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
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?
2
Sources of Magnetic Field
Charges at rest do not generate B. What would be our “best bet” for the
source of the magnetic field?
_
“magnetic point charge” (a magnetic monopole): have not been observed yet
(though its existence doesn’t contradict anything)
a moving single charge: the field is non-stationary, not good for
magnetostatics
steady (time-independent) current: our best bet
Sources of B:
_
+
Units: tesla, T
 Charges in motion: currents, orbital motion of
electrons in atoms.
 Ferromagnetic materials (electron spins – purely
quantum phenomenon).
 Time-dependent electric fields (we’ll consider
this source later in Electrodynamics).
3
Characteristic Magnetic Fields
−4
𝐵 ≈ 10 𝑇
rare-earth magnets
𝐵 𝑢𝑢 𝑡𝑡 1.4𝑇
Superconducting solenoids in LHC
𝐵 𝑢𝑢 𝑡𝑡 8𝑇
𝐵 ≈ 1 − 2𝑇
Superconducting solenoids
𝐵 𝑢𝑢 𝑡𝑡 30𝑇
4
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
5
Magnetic Field Lines
Magnetic field is a non-conservative
vector field. In general
� 𝐵 ∙ 𝑑𝑙⃗ ≠ 0
to be specified
later
Magnetic field lines are closed loops (no mag. monopoles):
6
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 !
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
12
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
13
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:
20
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 . 21
Mass Spectrometer
𝑚𝑣 2
= 𝑞𝑞
2
𝑣=
2𝑞𝑞
𝑚
𝑞𝑞 = 𝑞𝑞𝑞
𝐸
𝑣=
𝐵
𝑚𝑣
𝑅=
𝑞𝑞
22
Conclusion
Magnetostatics: 𝐵 ≠ 𝐵 𝑡
Sources of B: motion of charges (currents), orbital motion of
electrons in atoms, electron spins, time-dependent 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
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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.
24
Cosmic Rays in the Earth’s Magnetic Field
27
Experiment with the vacuum electron tube
28