Download Lecture 13. Magnetic Field, Magnetic Forces on Moving Charges.

Lecture 13. Magnetic Field, Magnetic Forces on Moving Charges.
Outline:
Intro to Magnetostatics.
Magnetic Field Flux, Absence of Magnetic Monopoles.
Force on charges moving in magnetic field.
1
Structure of the Course
Electrostatics
Electric fields
generated by
charges at rest
Ch. 21 - 26
𝒅
=𝟎
𝒅𝒅
Electromagnetism
E - electric fields
B – magnetic fields
Q – charges
I – currents
Φ - fluxes
0
Magnetostatics
Magnetic fields
generated by
time-independent
currents
Ch. 27 - 31
Electromagnetic 𝒅
≠ 𝟎 Ch. 32
𝒅𝒅
Waves
- covered in detail in more
advanced courses on
electrodynamics and optics.
2
Intro to Magnetostatics
Our goal: to describe the magnetostatic field the same way
we’ve described the electrostatic field.
� 𝐸 ∙ 𝑑𝐴⃗ =
𝑄𝑒𝑒𝑒𝑒
- always true
𝜀0
� 𝐸 ∙ 𝑑𝑙⃗ = 0
𝐹⃗ = 𝑞𝐸
- true only if the
fields are static
� 𝐵 ∙ 𝑑𝐴⃗ =?
� 𝐵 ∙ 𝑑𝑙⃗ =?
𝐹⃗ =?
Do you know a vector field 𝑎⃗ that has both ∮ 𝑎⃗ ∙ 𝑑𝐴⃗ = 0 and ∮ 𝑎⃗ ∙ 𝑑𝑙⃗ = 0?
3
Sources of Magnetic Field
Charges at rest do not generate B. What are the sources of the magnetic
field?
 “magnetic point charge” (a magnetic monopole): has not been observed
yet (though its existence doesn’t contradict anything)
 ferromagnetic materials (electron spins – purely quantum
phenomenon).
 Time-dependent electric fields (we’ll consider this source
later in Electrodynamics).
 Charges in motion: currents, orbital motion of electrons in
atoms.
In magnetostatics:
The B field a moving single charge is non-stationary, not good for
magnetostatics
Steady (time-independent) current: our best bet.
+
4
Characteristic Magnetic Fields
Units: tesla, T
−4
𝐵 ≈ 10 𝑇
𝐵 ≈ 1 − 2𝑇
Superconducting solenoids in LHC
𝐵 𝑢𝑢 𝑡𝑡 8𝑇
rare-earth magnets
𝐵 𝑢𝑢 𝑡𝑡 1.4𝑇
Superconducting solenoids
𝐵 𝑢𝑢 𝑡𝑡 30𝑇
5
Flux of the Magnetic Field
Flux of the
magnetic field:
Φ𝐵 = � 𝐵 ∙ 𝑑𝐴⃗
Units: T⋅m2=Weber, Wb
compare with
Φ𝐸 = � 𝐸 ∙ 𝑑𝐴⃗
No “magnetic point
charges” (magnetic
monopoles): have
not been observed
yet.
Magnetic dipole
Consequence:
for ANY closed
surface
� 𝐵 ∙ 𝑑𝐴⃗ = 0
Electric dipole
- always true, not only
in electrostatics but
also in electrodynamics
6
Magnetic Field Lines
Magnetic field lines are closed loops (no mag. monopoles):
Magnetic field is a non-conservative
vector field.
� 𝐵 ∙ 𝑑𝑙⃗ ≠ 0
to be
specified
later
7
Force on a Charge Moving in Magnetic Field
Lorentz
force
Heaviside
𝐹⃗ = 𝑞 𝑣⃗ × 𝐵
𝐹 = 𝑞𝑞𝑞𝑞𝑞𝑞𝜙
𝐹⃗ = 0 for charges
moving along 𝐵
𝐹 is max when
𝑣⃗ ⊥ 𝐵
Magnetic field lines are
not lines of force !
demonstration
8
Iclicker Question
Imagine that you are looking at the face of a CRT. The bright spot
indicating where the electron beam hits the face. You bring a
permanent magnet toward the CRT with its north pole oriented
upward. Which direction will the spot be deflected across the
screen?
A. up
B. down
𝐹⃗ = 𝑞 𝑣⃗ × 𝐵
C. the spot does not deflect
D. right
E. left
9
Iclicker Question
Imagine that you are looking at the face of a CRT. The bright spot
indicating where the electron beam hits the face. You bring a
permanent magnet toward the CRT with its north pole oriented
upward. Which direction will the spot be deflected across the
screen?
A. up
B. down
𝐹⃗ = 𝑞 𝑣⃗ × 𝐵
C. the spot does not deflect
D. right
E. left
10
Iclicker Question
When does a magnetic field exert a force on a charged particle?
A. Always.
B. Only when the particle moves exactly perpendicular to the
magnetic field lines.
C. When the particle is moving at a non-zero angle with respect
to the magnetic field lines.
D. When the particle is moving along the magnetic field lines.
E. When the particle is moving.
𝐹⃗ = 𝑞 𝑣⃗ × 𝐵
11
Iclicker Question
When does a magnetic field exert a force on a charged particle?
A. Always.
B. Only when the particle moves exactly perpendicular to the
magnetic field lines.
C. When the particle is moving at a non-zero angle with respect
to the magnetic field lines.
D. When the particle is moving along the magnetic field lines.
E. When the particle is moving.
𝐹⃗ = 𝑞 𝑣⃗ × 𝐵
12
No Work Done by Magnetic Force!
𝐹⃗ = 𝑞 𝑣⃗ × 𝐵
𝐹⃗ ⊥ 𝑑𝑙⃗
The work done by
the magnetic field
on a moving charge
=0
𝑑𝑑 = 𝐹⃗ ∙ 𝑑𝑙⃗ = 0
You cannot increase the speed of charged particles using magnetic field.
However, the B-induced acceleration is non-zero, it’s just perpendicular to
the velocity 𝑎⃗ ≡
𝑑𝑣
𝑑𝑑
.
However, there are many situations in which this statement appears to be
false: in a non-uniform magnetic field, a current-carrying wire loop would
accelerate, two permanent magnets would accelerate towards one another,
etc. In these cases, the kinetic energy of objects increases. Who does the
work? For discussion, see
http://van.physics.illinois.edu/qa/listing.php?id=17176
13
Iclicker Question
When a charged particle moves through a magnetic field, the
trajectory of the particle at a given point is
A. parallel to the magnetic field line that passes
through that point.
B. perpendicular to the magnetic field line that passes
through that point.
C. neither parallel nor perpendicular to the magnetic
field line that passes through that point.
D. any of the above, depending on circumstances
14
Iclicker Question
When a charged particle moves through a magnetic field, the
trajectory of the particle at a given point is
A. parallel to the magnetic field line that passes
through that point.
B. perpendicular to the magnetic field line that passes
through that point.
C. neither parallel nor perpendicular to the magnetic
field line that passes through that point.
D. any of the above, depending on circumstances
15
Cross Product
𝐹⃗ = −𝑒 𝑣⃗ × 𝐵
An electron moves along the z axis, the B field is in the x-y plane.
𝑣⃗ = 1𝑘� 𝑚/𝑠
𝐵 = 2𝚤̂ + 3𝚥̂ 𝑇
𝑣⃗
𝐹⃗ = −𝑒 𝑣⃗ × 𝐵 = −𝑒 1𝑘� × 2𝚤̂ + 3𝚥̂
𝑥� 𝚤̂
= −e 1𝑘� × 2𝚤̂ + 1𝑘� × 3𝚥̂
= −e 2𝚥̂ − 3𝚤̂ = 𝑒 3𝚤̂ − 2𝚥̂
𝑧̂ 𝑘�
𝑧̂ 𝑘�
𝑥� 𝚤̂
𝑦� 𝚥̂
𝑦� 𝚥̂
𝐵
𝚤̂ × 𝚥̂ = 𝑘�
𝚥̂ × 𝑘� = 𝚤̂
𝑘� × 𝚤̂ = 𝚥̂
16
Iclicker Question
A particle with charge q = –1 C is moving in the positive zdirection at 5 m/s. The magnetic field at its position is

B = 3iˆ – 4 ˆj T
𝑧̂
(
)
What is the magnetic force on the particle?
(
)
B. ( 20iˆ − 15 ˆj ) N
C. ( −20iˆ + 15 ˆj ) N
D. ( −20iˆ − 15 ˆj ) N
A. 20iˆ + 15 ˆj N
E. none of these
𝑣⃗
𝑥� 𝚤̂
𝑘�
𝐵
𝑦� 𝚥̂
𝚤̂ × 𝚥̂ = 𝑘�
𝚥̂ × 𝑘� = 𝚤̂
𝑘� × 𝚤̂ = 𝚥̂
𝐹⃗ = 𝑞 𝑣⃗ × 𝐵
17
Iclicker Question
A particle with charge q = –1 C is moving in the positive zdirection at 5 m/s. The magnetic field at its position is

B = 3iˆ – 4 ˆj T
𝑧̂
(
)
What is the magnetic force on the particle?
(
)
B. ( 20iˆ − 15 ˆj ) N
C. ( −20iˆ + 15 ˆj ) N
D. ( −20iˆ − 15 ˆj ) N
A. 20iˆ + 15 ˆj N
E. none of these
𝐹⃗ = −1
5𝑘� × 3𝚤̂ − 4𝚥̂
𝑣⃗
𝑥� 𝚤̂
𝑘�
𝐵
𝑦� 𝚥̂
𝚤̂ × 𝚥̂ = 𝑘�
𝚥̂ × 𝑘� = 𝚤̂
𝑘� × 𝚤̂ = 𝚥̂
𝐹⃗ = 𝑞 𝑣⃗ × 𝐵
= −15𝚥̂ − 20𝚤̂
18
Iclicker Question
A positively charged particle moves in the positive z-direction.
The magnetic force on the particle is in the positive y-direction.
What can you conclude about the z-component of the magnetic
field at the particle’s position?
�
A. Bz > 0
B. Bz = 0
C. Bz < 0
D. not enough information given to decide
𝑧̂ 𝑘
𝑣⃗
𝑥� 𝚤̂
𝐹⃗
𝑦� 𝚥̂
𝚤̂ × 𝚥̂ = 𝑘�
𝚥̂ × 𝑘� = 𝚤̂
𝑘� × 𝚤̂ = 𝚥̂
𝐹⃗ = 𝑞 𝑣⃗ × 𝐵
19
Iclicker Question
A positively charged particle moves in the positive z-direction.
The magnetic force on the particle is in the positive y-direction.
What can you conclude about the z-component of the magnetic
field at the particle’s position?
�
A. Bz > 0
B. Bz = 0
C. Bz < 0
D. not enough information given to decide
𝑧̂ 𝑘
𝑣⃗
𝑥� 𝚤̂
𝐹⃗
𝑦� 𝚥̂
𝚤̂ × 𝚥̂ = 𝑘�
𝚥̂ × 𝑘� = 𝚤̂
𝑘� × 𝚤̂ = 𝚥̂
𝐹⃗ = 𝑞 𝑣⃗ × 𝐵
20
Cyclotron Motion in Magnetic Field
Motion along
a circular orbit:
𝐹⃗ = 𝑞 𝑣⃗ × 𝐵
𝑚𝑣
𝑅=
𝑞𝑞
𝑣2
→ 𝑞𝑞𝑞 = 𝑚
𝑅
alternatively, by measuring R, one can
determine the ratio m/q if v (the kinetic
energy) is known.
𝑇=
2𝜋𝑅 2𝜋𝑚
=
𝑣
𝑞𝑞
- the period T is independent of v
If a charge has a
velocity component
along 𝐵:
21
Large Hadron Collider
𝑚𝑣
𝑅=
𝑞𝑞
𝑚 ≅ 1.6 ∙ 10−27 kg
𝑣=c
𝐵 = 8T
1.6 ∙ 10−27 kg ∙ 3 ∙ 108 𝑚/𝑠
𝑅=
≈ 0.38𝑚
1.6 ∙ 10−19 C ∙ 8𝑇
8.6 km
What’s wrong? In this form, the equation works only for nonrelativistic motion. To correct, you need to replace the “classical”
momentum 𝑚𝑚 with the relativistic one 𝑚𝑚/ 1 − 𝑣/𝑐 2 . 22
Mass Spectrometer
𝑚𝑣 2
= 𝑞𝑞
2
𝑣=
2𝑞𝑞
𝑚
𝑞𝑞 = 𝑞𝑞𝑞
𝐸
𝑣=
𝐵
𝑚𝑣
𝑅=
𝑞𝑞
23
Conclusion
Magnetostatics: 𝐵 ≠ 𝐵 𝑡
Sources of B: motion of charges (currents), orbital motion of
electrons in atoms, electron spins, time-dependent electric
fields.
Sources of B in “classical” magnetostatics: steady currents.
Absence of Magnetic Monopoles: ∮ 𝐵 ∙ 𝑑𝐴⃗ = 0
Force on charges moving in magnetic field: 𝐹⃗ = 𝑞 𝑣⃗ × 𝐵 .
Next time: Lecture 14: Magnetic Forces on
Currents.
§§ 27.6- 27.9
24