Physics 201
Professor P. Q. Hung
311B, Physics Building
Physics 201 – p. 1/3
Magnetic Fields
A proton has a speed of 106 m/s and enters a
region with a uniform magnetic field. It makes an
angle of 450 with respect to the direction of the
~ Find its subsequent motion.
magnetic field B.
What is a magnetic field?
Physics 201 – p. 2/3
Magnetic Fields
A proton has a speed of 106 m/s and enters a
region with a uniform magnetic field. It makes an
angle of 450 with respect to the direction of the
~ Find its subsequent motion.
magnetic field B.
What is a magnetic field?
Subsequent motion? Is there any force? Are
electrically neutral particles affected by a
magnetic field? Or only charged particles are
affected?
Physics 201 – p. 2/3
Magnetic Fields
Some facts
We have seen how magnets influence the
motion of particles which are charged, how
they interact with metallic objects, etc...
Physics 201 – p. 3/3
Magnetic Fields
Some facts
We have seen how magnets influence the
motion of particles which are charged, how
they interact with metallic objects, etc...
A magnet has a north pole and a south pole
which do not actually coincide with the
geographical poles (from the axis of rotation).
⇒Like poles repel and unlike poles attract
each other.
Physics 201 – p. 3/3
Magnetic Fields
Some facts
Just like the case of electrostatics, there is a
magnetic field surrounding a magnet.
Physics 201 – p. 4/3
Magnetic Fields
Some facts
Just like the case of electrostatics, there is a
magnetic field surrounding a magnet.
One can draw magnetic field lines starting
from the “North” pole and ending in the
“South” pole.
Physics 201 – p. 4/3
Magnetic Fields
Some facts
Just like the case of electrostatics, there is a
magnetic field surrounding a magnet.
One can draw magnetic field lines starting
from the “North” pole and ending in the
“South” pole.
~
We will denote a magnetic field by B.
Physics 201 – p. 4/3
Magnetic Fields
Magnetic force between two magnets
Physics 201 – p. 5/3
Magnetic Fields
Always two poles, North and South
Physics 201 – p. 6/3
Magnetic Fields
Magnetic field lines of a bar magnet
Physics 201 – p. 7/3
Magnetic Fields
Refrigerator magnet
Physics 201 – p. 8/3
Magnetic Fields
Magnetic field lines of the Earth
Physics 201 – p. 9/3
Magnetic Fields
A proton has a speed of 106 m/s and enters a
region with a uniform magnetic field. It makes an
angle of 450 with respect to the direction of the
~ Find its subsequent motion.
magnetic field B.
What if the proton enters the region with a
uniform magnetic field with zero speed?
Nothing happens!
Physics 201 – p. 10/3
Magnetic Fields
A proton has a speed of 106 m/s and enters a
region with a uniform magnetic field. It makes an
angle of 450 with respect to the direction of the
~ Find its subsequent motion.
magnetic field B.
What if the proton enters the region with a
uniform magnetic field with zero speed?
Nothing happens!
What if it enters that region with a velocity
which is either parallel to or antiparallel to the
magnetic field? It will experience no force.
Physics 201 – p. 10/3
Magnetic Fields
When neither of the previous two cases
applies, it is found that the force is directly
proportional to the magnitudes of the
magnetic field, the charge, and the speed.
Physics 201 – p. 11/3
Magnetic Fields
When neither of the previous two cases
applies, it is found that the force is directly
proportional to the magnitudes of the
magnetic field, the charge, and the speed.
It is found that this force is maximal when the
velocity of the proton is perpendicular to the
direction of the magnetic field.
Physics 201 – p. 11/3
Magnetic Fields
Force on moving charge
~ zero when ~v = 0 and/or when
F~ = q~v × B.
~
~v k B
Physics 201 – p. 12/3
Magnetic Fields
Force on moving charge
~ zero when ~v = 0 and/or when
F~ = q~v × B.
~
~v k B
Magnitude: F = qv B sin θ. θ: Angle between
~
~v and B
Physics 201 – p. 12/3
Magnetic Fields
Force on moving charge
~ zero when ~v = 0 and/or when
F~ = q~v × B.
~
~v k B
Magnitude: F = qv B sin θ. θ: Angle between
~
~v and B
F is maximal when θ = 900 .
Physics 201 – p. 12/3
Magnetic Fields
Force on moving charge
~ zero when ~v = 0 and/or when
F~ = q~v × B.
~
~v k B
Magnitude: F = qv B sin θ. θ: Angle between
~
~v and B
F is maximal when θ = 900 .
B = q v Fsin θ . Unit: 1 tesla (T) = 1 N/(A.m).
1 gauss = 10−4 T
Physics 201 – p. 12/3
Magnetic Fields
Force on a moving charge
Physics 201 – p. 13/3
Magnetic Fields
Force on a moving charge
Physics 201 – p. 14/3
Magnetic Fields
A proton has a speed of 106 m/s and enters a
region with a uniform magnetic field. It makes an
angle of 450 with respect to the direction of the
~ Find its subsequent motion.
magnetic field B.
Subsequent motion? Use Newton’s law:
θ
a = mFp = qvBmsin
p
Physics 201 – p. 15/3
Magnetic Fields
A proton has a speed of 106 m/s and enters a
region with a uniform magnetic field. It makes an
angle of 450 with respect to the direction of the
~ Find its subsequent motion.
magnetic field B.
Subsequent motion? Use Newton’s law:
θ
a = mFp = qvBmsin
p
For e.g. B = 0.5T :
a=
(1.6×10
√
6m
C)(10 s )(0.5T )( 22 )
1.67×10−27 kg
−19
= 3.4 × 1013 m/s2
Physics 201 – p. 15/3
Magnetic Fields
What happens to the acceleration if it were an
electron instead?
√
−19
6m
(1.6×10 C)(10 s )(0.5T )( 22 )
17
2
a=
=
0.6
×
10
m/s
.
9.11×10−31 kg
Physics 201 – p. 16/3
Magnetic Fields
What happens to the acceleration if it were an
electron instead?
√
−19
6m
(1.6×10 C)(10 s )(0.5T )( 22 )
17
2
a=
=
0.6
×
10
m/s
.
9.11×10−31 kg
Easier to accelerate electrons than protons
because they are lighter.
Physics 201 – p. 16/3
Magnetic Fields
How does the motion of the proton look like?
Since the velocity of the proton makes an
angle of 450 with respect to the direction of
~ it has a component
the magnetic field B,
~ This
v cos θ along the direction of B.
component is unchanged.
Physics 201 – p. 17/3
Magnetic Fields
How does the motion of the proton look like?
Since the velocity of the proton makes an
angle of 450 with respect to the direction of
~ it has a component
the magnetic field B,
~ This
v cos θ along the direction of B.
component is unchanged.
The perpendicular component ~vperpendicular is
changed in direction due to the above force
~ Combined with
⇒ Circular motion around B.
the unchanged parallel component, the
motion looks like a helix!
Physics 201 – p. 17/3
Magnetic Fields
How does the motion of the proton look like?
Physics 201 – p. 18/3
Magnetic Fields
What if the proton comes in with a velocity which
is perpendicular to the magnetic field?
In this case, there is no parallel component of
the velocity ⇒ circular motion in a plane
perpendicular to the magnetic field. ⇒
v2
F = qvB = m r = mω 2 r ⇒
Physics 201 – p. 19/3
Magnetic Fields
What if the proton comes in with a velocity which
is perpendicular to the magnetic field?
In this case, there is no parallel component of
the velocity ⇒ circular motion in a plane
perpendicular to the magnetic field. ⇒
v2
F = qvB = m r = mω 2 r ⇒
r=
mv
qB
=
P
qB :
Radius of circular motion
Physics 201 – p. 19/3
Magnetic Fields
What if the proton comes in with a velocity which
is perpendicular to the magnetic field?
In this case, there is no parallel component of
the velocity ⇒ circular motion in a plane
perpendicular to the magnetic field. ⇒
v2
F = qvB = m r = mω 2 r ⇒
r=
mv
qB
ω=
v
r
=
=
P
qB :
qB
m
Radius of circular motion
Cyclotron angular frequency
Physics 201 – p. 19/3
Magnetic Fields
What if the proton comes in with a velocity which
is perpendicular to the magnetic field?
In this case, there is no parallel component of
the velocity ⇒ circular motion in a plane
perpendicular to the magnetic field. ⇒
v2
F = qvB = m r = mω 2 r ⇒
r=
mv
qB
ω=
qB
v
=
r
m Cyclotron
2π
2πm
=
ω
qB
T =
=
P
qB :
Radius of circular motion
angular frequency
Physics 201 – p. 19/3
Magnetic Fields
Example of circular motion
An electron and a proton move in circular orbits
in a plane perpendicular to a uniform magnetic
~ Find the ratio of the radii of their circular
field B.
orbits when the electron and the proton have (a)
the same momentum and (b) the same kinetic
energy.
P
(a) r = mv
=
qB
qB ⇒
the same for both.
re
rp
=
Pe
Pp
= 1 because qB is
Physics 201 – p. 20/3
Magnetic Fields
Example of circular motion
√
2
P
(b) K = 2m
⇒ P = 2mK ⇒ rrpe = PPpe =
q
q
q
9.11×10−31 kg
2me Ke
me
−2
=
=
=
2.3
×
10
2mp Kp
mp
1.67×10−27 kg
Physics 201 – p. 21/3
Magnetic Fields
Some applications: Velocity selector
In some situations, it is important for the
charged particles to all move with the same
velocity. Let them pass through a region in
which there is a uniform magnetic field (say
pointing into the plane of the paper) and a
uniform electric field perpendicular to it (say
pointing downward).
Physics 201 – p. 22/3
Magnetic Fields
Some applications: Velocity selector
In some situations, it is important for the
charged particles to all move with the same
velocity. Let them pass through a region in
which there is a uniform magnetic field (say
pointing into the plane of the paper) and a
uniform electric field perpendicular to it (say
pointing downward).
~ points upward
For q > 0, q~v × B
~ points downward.
and q E
Physics 201 – p. 22/3
Magnetic Fields
Some applications: Velocity selector
~ and B
~ can be chosen so that these two
E
forces cancel ⇒ qvB = qE
E
⇒v=B
Physics 201 – p. 23/3
Magnetic Fields
Some applications: Electromagnetic flowmeter
as a Velocity selector
Physics 201 – p. 24/3
Magnetic Fields
Some applications: Mass spectrometer
After passing through a velocity selector, the
particles go through a second region with
uniform magnetic field B0 in the same
direction as the one in the velocity selector.
They bend around and using the expression
for the radius of curvature, one finds
rB0
rB0 B
m
=
=
q
v
E .
Physics 201 – p. 25/3
Magnetic Fields
Some applications: Mass spectrometer
After passing through a velocity selector, the
particles go through a second region with
uniform magnetic field B0 in the same
direction as the one in the velocity selector.
They bend around and using the expression
for the radius of curvature, one finds
rB0
rB0 B
m
=
=
q
v
E .
Measuring r, B0 , B, E, one can determine
m
q.
Physics 201 – p. 25/3
Magnetic Fields
Some applications: Mass spectrometer
Physics 201 – p. 26/3
Magnetic Fields
Magnetic force on wire with current
A wire with current contains flowing electrons.
Physics 201 – p. 27/3
Magnetic Fields
Magnetic force on wire with current
A wire with current contains flowing electrons.
In the presence of a magnetic field, each
electron experiences a magnetic force
~
F~ = −e~velectron × B
Physics 201 – p. 27/3
Magnetic Fields
Magnetic force on wire with current
A wire with current contains flowing electrons.
In the presence of a magnetic field, each
electron experiences a magnetic force
~
F~ = −e~velectron × B
Notice that by convention, the direction of the
current is opposite to that of the electrons i.e.
~ i.e.
~v+ = −~velectron . Equivalent to F~ = e~v+ × B
force on “positive” charges moving in the
opposite direction to that of the electrons.
Physics 201 – p. 27/3
Magnetic Fields
Magnetic force on wire with current
In a segment of a wire with cross section A
and length L, the number of charged particles
with a drift speed vd is nAL where n is the
number of particles per unit volume.
Physics 201 – p. 28/3
Magnetic Fields
Magnetic force on wire with current
In a segment of a wire with cross section A
and length L, the number of charged particles
with a drift speed vd is nAL where n is the
number of particles per unit volume.
The magnetic force on the wire is
~
F~ = (q~vd × B)nAL
Physics 201 – p. 28/3
Magnetic Fields
Magnetic force on wire with current
In a segment of a wire with cross section A
and length L, the number of charged particles
with a drift speed vd is nAL where n is the
number of particles per unit volume.
The magnetic force on the wire is
~
F~ = (q~vd × B)nAL
With the current being I = nqvd A, one gets
~ ×B
~
F~ = I L
~ vector in the direction of the current and
L:
with magnitude L.
Physics 201 – p. 28/3
Magnetic Fields
Magnetic force on wire with current
Its magnitude is
F = IL B sin θ
θ: Angle between the magnetic field and the
current.
Physics 201 – p. 29/3
Magnetic Fields
Magnetic force on wire with current
Its magnitude is
F = IL B sin θ
θ: Angle between the magnetic field and the
current.
It can be shown that the net force on a closed
current loop in a uniform magnetic field is
zero.
Physics 201 – p. 29/3
Magnetic Fields
Magnetic force on wire with current
Physics 201 – p. 30/3