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Physics 212
Lecture 12
Magnetic Force on moving charges
FB  q v  B
Physics 212 Lecture 12, Slide 1
Magnetic Observations
• Bar Magnets
N
S
N
S
S
N
N
S
• Compass Needles
N
S
Magnetic Charge?
N
07
S
cut in half
N
S
N
S
Physics 212 Lecture 12, Slide 2
Magnetic Observations
• Compass needle deflected by electric current
I
• Magnetic fields created by electric currents
• Magnetic fields exert forces on electric currents (charges in motion)
V
V
I
F
F
I
F
I
I
F
V
12
Physics 212 Lecture 12, Slide 3
Magnetism & Moving Charges
All observations are explained by two equations:
14
FB  qv  B
Lorentz Force
 0 I ds  rˆ
dB 
2
4 r
Biot-Savart Law
Physics 212 Lecture 12, Slide 4
Cross Product Review
• Cross Product different from Dot Product
– A●B is a scalar; A x B is a vector
– A●B proportional to the component of B parallel to A
– A x B proportional to the component of B perpendicular to A
• Definition of A x B
– Magnitude: ABsinq
– Direction: perpendicular to plane defined by A and B with sense given
by right-hand-rule
16
Physics 212 Lecture 12, Slide 5
Remembering Directions: The Right Hand Rule
y
FB  q v  B
x
F
B
qv
17
z
Physics 212 Lecture 12, Slide 6
Checkpoint 1a
Three points are arranged in a uniform magnetic
field. The B field points into the screen.
1) A positively charged particle is located at point A and is stationary.
The direction of the magnetic force on the particle is:
A
B
C
D
E
F  qv  B
The particle’s velocity is zero.
There can be no magnetic force.
18
Physics 212 Lecture 12, Slide 7
Checkpoint 1b
Three points are arranged in a uniform magnetic
field. The B field points into the screen.
A
B
C
D
E
1) A positively charged particle is located at point A and is stationary.
The direction of the magnetic force on the particle is:
F  qv  B
qv
F
21
X
B
Physics 212 Lecture 12, Slide 8
F  qv  B
Cross Product Practice
• Positive charge coming out of screen
• Magnetic field pointing down
• What is direction of force ?
A) Left
-x
B) Right
+x
C) UP
+y
D) Down
-y
E) Zero
y
x
B
24
z
Physics 212 Lecture 12, Slide 9
B field cannot change kinetic energy of a charge.
FB  q v  B  FB  v
FB
v
dWB  FB  dr  FB  v dt  0
No work done by B field  Kinetic energy is constant
v
But
is constant
v
can change !
Cyclotron Motion – uniform B field
F  qv  B  F  qvB
v v
x vx x x x x x
q
constant
x R x x x x vx x
2
v
Centripetal acceleration: a 
R
30
q
F
F
q
x x xF x x x x
xv x x x x x x
q
x x x x xv x x
Uniform B into page
Solve for R:
v2
qvB  m
R
x x x Fx x x x
mv
R
qB
Physics 212 Lecture 12, Slide 11
R
LHC
mv
p

qB qB
17 miles diameter
Physics 212 Lecture 12, Slide 12
Checkpoint 2a
The drawing below shows the top view
of two interconnected chambers. Each
chamber has a unique magnetic field. A
positively charged particle is fired into
chamber 1, and observed to follow the
dashed path shown in the figure.
A
B
C
D
F  qv  B
qv
.
B
34
F
Physics 212 Lecture 12, Slide 13
Checkpoint 2b
The drawing below shows the top view
of two interconnected chambers. Each
chamber has a unique magnetic field. A
positively charged particle is fired into
chamber 1, and observed to follow the
dashed path shown in the figure.
A
B
C
Observation: R2 > R1
R
36
mv
qB
B1  B2
Physics 212 Lecture 12, Slide 14
Calculation
A particle of charge q and mass m is accelerated
from rest by an electric field E through a
distance d and enters and exits a region
q,m
containing a constant magnetic field B at the
points shown. Assume q,m,E,d, and x0 are known.
What is B?
–
–
–
x0/2
E
d
enters here
exits here
XXXXXXXXX
X X X X X X X X X x0
XXXXXXXXX
XXXXXXXXX
B
B
Calculate v, the velocity of the particle as it enters the magnetic
field
Use Lorentz Force equation to determine the path in the field as a
function of B
Apply the entrance-exit information to determine B
Physics 212 Lecture 12, Slide 15
Calculation
A particle of charge q and mass m is accelerated
from rest by an electric field E through a
distance d and enters and exits a region
q,m
containing a constant magnetic field B at the
points shown. Assume q,m,E,d, and x0 are known.
E
d
enters here
What is B?
exits here
x0/2
XXXXXXXXX
X X X X X X X X X x0
XXXXXXXXX
XXXXXXXXX
B
B
• What is v0, the speed of the particle as it enters the magnetic field ?
2E
m
(A)
vo 
• Why??
–
–
2qEd
vo 
m
(B)
vo  2 ad
(C)
2qE
md
(D)
vo 
vo 
qEd
m
(E)
Conservation of Energy
•
•
Initial: Energy = U = qV = qEd
Final: Energy = KE = ½ mv02
•
•
a = F/m = qE/m
v02 = 2ad
Newton’s Laws
vo2  2
1 mv 2  qEd
o
2
qE
d
m
vo 
vo 
2qEd
m
2qEd
m
Physics 212 Lecture 12, Slide 16
Calculation
A particle of charge q and mass m is accelerated
from rest by an electric field E through a
distance d and enters and exits a region
q,m
containing a constant magnetic field B at the
points shown. Assume q,m,E,d, and x0 are known.
What is B?
x0/2
E
d
enters here
2qEd
vo 
m
exits here
XXXXXXXXX
X X X X X X X X X x0
XXXXXXXXX
XXXXXXXXX
B
B
• What is the path of the particle as it moves through the magnetic field?
XXXXXXXXX
XXXXXXXXX
XXXXXXXXX
(A) X X X X X X X X X
• Why??
–
(B)
XXXXXXXXX
XXXXXXXXX
XXXXXXXXX
XXXXXXXXX
(C)
XXXXXXXXX
XXXXXXXXX
XXXXXXXXX
XXXXXXXXX
Path is circle !
•
•
•
Force is perpendicular to the velocity
Force produces centripetal acceleration
Particle moves with uniform circular motion
Physics 212 Lecture 12, Slide 17
Calculation
A particle of charge q and mass m is accelerated
from rest by an electric field E through a
distance d and enters and exits a region
q,m
containing a constant magnetic field B at the
points shown. Assume q,m,E,d, and x0 are known.
E
d
enters here
2qEd
vo 
m
What is B?
x0/2
exits here
XXXXXXXXX
X X X X X X X X X x0
XXXXXXXXX
XXXXXXXXX
B
B
• What is the radius of path of particle?
R  xo
R  2 xo
(A)
(B)
1
R  xo
2
(C)
mvo
R
qB
(D)
vo2
R
a
(E)
• Why??
R
xo/2
XXXXXXXXX
XXXXXXXXX
XXXXXXXXX
XXXXXXXXX
Physics 212 Lecture 12, Slide 18
Calculation
A particle of charge q and mass m is accelerated
from rest by an electric field E through a
distance d and enters and exits a region
q,m
containing a constant magnetic field B at the
points shown. Assume q,m,E,d, and x0 are known.
What is B?
2
B
xo
2 mEd
q
(A)
vo 
2qEd
m
E
d
enters here
R  1 x0
2
E
B
v
m
BE
2qEd
(B)
(C)
1
B
xo
exits here
x0/2
XXXXXXXXX
X X X X X X X X X x0
XXXXXXXXX
XXXXXXXXX
B
B
2 mEd
q
B
(D)
mvo
qxo
(E)
• Why??


F  ma
vo2
qvo B  m
R
m vo
B
q R
m 2
B
q xo
B
2
xo
2 qEd
m
2 mEd
q
Physics 212 Lecture 12, Slide 19
Follow-Up
A particle of charge q and mass m is accelerated
from rest by an electric field E through a
distance d and enters and exits a region
q,m
containing a constant magnetic field B at the
points shown. Assume q,m,E,d, and x0 are known.
What is B?
2
B
xo
x0/2
E
d
enters here
2 mEd
q
exits here
XXXXXXXXX
X X X X X X X X X x0
XXXXXXXXX
XXXXXXXXX
B
B
• Suppose the charge of the particle is doubled (Q = 2q),while keeping the
mass constant. How does the path of the particle change?
(A)
X X X X X X X X XX X X X X X X X XX X X X X X X X X
X X X X X X X X XX X X X X X X X XX X X X X X X X X
X X X X X X X X XX X X X X X X X XX X X X X X X X
X X X X X X X X XX X X X X X X X XX X X X X X X X X
Several things going on here
(B)
(C)
1. q changes  v changes
2. q & v change  F changes
3. v & F change  R changes
Physics 212 Lecture 12, Slide 20
Follow-Up
A particle of charge q and mass m is accelerated
from rest by an electric field E through a
distance d and enters and exits a region
q,m
containing a constant magnetic field B at the
points shown. Assume q,m,E,d, and x0 are known.
What is B?
2
B
xo
x0/2
E
d
enters here
2 mEd
q
exits here
XXXXXXXXX
X X X X X X X X X x0
XXXXXXXXX
XXXXXXXXX
B
B
• Suppose the charge of the particle is doubled (Q = 2q),while keeping the
mass constant. How does the path of the particle change?
How does v, the new velocity at the entrance,
compare to the original velocity v0?
v
(A) v  o
2
• Why??
v
(B) v  o
2
(C) v  vo
1 mv 2  QEd  2 qEd  2 1 mv 2
o
2
2
(D) v  2vo
v 2  2vo2
(E) v  2 vo
v  2 vo
Physics 212 Lecture 12, Slide 21
Follow-Up
A particle of charge q and mass m is accelerated
from rest by an electric field E through a
distance d and enters and exits a region
q,m
containing a constant magnetic field B at the
points shown. Assume q,m,E,d, and x0 are known.
What is B?
2
B
xo
x0/2
E
d
enters here
2 mEd
q
exits here
XXXXXXXXX
X X X X X X X X X x0
XXXXXXXXX
XXXXXXXXX
B
B
• Suppose the charge of the particle is doubled (Q = 2q),while keeping the
mass constant. How does the path of the particle change?
v  2 vo
How does F, the magnitude of the new force at the entrance,
compare to F0, the magnitude of the original force?
(A) F  Fo
2
(B) F  Fo
(C) F  2 Fo
(D) F  2 Fo
(E) F  2 2 Fo
• Why??
F  QvB  2 q  2 vo  B
F  2 2 Fo
Physics 212 Lecture 12, Slide 22
Follow-Up
A particle of charge q and mass m is accelerated
from rest by an electric field E through a
distance d and enters and exits a region
q,m
containing a constant magnetic field B at the
points shown. Assume q,m,E,d, and x0 are known.
What is B?
2
B
xo
x0/2
E
d
enters here
2 mEd
q
exits here
XXXXXXXXX
X X X X X X X X X x0
XXXXXXXXX
XXXXXXXXX
B
B
• Suppose the charge of the particle is doubled (Q = 2q),while keeping the
mass constant. How does the path of the particle change?
v  2vo F  2 2 Fo
How does R, the radius of curvature of the path, compare to
R0, the radius of curvature of the original path?
R
(A) R  o
2
(B) R 
Ro
2
(C) R  Ro
(D) R  2 Ro
(E) R  2 Ro
• Why??
v2
F m
R
v2
Rm
F
2vo2
vo2
R
Rm
m
 o
2 2 Fo
2 Fo
2
Physics 212 Lecture 12, Slide 23
Follow-Up
A particle of charge q and mass m is accelerated
from rest by an electric field E through a
distance d and enters and exits a region
q,m
containing a constant magnetic field B at the
points shown. Assume q,m,E,d, and x0 are known.
What is B?
2
B
xo
x0/2
E
d
enters here
2 mEd
q
exits here
XXXXXXXXX
X X X X X X X X X x0
XXXXXXXXX
XXXXXXXXX
B
B
• Suppose the charge of the particle is doubled (Q = 2q),while keeping the
mass constant. How does the path of the particle change?
(A)
X X X X X X X X XX X X X X X X X XX X X X X X X X X
X X X X X X X X XX X X X X X X X XX X X X X X X X X
X X X X X X X X XX X X X X X X X XX X X X X X X X
X X X X X X X X XX X X X X X X X XX X X X X X X X X
(B)
(C)
Ro
R
2
Physics 212 Lecture 12, Slide 24