Concept 19.1: The cross product combines
two vectors to give a third vector.
v
v r r
B⊥
B
C = A× B
v
C = A B sin φ = A B⊥
φ
A
19. Magnetic force on a moving
charge
• Review the cross product
• Lorentz force
• Motion of charged particles in E
and B fields
v
The direction of C is perpendicular to both A and B ,land is
given by the “right-hand-rule.” Here, it points out of the
page.
The cross-product is used to define torque as a vector.
r
Serway and Beichner
Sections 29.1, 29.4-.5
1
v
B
r r
F ⊥B
r r
2) F ⊥ v
1)
3) For a negative charge, the
force is the opposite direction,
since q is a negative number.
+
θ
r
V
charge
ω
2
Physics 1E03 Lecture 19
Concept 19.2: Experiments show that a charge
moving in a region where there is a magnetic field
r
experiences a force.
F
q
r
To represent rotation as
a vector, also use a “right-hand-rule”.
Physics 1E03 Lecture 19
A moving charge is like
a current with a very low
charge density. It should
have a magnetic force on
it. Experiment shows:
r
τ = r ×F
cross product
Quiz of concept 19.2
An electron is injected into a large
region of constant magnetic field
Bxi with an initial velocity vyj. The
trajectory of the electron
a) curves into the page
b) circles in the xy plane
c) makes a circle out of the page, in
the yz plane
d) curves out of the page.
r
B
r
v
y
x
e-
r
r r
F = qv × B
velocity
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Quiz of concept 19.2
Quiz of concept 19.2
r
B
An electron is injected into a large
region of constant magnetic field
Bxi with an initial velocity v at an
angle θ to the x axis. The
trajectory of the electron
r
v
y
θ
e-
x
a) circles in the xy plane
b) makes a circle out of the page, in
the yz plane
c) drifts along the direction of B
d) both b) and c)
a)
b)
c)
d)
Physics 1E03 Lecture 19
N = C • m • (Τ) = A • m • T
s
N
Also…
T=
1 gauss (G) = 10-4 T
A•m
Typical Fields
Earth’s Field
~ 1 x 10-4 T (1 gauss)
Strong Fridge magnet
~ 10-2 T (100 G)
Big Lab Electromagnet
~4T
B
r
v
y
θ
e-
x
increases
stays the same
decreases
more information is needed
5
Concept 19.3: The force on a particle can be
used to define the units of the magnetic field,
the Tesla (T).
r
r r
F = qv × B
r
An electron is injected into a large
region of constant magnetic field
Bxi with an initial velocity v at an
angle θ to the x axis. As the
electron spirals along the field line,
the energy of the electron
Physics 1E03 Lecture 19
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Concept 19.4: In a region with both electric
and and magnetic fields, a charged particle
moves according to the combined forces.
r
r
r r
F = qv × B + qE
Force doing no work,
acts perpendicular to
v and B.
Force accelerating the charge
in the direction of E, can do work.
Superconducting magnet ~ 20 T
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Quiz of concept 19.4
An electron beam passes between the plates of a capacitor.
The plates are separated by a distance d and have a uniform
magnetic field B between them. The potential between the
plates is ∆V.
In order to pass between
the capacitor plates with a
constant velocity, the
electrons
∆V
a) must have velocity v =
dB
b) must have an initial
velocity parallel to the
plates
c) both a) and b) above
d) neither a) nor b) above
+ + + + + + + +
- - - - - - - -
Summary
• since currents feel a force from a magnetic field, so
do individual moving chargesr
r r
• the force on a moving charge is F = qv × B , it is
perpendicular to both velocity and B
• since the force is perpendicular to the motion, the
speed of the charged particle does not change
• in a constant magnetic field, charged particles travel
in a circle or in a helix.
• the units of B are Tesla, T=N/Am
Practice problems: Chapter 29 #5, 9, 34, 41
Next lecture: read sections 29.2-.3
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