1
Physics 1B
Class Notes©
Walter Gekelman
Week 10
Magnetism
We have studies the force law for a charge in an electric field
F
=
q
E
.
If a magnetic
field is present the generalized law, called the Lorentz equation is:
(1)
F
=
q
E
+
v
×
B
( )
The magnetic field is in tesla (or weber/m
2
) in MKS and in Gauss in cgs systems of units.
1 tesla = 10
4
Gauss.
This is a velocity dependant law.
The only other time we saw velocity occur in a force
law is when frictional drag is present in the equation of motion of a falling object.
In
rectangular coordinates:
(1a)
or
The magnetic field now gives us a “preferred direction” in space as form equation (1)
there is no magnetic force on particles moving along the magnetic field. We can write
the velocity as
v
=
v
+
v
⊥
.
Now
(2)
F
=
q
E
+
v
+
v
⊥
( )
×
B
( )
=
q
E
+
v
⊥
( )
×
B
( )
Let us assume that
B
=
B
0
−
ˆ
k
( )
and there is no electric field.
Then
(3)
From the z component we get
m
d
v
z
dt
=
0
;
v
z
=
v
z
0
.
The z component of the velocity is
not changed and remains whatever it was when the magnetic field appears.
The other
components are:
m
d
v
x
dt
=
−
v
y
B
0
;
m
d
v
y
dt
=
v
x
B
0
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Take the derivative of the equation for
d
v
x
dt
and substitute the second equation to get
Now there are two equations for the x,y velocity components:
(4)
If we define
ω
c
=
qB
0
m
we get two equations that we have seen before.
The solutions
are:
(5)
v
x
=
A
cos
c
t
( ) +
D
sin
c
t
( )
A and B are constants that depend on the initial conditions.
Suppose at t=0
v
=
v
x
.
This
means that D=0.
v
x
=
A
cos
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 Spring '10
 Corbin
 Magnetic Field, Electric charge, dvy dvx

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