need to account for q. This is an electron, so q is
negative. Therefore, the magnitude of the force is (e v B) and the
direction is –Y.
24-8
Uniform Circular Motion
As the electron turns, so does the force vector.
 Speed stays constant because acceleration is
always perpendicular to velocity.
 The electron travels in a circle at a constant speed.
24-9
The Radius of the Circle
F
r
v
Although the directions of the vectors are
changing, the magnitudes stay the same.
v2
F  ma  m
r
F  qvB
v2
qvB  m
r
v2
mv
rm

qvB qB
24-10
The Period and Frequency
F
r
v
The circumference of the circle is 2  r.
Distance 2  r

Time
T
mv
2
2r
qB 2m
T


v
v
qB
v
f
1
qB

T 2m
  2f 
qB
m
24-11
Bubble Chamber
The red and green lines in the figure
to the left are tracks of charged
particles in a bubble chamber. Each
charged particle makes a trail of
tiny bubbles as it moves in the
chamber. There is a magnetic field
of 1.0 T directed into the page.
mv
r
qB
What are the signs of the charges of the particles?
Why do they spiral inward?
What are they?
What created them at the points where the tracks start?
24-12
The Aurora
 There is no acceleration in the direction of the
magnetic field line. (Why?)
 The component of velocity in the direction of the
field line remains constant. (Why?)
 The component of velocity at a right angle to the
field line continually changes direction. (Why?)
The result is that the charged particle (electron)
travels in a spiral path along the magnetic field line,
giving off light when it hits the atmosphere.
24-13
The Aurora
As Seen from the Space Shuttle
24-14
Class #24
Take-Away Concepts
1.
The Lorentz
(magnetic)
force on a moving, charged particle:



F  q vB
2. The Lorentz force cannot change a particle’s speed, only the
direction of its velocity.
3. Radius and angular frequency of a charged particle in uniform
circular motion in a magnetic field:
mv
r
qB
qB

m
24-15
Class #24
Problems of the Day
___1. A charged, non-magnetic particle is moving in a uniform
magnetic field. Which of the following conditions (if any)
would cause the particle to speed up?
A) The velocity of the particle is at a right angle to the magnetic
field.
B) The velocity of the particle is in the same direction as the
magnetic field.
C) The velocity of the particle is in the opposite direction as the
magnetic field.
D) Any of the above (A-C) would cause the particle to speed up.
E) None of the above; the magnetic force cannot cause the
particle to speed up.
24-16
Answer to Problem 1 for Class #24
The answer is E.
The Lorentz force cannot change the speed of the particle because
it is always at a right angle to the velocity of the particle. The only
change in the velocity vector due to this force is the direction.
24-17
Class #24
Problems of the Day
2. An electron is traveling in a vacuum tube at 1.4 x 107 m/s in a
horizontal direction toward the south. There is a constant
magnetic field in the tube with a magnitude of 0.5 gauss. The
direction of the magnetic field is toward the north and 30º down
(toward the ground). What are the magnitude and direction of the
magnetic (Lorentz) force on the electron? (1 T = 10,000 gauss.)
Up
v
S
N
-e
30°
B
Down
24-18
Answer to Problem 2 for Class #24
 
First, the direction: v  B is out of the page by the right-hand rule.
The charge is negative, so the force is into the page (West).
The magnitude is e v B sin(180°–30°) =
(1.6 x 10–19) (1.4 x 107) (0.5 x 10–4) sin(150°) =
5.6 x 10–17 N
24-19
Activity #24
Lorentz Force and Magnetic Field
Objective of the Activity:
1.
2.
Consider the implications of the Lorentz force on speed and
direction of a charged particle.
Determine the direction and magnitude of the magnetic field at
your table in the classroom using a compass, a coil of wire, a
power supply, and a current meter.
24-20