Download Magnetic Force on Moving Charged Particles.

Today’s agenda:
Magnetic Fields.
You must understand the similarities and differences between electric fields and field lines,
and magnetic fields and field lines.
Magnetic Force on Moving Charged Particles.
You must be able to calculate the magnetic force on moving charged particles.
Magnetic Flux and Gauss’ Law for Magnetism.
You must be able to calculate magnetic flux and recognize the consequences of Gauss’ Law
for Magnetism.
Motion of a Charged Particle in a Uniform Magnetic Field.
You must be able to calculate the trajectory and energy of a charged particle moving in a
uniform magnetic field.
Magnetic Fields and Moving Charges
A charged particle moving in a magnetic field experiences a
force.
The following equation (part of the Lorentz Force Law)
predicts the effect of a magnetic field on a moving charged
particle:
F = qv  B
force on
particle
velocity of
charged particle
Oh nooo! The
little voices
are back.
magnetic field
vector
What is the magnetic force if the charged particle is at rest?
What is the magnetic force if v is (anti-)parallel to B?
Vector notation conventions:

 is a vector pointing out of the paper/board/screen (looks
like an arrow coming straight for your eye).

 is a vector pointing into the paper/board/screen (looks
like the feathers of an arrow going away from eye).
Direction of magnetic force--Use right hand rule:
fingers out in direction of v, thumb perpendicular to them
rotate your hand until your palm points in the direction of B
(or bend fingers through smallest angle from v to B)
thumb points in direction of F on + charge
Your text presents two alternative variations (curl your fingers,
imagine turning a right-handed screw). There is one other
variation on the right hand rule. I’ll demonstrate all variations in
lecture sooner or later.
still more variations: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfor.html
Complication: your Physics
1135 text uses the right
hand rule shown in the
figure, so we’ll use the
same rule in Physics 2135.
There are a number of
variations of this rule.
Unfortunately, most of the Youtube videos I find say to use your
palm for A , your thumb for B , and your outstretched fingers for
A  B . This includes the MIT Open Courseware site.
I’ll show you later that both ways are correct. I am going to use
our textbook’s technique. You can use whatever works for you!
F = q vB sinθ
Direction of magnetic force:
F
+

Fingers of right hand out in direction of velocity, thumb
perpendicular to them. Bend your fingers until they point
in the direction of magnetic field. Thumb points in
direction of magnetic force on + charge.
B

-
v
F
B
v
Force is “down”
because charge is -.
v
F?
-

B
I often do this: fingers out in direction of velocity, thumb perpendicular to them. Rotate your hand until your palm points in the
direction of magnetic field. Thumb points in direction of magnetic force on + charge.
“Foolproof” technique for calculating both magnitude and
direction of magnetic force.
F = qv  B

 ˆi


F = q det  v x

B

 x
ˆj
vy
By
kˆ  

v z 
B z  

http://www.youtube.com/watch?v=21LWuY8i6Hw
All of the right-hand rules are just techniques for determining the direction of vectors
in the cross product without having to do any actual math.
^
Example: a proton is moving with a velocity v = v0j in a region
^
of uniform magnetic field. The resulting force is F = F0i. What is
the magnetic field (magnitude and direction)?
To be worked at the blackboard.
Blue is +, red is -. Image from http://www.mathsisfun.com/algebra/matrix-determinant.htm
^
Example: a proton is moving with a velocity v = v0j in a region
^
of uniform magnetic field. The resulting force is F = F0i. What is
the magnetic field (magnitude and direction)?
v = v 0 ˆj
F = F0 ˆi
F = qv  B

 ˆi


ˆ
F = F0 i = q det  v x

B

 x
ˆj
vy
By
kˆ  

v z   =  e 
B z  


 ˆi


det  0

B

 x
ˆj
v0
By
kˆ  

0 
B z  

F0ˆi =  e  ˆi  v 0  Bz  B y  0   ˆj 0  B z  B x  0   kˆ 0  B y  B x  v 0 


^
Example: a proton is moving with a velocity v = v0j in a region
^
of uniform magnetic field. The resulting force is F = F0i. What is
the magnetic field (magnitude and direction)?
F0 ˆi = ev 0B z ˆi  0jˆ  eB x v 0kˆ =  ev 0B z  ˆi +  0  ˆj+  eB x v 0  kˆ
F0 = ev 0B z
F0
Bz =
ev 0
and
and
0 = eB x v 0
Bx = 0
What is By???? It could be anything. Not enough information is
provided to find By. We can’t find the magnitude and direction
of the magnetic field using only the information given!
^
Example: a proton is moving with a velocity v = v0j in a region
^
of uniform magnetic field. The resulting force is F = F0i. What is
the minimum magnitude magnetic field that can be present?
Bz =
F0
ev 0
By = 0
B min =
and
Bx = 0
for a minimum magnitude magnetic field, so...
F0
ev 0
Homework Hint
F = q vB sinθ
A charged particle moving along or opposite to the direction of
a magnetic field will experience no magnetic force.
Conversely, the fact that there is no magnetic force along some
direction does not mean there is no magnetic field along or
opposite to that direction.
It’s OK to use F = q vB if you know that v and B are
perpendicular, or you are calculating a minimum field to
produce a given force (understand why).
Alternative* view of magnetic field units.
F = q vB sinθ
F
B=
q v sinθ
N
N
=
B  = T =
C  m/s A  m
*“Official” definition of units coming later.
Remember, units of field are
force per “something.”