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.
Our goal: to describe the magnetostatic field the same way we’ve described the
electrostatic field
� 𝐸 ∙ 𝑑𝐴⃗ =
𝑄𝑒𝑒𝑒𝑒
𝜀0
� 𝐸 ∙ 𝑑𝑙⃗ = 0
𝐹⃗ = 𝑞𝐸
- always true
- true only if the
fields are static
� 𝐵 ∙ 𝑑𝐴⃗ =?
� 𝐵 ∙ 𝑑𝑙⃗ =?
𝐹⃗ =?
- today
- Ch. 28
- today
1
Magnetic Field B
Sources:  Charges in motion: currents, orbital motion of electrons in atoms.
 Electron spins (an internal degree of freedom of quantum particles).
 Time-dependent electric fields (we’ll consider this source later in
Electrodynamics).
Characteristic Magnetic Fields:
Units: tesla, T
𝐵 ≈ 10−4 𝑇
rare-earth magnets
𝐵 𝑢𝑢 𝑡𝑡 1.4𝑇
Superconducting solenoids in LHC
𝐵 𝑢𝑢 𝑡𝑡 8𝑇
𝐵 ≈ 1 − 2𝑇
Superconducting solenoids
2
𝐵 𝑢𝑢 𝑡𝑡 30𝑇
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
3
Magnetic Field Lines
Do you know a vector field 𝑎⃗ that has both ∮ 𝑎⃗ ∙ 𝑑𝐴⃗ = 0 and
∮ 𝑎⃗ ∙ 𝑑𝑙⃗ = 0?
Magnetic field is a non-conservative
vector field. In general
� 𝐵 ∙ 𝑑𝑙⃗ ≠ 0
Magnetic field lines are closed loops (no mag. monopoles):
4
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 !
5
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
6
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
7
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
𝑑𝑣
. An accelerated charge (e.g. moving with constant
the velocity 𝑎⃗ ≡
𝑑𝑑
speed along a curved trajectory) loses its energy by radiating
electromagnetic waves.
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
8
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
9
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
10
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 deflect?
A. up
B. down
𝐹⃗ = 𝑞 𝑣⃗ × 𝐵
C. the spot does not deflect
D. right
E. left
11
Iclicker Question
Imagine that you are looking at the face of a CRT. The bright spot
indicating where the electron beam hits the face is in the center
of the screen. You bring a permanent magnet toward the CRT
vertically from above. The magnet is oriented vertically with its
north pole downward. Which direction will the spot deflect?
A. up
B. down
𝐹⃗ = 𝑞 𝑣⃗ × 𝐵
C. the spot does not deflect
D. right
E. left
12
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
13
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 = −15𝚥̂ − 20𝚤̂
𝑣⃗
𝑥� 𝚤̂
𝑘�
𝐵
𝑦� 𝚥̂
𝚤̂ × 𝚥̂ = 𝑘�
𝚥̂ × 𝑘� = 𝚤̂
𝑘� × 𝚤̂ = 𝚥̂
14
Cyclotron Motion in Magnetic Field
Motion along
a circular
orbit:
𝑣2
𝐹⃗ = 𝑞 𝑣⃗ × 𝐵 = 𝑞𝑞𝑞 = 𝑚
𝑅
𝑚𝑣
𝑅=
𝑞𝑞
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:
15
Mass Spectrometer
𝑚𝑣 2
= 𝑞𝑞
2
𝑣=
2𝑞𝑞
𝑚
𝑞𝑞 = 𝑞𝑞𝑞
𝐸
𝑣=
𝐵
𝑚𝑣
𝑅=
𝑞𝑞
16
Conclusion
Magnetostatics: 𝐵 ≠ 𝐵 𝑡
Sources of B: motion of charges (currents), orbital
motion of electrons in atoms, electron spins, timedependent electric fields.
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
17